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The Calculus of Themodynamicss

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The Calculus ofThennodynarnics
5
In this chapter we develop functional property relationships for simple systems. These
relationships are completely general and are a direct consequence of applying the
postulates and laws of thermodynamics discussed in Chapters 2-4. They also reflect
behavior consistent with the calculus of multivariable functions.
These functional property relationships are fundamentally different than physical
property models, such asPyrN equations of state, which represent practical constitutive
equations specific for each pure substance or mixture composition. In order to obtain
working physical property equations, experiments are performed and data correlated or
suitable estimation methods are utilized to predict property values (see Chapters 8 and
13). Alternatively, molecular models using statistical thermodynamics may be used,
albeit with some approximations (see Chapter 10).
A major objective of this chapter is to introduce techniques for recasting derived
properties and their derivatives into forms that can be directly evaluated in tenns of
primitive properties which are measurable. Our starting point is to introduce a
fundamentaJ operating equation for simple systems that establishes the functional
dependence of internal energy U on the variables entropy§.. volume V, and moles (or
masses) of all components present N1,. .. , N,. This Fundamental Equation has its origin
from the combined first and second laws for open simple systems that was derived in
Section 4.7. The S, V, N1, ... , N11 coordinates of that equation were used extensively by
J.W. Gibbs in his classic work on thermodynamics in the late 1800's, and as a result they
are commonly referred to as Gibbs coordinates. They represent a set of n + 2 natural or
canonical variables that provide the necessary and sufficient information required to
describe the thermodynamic state of a simple system.
By employing methods from calculus and geometry 1 Gibbs was able to show, in a
completely rigorous mathematical mannert how various thermodynamic properties were
related. He clearly laid the groundwork for understanding the functional interrelationships that exist among the so-called potential functions. U, HlA, and G. Without
the computational power and graphical display methods that we have today, Gibbs had
to conceptualize the structure of these property inter-relationships in his mind without
being able to illustrate them graphically. He used only narratives with equations in his
writings to describe his findings. In part, this restriction has limited how fast we have
been able to understand the details of Gibbs• work. It has taken over a century of study
to fully appreciate the significance of his contributions. Interested readers should consult
Section 5.1 The fundamental Equation in Gibbs Coordinates
125
the papers by K. Jolls and his co-workers listed at the end of the chapter for a
comprehensive discussion of Gibbs' contributions in this area. In the sections that follow,
we will employ graphical methods to illustrate some of the geometric aspects of the
Fundamental Equation and its transformations. In addition, several methods of
manipu1ating functions and partial derivatives will be described and applied to problems
frequently encountered in classical thermodynamics.
5.1 The Fundamental Equation in Gibbs Coordinates
As will be seen presently, the relationship
U = /u(§_, V, Np· .. , N,J
(5-1)
completely describes all of the stable equilibrium states of a simple system containing
n components. By solving Bq. (5-1) explicitly for~. an alternative form is obtained:
§.=f"-U,V,N1, • ••• N,)
(5-2)
Either relationship is called the Fundamental Equation: Eq. (5-1) is termed the
energy representation and Bq. (5-2) the entropy represen1ation.
The Fundamental Equation is represented by a (hyper) surface in (n + 3} dimensional
space. The points on this surface represent stableequilibriumstates of the simple system.
Quasi-static processes can be represented by a curve on this surface. Processes that are
not quasi-static are not identified with points on this surface. (Recall that derived
properties such as U and§. are not defined for non-equilibrium states.)
Then+ 2 ftrst-order partial derivatives or the Fundamental Equation correspond to
traces in the respective coordinate walls of plan.es tangent to the thermodynamic
surface 1• The significance of these tangent plane traces can be seen by expressing the
Fundamental Equation in differential fonn. For the energy representation, Eq. (5-1),
du;:
as
(dful
-
_')I
(dful
d~+ av
-
-~
~ (dfu]
aN.
dV+ ~
1=1
I
dN;
(5-3)
~~J[J]
where subscript fvj[i] indicates that all N1 except j =i are held constant in the
differentiation. If we compare Eq. (5·3) to the combined Jaw for a simple system (see
Section 4.7). namely
n
dU=Td~ -PdV+
L Jl.;dN;
(5-4)
i=l
it is clear that
l The lenn plane is used in the generic sense here. Even for a pure fluid there arc three independent
variables and three partial derivatives. ThuK the tangent plane is re.a.Uy a three-dimensional
hyperplane.
The Caladus of Thenno~amics Chapter 5
126
(5-6)
and
(Jful
(
()N,
=lli =g~, V, Np ... , Nn)
(5-7)
_.X:. N1 [tl
where the functions 8T• gp. and gi could be obtained directly from the Fundamental
Equation if it were available. Equations (5-5) through (5-7) represent a particular set of
equations ofstate expressed in Gibbs~ V N cooroinates. As shown later in Sections 5.2
and 5.3, only two of these three equations of state for a pure substance are independent.
The second-order partial derivatives of the Fundamental Equation are also related to
quantities that can be measured experimentally. For example, for a pure materialt there
are four second-order partial derivatives at constant mass or moles:
~- = ;sfL(~~)
] =(~~lN
-Y..NV' l ~
~;=aav[(~~J~..N ]~,II =-(~~t
~
-
:~y=:~[(~~L
(5-8)
(5-9)
-
L=-(~L
a~~~= :v[(~~L L=(~~1
(5-10)
(5-11)
Of these four derivativ~ only three are independent because the last two are related by
the reciprocity theorem of Maxwell (see Section 5.3, part 4).
For an n-component mixture there are n + 1 ftrst-order and (n + 2Xn + 1)/2 secondorder independent partial derivatives of the Fundamental Equation given by Eq. (5-1)
orEq. (5-2). These derivatives are particularly important because they form a basis for
all other partial derivatives involving thermodynamic properties. That is, any partial
derivative can be expressed in tenns of an independent set of first-order and second·order
derivatives of any form of the Fundamental Equation. Proof of this statement fo1· a pure
material is given in Section 5.3.
At this point. it is interesting to consider what the shape of the internal energy sutface
would look like in~ V N coordinates. Since even for a pure material this would entail a
Section 5.1 The Fundamental Equation in Gibbs Coordinates
127
four-dimensional plot, one is forced to reduce the dimensionality of the system in order
to visualize the surface in three dimensions. For a pure system, one can propose a 3-D
plot of intensive U as a function of Sand V. In fact, James Clerk Maxwell in the late
1800's was so intrigued with Gibbs' work that he actually made a 3-D model of the USV
surface for water. He sent a plaster replica of his model to Gibbs at Yale, where it remains
on display today. Figure 5 .la is a photograph of a more recent 3-D USV surface for water
constructed by Clark and Katz (1939) while Figure5.1bis a2-D projection of the surface
showing several important features. Note, for example, the point of triple tangency that
corresponds to the three-phase, solid-liquid-gas, triple point equilibrium for water. We
will have much more to say about such geometrical aspects and their relationship to the
characterization and limits of stable equilibrium states in Section 5.7. With more
accurate thermodynamic data available and computational tools for correlating and
manipulating these data, it is possible to generate USV surfaces as computer-generated,
3-D plots for a wide range of pure materials.
Jolls (1990), Coy (1993) and coworkers (1991, 1992) at Iowa State havedonejusl
this for a number of pure compounds using a well-tested PVTN equation of state
developed by Peng and Robinson (see Section 8.4) and a temperature-dependent
correlation for the heat capacity in an ideal gas state. The actual methods for generating
values of U for specified va1ues of Sand V are straightforward and described in detail
in Sections 8.1 and 8.2 [see also Coy (1993)]. Figure 5.2 shows two fundamental USV
surfaces~ne for an ideal gas and the other generated by J olls and co-workers for
ethylene using the Peng-Robinson equation of state as a modeJ. It is important to keep
in mind that the values of S, V and U are not absolute, they depend on the specification
and U 0 .
of reference state conditions,
E"ample 5.1 actually derives the governing equation for U for a monatomic ideal gas
that is plotted in Figure 5.2a. The USV surface is convex with respect to variations inS
as the second derivative is always positive for the exponential dependence of U on S
shown in Figure 5. 2a. Later we will invoke even stronger arguments that this convexity
must be true in general. Although the actual functional dependence of U on V will vary
from substance to substance, the general power 1aw decrease depicted in Figure 5.2a
will be followed for a11 fluids in the dilute gas region where nearly ideal behavior is
expected.
The important thing to remember is that the graphical representation of the USV
surface encodes all the thermodynamic infonnation about a pure substance as it is
equivalent to the Fundamental Equation itself. The geometry of the surface represents
all single phase stable equilibrium states, aU two-phase coexistence points, and all triple
points; it also can be used to identify limits of stability such as critical and spinodal points
(see Chapter 7). A major difficulty, of course, is that a significant amount of physical
property information is required to construct the Gibbs USV surface for a pure substance,
e.g., the steam tables for water, or a robust PVTN equation of state that models the
behavior of a compound adequately in all states of aggregation.
sn
The Calculus of Thermodynamics Chapter 5
128
u
VAP'OR
Figure S.la Three-dimensional model of the USV surface for pure water [from
Clark and Katz (1939), with permission of the Royal Society of Canada].
u{;
Ts.L < Ttp
-
Metastable region
c::=J Unstable region
p ""I Steble region
Figure S.lb A two-dimensional projection of the USV surface for pure water shown
in Figure 5. t a.
Section 5.1 The Fundamental Equation in Gibbs Coordinates
129
t
VIC!'
\~
T=llopet, ~(()U1
as~,.
-P=slo~ =(au~
iY J~
Figure S.2a The USV surface from the Fuodamental Equation for a pure monatomic
ideal gas referenced to lJ>, ~.and V' (see Example 5.1).
Figure S.2b A predicted USV surface for the fluid phases of pure ethylene.
Volumetric PVT properties from the Peng-Robinson equation of state and heat
capacities from a fitted ideal gas state equation. (Vnuu = 15 V,J [adapted from Jolls
and Coy (1992)].
It should be clear that Fundamental Equations would be of great use if they were
generally available. The problem is that the complete form of the Fundamental Equation
is not specified by classical thermodynamics; each substance has its own peculiarities
The Calculus of Thennodynamics Chapter 5
130
that are reflected in different functionalities of the Fundamental Equation. Thus, there
is no single Fundamental Equation governing the properties of all materials.
The postulates of classical thermodynamics place some restrictions on the form of
the Fundamental Equation. Let us examine the differential form of the Fundamental
Equation in the energy representation, Bq. (5-3). Since !f. must be first order in mass or
mole number, we can apply Euler's theorem (see Appendix C) to obtain the integrated
form of Eq. (5-3). In differential form, dU is given by:
du = (oful
as d§. + (Clfu]
av
-
-
,N
-
ful
0
~
(
dV + £..J aN.
'S,N
i=l
-
1
dN;.
(5-12)
V.N [i]
-•-J
or as we demonstrated earlier
n
dU = Til§.- PdV +
L t.t,J.Nj
(5-13)
i::l
which can be directly integrated using Euler's theorem recognizing, of course, that T.
P. and ~are intensive properties and kept constant. The result is
U= T~- PV+ ~ llf/;
(5-14)
Equation (5-12) is a linear partial differential equation of the first order. Therefor~
the solution must be of the form
U=x[{;. >·]
(5-15)
where x, y, z, ... can be ~. V, N 1
N, or any permutation of these variables. For a
one-component system, it is often convenient to choose x == N, y = ~. and z = V~ we then
obtain
p
• ••
(5-16)
or, since U =NV.
U=g(S, V)
(5-17)
The only other requirements our prior developments placed on the form of the
Fundamental Equation are that U should be a single-valued function of~. V, and N (see
Postulate I), and that. (()fuf()~Y...JJ::: Tshould be nonnegative.
5.2 Intensive and Extensive Properties
At this point, let us digress to a special case of Postulate I. In Eq. (5-17), we see that
for a one-component system. only two properties, Sand V, are required to obtain the
specific energy of a system. This is in no way a violation of Postulate I; by delineating
Section 5.2 Intensive and Extensive Properties
131
the specific properties (expressed in tenns of unit mass or mole number), we have
determined the .. intensity" of the system but notthe "extent'' ofthe system. To specify
the system completely (e.g., so that it can be reproduced by others), we must specify the
mass of the system in addition to S and V.
The variables that express intensity of the system are zero order in mass and are called
intensive variables. Variables that relate to extent of the system are first order in mass
and are called extensive variables.
We now prove that for a single-phase system of n components, any intensive property
can be defmed by the values of n + 1 other intensive properties.2 Let us call
b, c 1, c 2, •• •, cn+1 intensive properties of a single-phase simple system containing n
components. In general, we can express b as a function of n + 2 other properties
according to Postulate I. Let us choose these n + 2 as ca. c2, •. . , en+ I• and the total moles
(or mass) N. Thus,
db =
(::,l
dc1 + ...
+
[a~:~l
. [I], N
J
dcn+1 +
. [n+I], N
(~!l
dN
(5-18)
J.L •... ,cn+l
j
Integrating Eq. (5-18) by using Euler's theorem (see Appendix C), we have
(dNdbl
N= 0
(5-19)
L, ... ,c..l
Since N can be nonzero, ("dbl'dN)c ..... c must be zero. Therefore, Eq. (5-18) reduces
to a function of n + 1 intensive variab~es.
course, these n + 1 intensive variables must
be independent, so that we clearly cannot use all of then mole fractions x 1, • •. , xn. We
could, however, use n- 1 mole fractions in addition to two other intensive variables to
obtain the ~uired n + 1.
Note that this result is valid because we Jirnited the original set of n + 2 variables to
include only one extensive variable; if we had included two extensive variables in the
original n + 2 set, no partial derivative would have to be zero. Thus. we could state as a
corollary to Postulate t
Ot
For a single-phase simple sys'U~ the change ofan} intensive 'Hlrilzble can be expressed
as a function ofany n + 1 other independent intensive Wlrilzbles.
We shall use this corollary frequently in Chapter 9 in dealing with the properties of
mixtures. At this point, a word of caution is in order when dealing with intensive and
extensive variables in partial derivatives. For example. if for a pure material we express
U as a function of Sand V, then using Eqs. (5-18) and (5-19), we find (oUiaN) 5 v= 0 .
•
But (CJ!l/'dN)s, vis not zero~ from Eq. (5-7) applied to a pure material.
2 The proof is restricted to a single-phase system to allow us to choose any n + 2 variables as an
independent set The proof can be extended to specific cases of composite simp~e systems provided
that the n + 2 variables chosen fonn an independently variable set.
The Ca/wlus of Thermodynamics Chapter 5
132
aul =~
(C>N1¥
(5-20)
= u + N(auj =J.L
(5-21)
Since U = NU, we also have
(aul
i)N _.f
oN
_k,y
Since U is not equal to ~ (dU/dN)s vis not equal to zero. Thus, each of the three
(iJUidN}s,v have different connotations. The
derivatives, (i1Uii1N)s,v~ (C)U/i)N)£V•
flrst represents the change in the -specific energy-as we add more material while
maintaining constant specific entropy and specific volume. Since we are holding two
intensive variables constant during the process, all other intensive variables (e.g., T, P,
etc.) for the pure material must remain unchanged. The only way to conduct the process
is to enlarge the system in direct proportion to the added mass. The second and third
cases, however, represent changes in the total and specific energy during a process in
which we ma]ntain constant total entropy and total volume. Since we are adding mass
to the system, the only way to keep total entropy and total volume constant is to change
the specific entropy and specific volume (e.g., by varying T and P during the addition
of mass). Thus, the specific energy changes as the state of the system is varied. The total
energy changes because both the specific energy and mass vary.
and
Emmple5.1
In the entropy representation, the Fundamental Equation for a monatomic ideal gas (such
as He, Ne, Kr. or Ar at low pressure) is
(5-22)
where SO. r.f', and V' are constants representing values n a reference a- base state. From
the energy representation in the fonn of Eq. (5-l)~ determine the three equations of state
in the fonn ofEqs. (5-5) through (5-7).
Solution
r (~S-S'YR)
Solving Eq. (S-22) explicitly for U yields
U =lf (
~
exp
(5-23)
and, thus,
213
U: Nlf (
~)
exp
(~S--S"))R)
(5-24)
Section 5.3 Methods tor Transfonning Derivatives
133
Using Bq. (5-24). U= U/N was plotted in Figure 5.2a as a function of Sand V. The
equations of state can be found directly by partial differentiation of Eq. (5-23) or (5-24):
l
(~S-sjl
R)
3
(5-25)
exp C1<S--S')/R)
(5-26)
213
l(aul (au1 3l (V"J
- P • (~~1 = (~ =- ~ lf~la
00
r=
=
=l ( ]
CJ§. _.N as~
R v
exp
213
~=(~~t =a~[Nr'(~fexp (~rs;jL
=1~r exp(fs:>) (i -;!)
(5-27)
The results given in Bqs. (5-25) through (5-27) may be simplified if Eq. (5-22) is used to
evaluate the exponential terms. If thts is done, then
T ~ 2U . P
3R '
=2U
. and " =
3V '
t"'
u
(53 _32 Rs)
(
5. 28)
In fact, the assumptions behind the development of Eq. (5-22) are that U = ~ RT and
PV=Kl'.
5.3 Methods for Transforming Derivatives
Often one is faced with the problem of evaluating the magnitude of a particular partial
derivative or integral involving non-measurable, derived thermodynamic properties. For
example, suppose we needed a specific value for the entropy change M of a pure fluid
between two well-defined states (T1• P 1) and (T2~ P2). The calculus of continuous
functions in two variables works to specify the intensive state for a pure component
system with n + 1 = 2 degrees of freedom. Thust
AS= *~tdP+ J(~}u
(5-29)
The second integral involves (as/a1)p which equals CIT. Presumably, values of
cp may be tabulated for the substance of interest at particular temperatures and pressures.
But what about the first integral that requires evaluation of (dS/aP)T which is not readily
available? We need some convenient way to express this partial derivative in terms of
measured or tabulated properties.
Again, by adhering to the calculus of continuous functions, we have several tools for
manipulating these derivatives. Some of the more important ones for applications in
thermodynamics are listed below. For a general function F(~ y) of two variables~
The Calculus d Thermodynamics Chapter 5
134
1. DeritJative inversion
(5-30)
For example,
asj
1
(5-31)
(aP )r =(dPidS}r
2. Triple produ.d (xyz-1 rule)
l
(~l(~ 1~
(S-32)
=-l
For example,
(t~l(~~u~~1 = - I
3.
(5-33)
Chain rule expanswn to add another independent variable Cl>
(aFl =
dy
((JFl (aq,l
Oy
(iJF!dci>)x =
(dyldel>).x
del>
(5-34)
For example, set <f)= T
as 1= <as;anp = c;r =l
(aH)
(oHia1)p
cp
(5-35)
r
4. MaxweU redprocizy rekdiotult.ip
Maxwel1 's reciprocity theorem states that the value of 2nd or higher order derivatives is
independent of the order of the differentiation for a smoothly varying, continuous
function F(x, y)
l (d(d:dy),l
(d(d~~xly
=
(5-36)
or representing these 2nd order derivatives in abbreviated form;
FXJ=Fyx
(5-37)
The relationships cited above in (1) - (4) can easily be extended to larger
functionalitiesl for example where n + 2 variables exist such as in the Fundamental
Equation.
Our original problem of evaluating the integral containing (oS/aP) 1 can be made
tractable using a Maxwell relation in the appropriate thermodynamic variables. Later in
this chapter we will estab]ish methods to select the appropri.ate variables for tl)ese
transformations, but for now, leCs employ the Gibbs free energy function, G. For a pure
system, G is written in intensive form as a function of Tand P, G = j{T,P), and
dG=-SdT+ VdP
(5-38)
Section 5.3 Methods fOI' Transforming Derivatives
135
where
Thus,
_(acaa/dT)pl- (acac;aP)r1~ _
Grp-
(Jp
-
dT
-GPT
(5-39)
or
(5-40)
so we can use PVT properties or a fitted PVT equation of state to determine (CJSI'dP)T
and then evaluate the integral directly. Earlier in Section 5.1, we stated that the 2nd
derivatives of Uwith respect to§. and V[see Eqs. (5-10) and (5-11)] were related by
Maxwell's reciprocity theorem. Now we can show that these 2nd derivatives are indeed
equal to one another.
uy_~=
a2 u a2u
asav = avas=~~
(5-41)
where we have used the abbreviations introduced earlier for the 2nd derivative. By taking
the reciprocal ofEq. (5-40), we immediately see that
u~~=-(~lN =(~~t=~y
(5-42)
recognizing, of course, that we have returned to an extensive fonn for U.
In addition to the derivative manipulations characterized above, it is also useful to
illustrate some of the different properties of intensive and extensive variables. A partial
derivative may involve intensive and extensive variables. For a pure material, only
n + I = 2 intensive variables are independent; hence, a partial derivative involving only
intensive variables b, c. and d can be expressed as (oblac)d, where it is implied that N
is constant. That is,
(5-43)
We now show that for a pure material, any partial derivative involving extensive
variables can always be reduced to expressions involving partial derivatives of entirely
intensive variables.
Now consider the derivative (()b/CJc)d. ~where one of the four variables is extensive.
) (i). If c is extensive, then
(5-44)
The Calculus of Thermodynamics Chapter 5
136
The proof follows froiD applying Euler's theorem ro b = f(t;_, d. e):
0=
(db1
ac- -c
.~
l(h). If b is extensive, then
l. (iJc/~!!)d,
(~~
=
e
~ ~
= =
(5-45)
l(iii). If d (or~) is extensive. then
(5-46)
Since
and
(~~l J~~J
l
c,t.
,e
then
abde I1
(
(aN/oc)b.~
e:::-
=d
(ahl
(iJN/ob)c, e = oc
N=
(abl
de
Thus, if only one variable is extensive. the partial derivative is ftnite and nonzero only
if d ore is extensive. The extensive variable may be deleted to yield a partial involving
only intensive variables.
Now consider the three cases in which two of the four variables are extensive.
2(i). If band care extensive, then
(5-47)
which follows directly by applying Euler·s theorem to!!_= f(£, d, e):
~=(~!l}
and !!.=Nb,
~:Nc
2(ii). If b aod d (or e) are extensive, then
(~!
Since, expanding !! == Nb,
LN[(~~1-!(~:),]
=
(5-48)
Section 5.4 Jacobian Transfofmations
137
(!~1. =N(~!L +b(~~L
(5-49)
Eq. (5-46) may be used to reduce (ob/CJc)~.- The last tennis reduced as follows:
2JN1 _- (i)gjoc)N
(OC -' e- (d#ON)C,
e
= _ N(CJdloc)t!
d
t!
2(fu). If d aDd e are extensive, then
(5-50)
The numerator and denominator can each be reduced by applying the results of case 2(ii).
Any partial derivative involving three extensive variables can now be reduced to
partials involving two extensive variables by using Eq. (5-49) followed by one or more
of the steps iHustrated above. Similarly, partials involving four extensive variables can
be reduced to three, etc. The net result is that any partial derivati11e for a pure material
can be expressed in terms ofpartials invol.ving only three inten.rille variables.
5.4 Jacobian Transformations
A useful technique for manipulating thermodynamic properties and their derivatives
involves a transformation using Jacobians or functional determinants. For the derivatives
of functions of 2 variables./ (x, y) and g(x, y ), the ]acobiJln is defined as:
a{f, g)=
Cl(x, y) -
(!),
(~).
-(211(E.s_l_(.?ti(M}
(~) (~)
J ay1
-
)'
ax
Oy
(5-51)
dx
%
Certain properties of Jacobians make them particularly useful for transforming
derivatives of thermodynamic variables. These include:
1. Transposition
au; s)
acg. I>
(){x, y) =- d(.r. y)
2. ln11ersion
d([. g) 1
O{x, y) - o(x. y)
(5-53)
CJ(f, g)
3. CluJin rule expllllsion
C)(£ g) _ d(/, g) d(Z. w)
a(x, y) - o{z. w) o(x, y)
(5-54)
The Calculus of Thetmodynamics Chapter 5
138
where z and w are two additional variables. Another important property is the
simplification that occurs to Eq. (5-51) if we are only interested in evaluating the first
partial derivative of a functionjtc~. g) with respect to z at constant g:
(Etl-
a(f; g)
(5-55)
dZ - o(z, g)
By applying the chain rule expansion and inversion property [Eqs. (5-54) and (5-53)],
Eq. (5-55) becomes:
act&)
(dZ
~l= d(x,y)
(5-51)
a(r, g1
acx. y)
The properties of two-variable Jacobians are easily extended to functions in m
variables xi(i=l, ..., m),j,{x1, x 2, ... , x,J. The Jacobian is
O(fl,f2, ...• f,.)
(5-58)
acx •• ~····· xm)
or in determinant form
atl
ax.
att
a~
ox2
axm
(5-59)
ofm
dfm
at"m
axl
ax2
oxm
where each partial derivative holds all X; constant except the one involved in the
differentiation. Let's consider a few examples to illustrate how Jacobians are used in
thermodynamics.
&ample5.2
Consider the isenthalpic (Jou]e-Thompson) expansion of a fluid across a well-insulated
va]ve. Here, the derivative
(~~l., Joule-Thompson coefficient= aH
is of interest. Express a.H in terms of measurable properties.
Solution
For a pure fluid, the specific enthalpy, H = f(T. P), so we can use Eq. (S-57) wtth /= T.
8 = H, and z=P. Thus,
Section 5.4 Jacobian Transformations
139
d(T, H)
d(X, ~)
()(P, H)
d(x. y)
(5-60)
Now we need toselectxandy. Although we have many choices, frequently.as in this case,
we want to obtain expressions in terms of measurable properties like T, P. V,etc. A good
first guess for x and y would be to use T and P since they are in the derivative of interest.
Thus,
d(T, H)
d(T, P)
d(P,H)
(5-61)
Cl(T, P)
Using the tranrposition property [Eq. (5-52)] and Eq. (5-55), Eq. (5-61) is quickly
simplified to:
arj
(~~~
(~~~
_ o(P, n _ )r ___
)r
d(H, 7)
(dP 1- -a(H, P)- joHj iJ(T,P)
(5-62)
-CP
lar ~
In this instance, we could have used the triple product directly (Eq. (5-33)) to obtain Eq.
(S-62). However, such direct lransformations are not always possible. For example, in
Section 5.9 we will demonstrate how Jacobian transformations are of considerable use in
manipulating complex partial derivatives of multivariable functions. Before we try ID
evaluate (()H/c)Phin Eq. (5-62) in terms of measurable properties. let's consider another
example.
Example 5.3
The jsentropic (reversible. adiabatic) expansion of a pure fluid iovolves the derivative:
((}TloP)s = isentropic coefficient
=0-s
Express the isentropic coefficient in terms of measurable properties.
Solution
The as derivative can also be reconstructed using 1acobians
arj = acr. P)
(aP) o(P, ~
a~.~
(~I
aP
_ ~ _ -T(as(
_ as( - -cp - cp aP )r
as
o(T, S)
:.
ar>
r
(5-63)
The Calculus of Thermodynamics Chapter 5
140
In order to express cx5 (or o.H} in tenns of measurable properties we must convert
(CJS/aP)T [or (i)H/clP}r] to another form. To start with. consider the closed system
combined Law equation which is equivalent to the intensive fonn of the Fundamental
Equation in differential fonn
(5-64)
dU=TdS-PdV
By rearranging, apd differentiating with respect to P at constant T. we get
asj 1 (au( P (av(
(aP )r =T l ()p )r + T ()p )r
(5-65)
Again. we end up with a cumbersome derivative @UioP)T' Some readers may recognize
that we need to introduce an auxiliary potential function that has as its natural variables P
and T. This is the Gibbs free energy G =f(T. P) wtUch is defined as:
G= U+ PV-TS
So by chain rule expansion of the differentials:
dG=dU + PdV+ VdP- (TdS+Sd1)
Thus. by substituting the expression for d.U given in Eq.
(5-64}~
(5-66)
dG becomes
(5-67)
dG=-SdT+ VdP
And, by usiog Maxwell's reciprocity theorem, we obtain
(!!1 =-(:~t
(5-68)
Now we can revise Eq. (5-63) for as to
~s=(~!l:+ ~(~~1
(5-69)
Let"s see if we can use this result to help us evaluate the derivative (iJH/aP) 1 in the O.n
expression [Eq. (5-62)]. Here we begin with the definition of H U + PV, differentiate.
and use the closed system combined taw as we did before to obtajn
=
dH == TdS + VdP
(5-10)
Thus. by differentiating with respect toP at constant T, we get
(~~1 =r(~~1+ v
(5-71)
Because we already have an expression for (iJStaP)r in terms of measurable PVT
properties (Eq. (5-68)), we can express the Joule-Thompson coefficienl CJ"H as
ijT
-[T(- dVIo1)p +
V]
~n:~Pl:-----c-p_____
where only PVT properties and
c, values are needed to evaluate et.H-
. (5-72)
141
Most vapor compression refrigeration systems in domestic use employ
Joule-Thompson expansions to cool the circulating refrigerant. Thus. in these practical
situations. ~ must be> 0 to obtain cooling as the pressure is lowered across the valve.
What happens if the fluid being expanded is an ideal gas? In this case. a,;.= 0; thus.
fluids whose PVJ' properties closely approximate ideal gas behavior would not be good
choices for refrigerants when Jou]e..Thompsen expallsions are used. An alternative is to
use an adiabatic turbine to expand the fluid to a lower pressure. Here. a 9 is the appropriate
sealing parameter as it provides the maxbnum cooling effect for the limiting c:ase of a
reversible, adiabatic or isentropic expansion. Now. even an ideal gas will cool upon
e:xpansion.
5.5 Reconstruction of the Ftrtdamental Equation
The Gibbs Pun.damental Equation where U=/~. V.N., ...,N,J provides the
necessary and sufficient information needed to describe all the stable equilibrium states
of any simple system. Unfortunately.~ V N coordinates are not completely amenable to
direct measurement-for example. nodired.-reading entropy meters e:xistl Thus aforma1
transformation is required that preserves the encoded information content of the
Fundamental Equation while expressing the functiona1 depeodence in variables other
than§.. v. and.N1 (i=l .... , n). Once we have this procedure, we can be sure that all other
infonnation of interest to us in classical thermodynamics ean be obtained from the
transformed equation. The Legendre tnnsfonnation provides a rigorous mathematical
route to achieve the desired reconstruction of the Ftmdamental Equation.
As shown in Section 5.1. if the Fundamental Equatioo were known. the properties T.
P, and J.a.,. could be determined by partial differentiation as expressed in the equations
of state in Bqs. (5·5) through (5-7). Altemativdy. the Pondamental Equation c.an be
recovered, if alJ the equatiODS of state are known, by substituting these equations into
Bq. (5-13) or the common integmtedform (Bq. (S-14)]
(5-73)
As shown in Section S.2. then+ 2 intensive variables. T. P, and 1Ji (i:=l ..... n), which
are expressed explicitly by the equations ofstate, are not all independently variable. Any
one of these variables can be expressed in diffes:ential fonn in tenns of the other n + 1
variables and. upon integration, an expression connecting the n + 2 variables can be
determined to within an arbitruy constant. It thus follows that only n + 1 equations of
state are necessary to determine the.Pundamenta1 Equation to within an arbitrary constant
(equivalent to specifying one or more reference state conditions). In Bxample 5.1. ifEq.
(5-28) is combined with Bq. (5-22) to eliminateS, p. can be expressed in terms of
reference state constants ll'. V' and S'; and variables T. V.
As an example, let us consider a pure material. The intensive Gibbs free energy or
chemical potential, G = JL, can be expressed as a function of T and P:
The Calculus of Thermodynamics Chapter s
142
d~=(~}iT+(~}lP
(5-74)
=-Sa nd (dJ,l./aP)r= v. If
In Section 5.4 [see Eq. (5-67)], it was shown that (dWOT}p
ltaneously to obtain
Eqs. (5-5) and (5-6) were known, we could solve these simu
(5-75)
S = g(T, P)
and
(5-76)
ly in analytical form, but
These equations are usually available. although not necessari
us assume that we have
the entropy is known only to within an arbitrary constant. Let
available Eqs. (5-76) and (5-77):
(5-77)
V=g '(T, P)
ition. Substitution into Eq. (5-74)
T=T', P =po, V = VO, and
and integration from an arbitrary reference state for which
0
f.L = f.L leads to
(5-78)
J.l = J.L0 - S"(T - T')- IT g''(T, P) dT + I' g'(T. P) dP
'f
P. Although this equation
which is the desired relationship for J.1 as a function ofT and
, (S-6), and (5-78) are
contains two arbitrary constants (S' and J.L~. when Eqs. (5-5)
tal Equation, these two
substituted into Eq. (5-73) in order to obtain the Fundamen
find
arbitrary constants appear as a sum; in particular, we would
(5-79)
where~ is an arbitrary constant or reference state cond
r
or
(5-80)
JJ.".>. it is clear that only
Of the three arbitrary constants in Eq. (5-80) (i.e.• UO, S", and
s of U and Sat the reference
two can be chosen independently. Thus. we can set base value
base value for J.l is uniquely
state for which T=T', P;;; pt', V =V', but having done so, the
specified by Eq. (5-80).
5.6 Legendre Transfonnations
Eq. (5-1), the properties
In the energy representation of the Fundamental Equation,
always an appropriate
§.. V, N1, . .. , Nn are treated as independent variables. This is not
re can be measured much
set of independent parameters. For example, since temperatu
, ••.• Nn as the independent
more conveniently than entropy. we might like to use T, V. N 1
express a propertysuch as
variables. For a single-phase simple system, we can always
, we know that a
U in terms of n + 2 other properties such as T, V, N" .. ., N,. Thus
func tion / exists such that
(5-81)
I
I
Section 5.6 Legendre T1ansfonnations
143
and
dU
=(~l
d:r +
'£N
(£G- I.fr
dV +
.N
i (!kl
.r=l
'
.Y...N, [•]
(5-82)
tiN;
Given the Fundamental Equation, the function of Eq. (5-81) can be found by
differentiating Eq. (5-1) to obtain Eq. (5-5),
T= g-M_, V, N1, ... , N,J
and then solving Eqs. (5-1) and (5-5) simultaneously in order to eliminate J... The result
is an equation of the form
U=f(T, V,
N,N,.)={(~) ,
V ,N1, .. .
1, ...
,N.]
(5-83)
Y..N
Although Eq. (5-83) is cf the fonn desired (i.e.• Eq. (5-81)), the information content
ofEq. (5-83) is less than that of the Fundamental Equation. Equation (5-83) is a partial
differential equation that can be integrated to yield the Fundamental Equation only to
within an arbitrary function of integration.
We must now ask whether or not there are other functions with the same information
content as that of the Fundamental Equation but with independent variables different
than §_, V, N1, •• •1 Nn. The answer is that there are such functions if we are willing to
restrict ourselves to a set of independent variables in which we choose only one from
each of the following (n+2) pairs: {~. T), { V. P]. (Ni, P,;} for i=l, n. These pairs of
variables are usually referred to as conjugate coordinates. Note that there is one
extensive (e.g.,§., V, Ni) and one intensive (e.g., T, P, Jli) variable in each conjugate pair.
This grouping of variables is a natural consequence of the original formulation of the
Fundamental Equation in Gibbs coordinates.
As mentioned in Section 5.3, conjugate pairs of additional natural variables may be
added to the Fundamental Equation to account for non-PV work effects. Table 5.1 lists
the pairs of conjugate coordinates that are commonly encountered in problems of
classical thennodynamics.
Table 5.1 Coojugate Coordinates
Type
&tensive Para01eter
Intensive Parameter
Reversible heat flow
T
Pressu~ Voaume work
-P
Mass flow enthalpy and entropy
Stufacedefonnation work
Generalized work
a
The Calculus of Thermodynamics Chapter 5
144
To formulate these functions, a Legendre transform is employed. Such a
transformation stems from a basic theorem in line geometry, and although the rigorous
proof is no simple task, the results are easy to apply. The basic principle is that a curve
consisting of a locus of points can be described completely by the tangent lines that form
the enve]ope of the curve. Lef s consider a simple case to illustrate what Legendre
transfonns are and how they are used. We define a basis function yC0>==f(x) with only
one independent variable. The graph of / 0) versus x shown in Figure 5.3 is indicative
of a well-behaved, continuously differentiable function. For any value of x~ a straight
line tangent to the curve has a defined slope and intercept with the y-axis as shown on
the figure. If we call the slope; =dymldx and the intercept yCI>. we can construct an
infinite set of line tangents to the original curve yCO> = f(x) such that:
y(O) = 9: + yC I)
( 5-84)
for all values of x. By knowing values of y<J) and~ for every value ofx we can reconstruct
the original function y<O>. The Legendre transformation is carried out by solving
Eq. (5-84) fory<•> and inverting the functional relationship between~ and x. Thus,
yO>= yro> _ = f[~]
(5-85)
x;
Now y<•> can be represented as a function of~ with its derivative slope given by,
-x=
#')
~
and tangent intercept equal to y<0>. The function yO> is the Legendre transform in one
dimension. It clearly contains the same information content as our original basis function
y<O> since one can reproduce the original function plotted in Figure 5.3 from the set of
line tangents.
An analogous approach can be used for functions in two variables, y<O> = f(x 1, x2).
Here there are two characteristic slopes corresponding to partial derivatives:
~. =(:i~l
and
~=~=l
(5-86)
i
The transforms y(l) = y(O)- x1 ; 1 and 2>~ yCO>- Xt l;l -~~are shown in Figure 5.4.
In this 2-D case the transforms are obtained from planar tangents to the basis function
surface rather than from line tangents to a curve.
Now we can generalize the results tom variables, where the basis function,
y0) = f(x 1, .. •, x,J
(5-87)
represents an m-dimensional surface. There are m f.trSt-order partial derivatives of ;/0>
with respect to each of the m independent variables, x1, ••• , xm. Defming these derivatives
as ~i•
(5-88)
Section 5.6 Legendre Transformations
145
Family of line tangents
~
y.o> = ;x + yl}
t
y.o>
{
yeo>= f(x)
yO>= y.o>-;x
!~"' df lldx "'f'(x)
0
~ = f'(x al'\ at Xa
I
1
Figure 5.3 Reconstructing a function y =j(x) using a family of ltne tangents-the
one-dimensional Legendre Transform concept.
or
(5-89)
where the symbol [x;] in the subscript of the partial derivative indicates that X; is not held
constant. It follows that the variation of yCO) with x 1 could be described by the envelope
of tangents in the / 01 - x 1 plane. If yC11 is the intercept of the tangent corresponding to ~~,
Y(l)(~ 1 • Xl•· ..• xm) = /
0
1
) - ~ x,
(5-90)
The function / >(~ 1 • x 2, . ..• x,J is called the first Legendre transform of / 0) with
respect to xa. In other words, a Legendre transform results in a new function in which
one or more independent variables is replaced by its slope. There are obviously m
different first transforms, depending on the ordering of the variables x 1, .•• , x111•
Higher·order transforms are defined in a similar manner; thus,
y(t)(~J~· .. , ~ xk+l•· ... x,J is the kth Legendre transfonn:
1
k
Yc~r.> =,CO> -
L ~ri
(5-91)
i=l
The total differential of the original basis function
as
1ft
dy(O)
=I, ~i dx.i
i=l
y(o)
can, of course, be expressed
The Calwlus of Thennodynamlcs Chapte1 5
146
E
2
Figure 5.4 Geometric construction for a two-dimensional Legendre transformation
using y(O) =f[.xp -s]. Two first orrler transforms yU> =IJ~1 , xJ at© and
y<ll f[x 1, ~]at (M) and one second order transform y<1 = /[~~· ~] at ®, are shown.
=
and the differential of the kth transform can be obtained by differentiating Eq. (5-91)
using the chain rule and substituting Eq. (5-92) to give
/c
m.
/c
~ ~ dx- ~X~.-~ ~.flx.
dy(i)_
- ~ .,;
i
~ .~.
~ ~.
'
pl
i=I
i•)
which simplifies to
k
dy(/c)
m
=- ~ xi~~+~ ~; dxi
l::l
(5-93)
r-kt-1
Since y<lcl is a function of~~~· ..• ~"' xk+b· . . , xm, it follows from Eq. (5-93) that:
For transformed varUlbles (1~ i 'Sf< )1
(ak:)
l.. .
r;J,....
~,.x..,..... x. ; -x,
•(5-94)
Section 5.6 Legendre Transtormations
147
(5-95)
Equation (5-94) is sometimes called the inverse Legendre transform. Equations (5-95)
and (5-94) clearly show the canonical relationship between conjugate coordinates
{x;. ~t} that is similar to forms found in classical mechanics. Eq. (5-95) is applicable for
all cases where i > k, and thus one may generalize the result as
a,<'-•> ay<i-'-> == ... ity<O> == ; .
'::!
dXI
dX;
(5-96)
1
dX;
The partial derivatives in Eq. (5-96) were expressed without indicating the set of
variables to be held constant. However, it is clear from the discussion above that the
degree of the transform determines the set. and the only exception would be that variable
used in the actual differentiation. For example, the restraints on the term ayO-l>/c)x;
would be that ~ 1 , ••. , ~-~, xi+l , . ..• x, would be held constant.
To illustrate the application of these relations. letyC«J be the total internal energy of a
simple system U; then the Fundamental Equation would be given in Bq. (5-1). Suppose
that we desired the transform yC2>. Using the ordering of variables given in Eq. (5-1), we
can organize the transformation using the table below:
'YP,))=
..ti
I
.,,., =G
u
~;
~
X;
T
T
-s
I
2
s
v
-P
2
-P
-V
3
Nt
J.L,
3
N•
J.l,
n-t-2
N11
J1
n-t-2
N,.
11
with this ordering of variables:
(2)_/(r. ~ N
- ~l• ~' 1•· .. , Nn)
Y
{5-97)
with
(5-98)
(5-99)
The Calculus of Thermodynamics Chapter 5
148
Then, with Eq. ( 5-91 ),
y<2) = U- ~- (- PYJ::: G
(5-100)
where G is the total Gibbs energy. The analogs ofEqs. {5-92) and (5-93) are Eqs. (5-4)
and (5-101).
n
dyo.) == dG ==- §. ar + v dP +I, J.L, dN,
(5-101)
i=l
where the chemical potential ll; can be defined in several ways as
lli
=(:~]~f.N,
I
(5-102)
•... ,[Ni), •.. .N,.
cr equivalently as
~i=(:~.l
(5-103)
r :,P.Nt·· .•,[N~ •.. -.N,
The significance of the Legendre transfonn is thus evident. The important
thennodynamic property G is simply a partial Legendre transform of the energy U from
(§_, V, N 1, ... , N,J space to (T, P, N 1, ..., N,J space. Equation (5-101) is also a
Fundamental Equation and no loss in infonnation content has resulted in going from the
U to the G representation.
We could have started our transformation in reverse. Referring to the table of x; and
~; given above, we could have defined a new basis function as G = y<2>~ y•(O) and
transformed back to U = y *(Z) :: y<0>, that is, to the old basis function. Again, there would
be no loss of information content.
The totm Legendre transform ofEq. (5-1) is
n
yCrt+l)
= U- T§. + PV- 2, 11/Ji
=0
[by Eq. (5-73)]
(5-104)
i=l
Thus, Eq. (5-93) becomes
n
dy(n+l)
= 0 =- §. dT + v dP -
L Ni $;
(5-105)
i=l
Equation (5-105) is known as the Gibbs-Duhem equation. Later in Chapter 9 on
mixtures and in those chapters that follow, we will utilize the Gibbs-Duhem relationship
extensively.
It is important to point out certain generalities in the use of Legendre transforrnations.
We introduced a general functional equation [Eq. (5-87)] with arbitrary ordering of
x 1,.. ·~ xm. We noted that for each xi there was a conjugate coordinate variable, ~i [Eq.
(5-88)]. We then illustrated that one could readily derive a functional relation with the
same information content wherein we replaced independent variables x 1•••• , xk by
Section 5.7 Graphical Representations of Thermodynamic Functions and Their Transforms
149
~ 1 .... , ~. This was the ktb Legendre transform defmed in Eq. (5-91) and shown in
differential form in Eq. (5-93).
As we demonstrated in the example above, one may also redefine this kth Legendre
transform as a new y(O) basis function if care is taken in defining the correct independent
variables and conjugate coordinates. For example, we have shown that beginning with
Eq. (5-l), we obtained the Gibbs energy potential function by a Legendre transform of
U into T, P, N 1, .•. , Nn space. We could now use the Gibbs energy function as our basis
function/0>, but the independent variable set (x,, ... , xnJ would be (T, P, N 1, ••. , N ,J with
arbitrary ordering. The conjugate coordinate variables ; 1, . .. , ~m would still be defined
by Eq. (5-88); for example, for the variable T, ~r= (dGid1)p N;:;;; -S~ for P, ~P = V, and
for Ni, ~i ~ J.l.J In fact, we may select any Legendre t~sfor~ as th~ y<O) basis function
by redefming the independent variable set. Examples illustrating the use of Legendre
transforms for manipulating partial derivatives are found at the end of Section 5.8.
If we had started the mathematical transform development with the entropy fotm of
the Fundamental Equation ~ = fi!:!.. V, N 1, ••• , N11 ), we would have produced an
equivalent set of transfonns that are called Massieu-Planck functions, which are
sometimes easier to use than Legendre transformations, e.g., when working in r 1
coordinates [see Debenedetti (1986)]. Additional treatments of Legendre transformations are given by Callen (1985), Alberty (1994), Aris and Amundson (1973) , and Tisza
(1966).
5.7 Graphical Representations of Thennodynamic Functions and Their
Transforms
Beginning with the Fundamental Equation for U [Eq. (5-1)], we can now transform
one or ali of the independent variable set~. V, N1, •• ., N,.. Let us choose only the variable
V and reorder so that V represents x 1• Then
y<O::::y< 0)-~x =V-(-P)V=U+PV:H
11-
----
(5-106)
where this particular Legendre transform is called the enthalpy. We note that
H = j(P, §., N 1, .. . , Nn)
(5-107)
We call Eq. (5-107) a Fundamental Equation in the same way that we refer to Eq.
(5-1)~ then P is x 1, §.is~. etc., and ~ 1 = (dHI()P) ~ V. We can recover Eq. (5-1) by
carrying out a Legendre transform assuming that Eq. (5-1 07) is the yCO) function; that is,
y<'> == y<O)- ~ 1 x 1 =H- (YJP =U
(5-108)
This transform can be readily shown in Figure 5.5 for a common pressure-enthalpy
diagram. If a curve of constant entropy is considered, the slope is V. The intercept of
this tangent of the enthalpy axis is, as shown, equal to the internal energy V.
The CaJrulus of Thermodynamics Chapter 5
150
t
I
I
••
I
I
-~--------J
~
{
0
p----.
Figure S.5
With internal energy as the basis function, there are n + 2 pennutations of first
Legendre transforms: the two common potential functions, H(P. §.. N 1•... , Nn) and
A(T, V, N 1,. .. , N,.). and n other functions for the independent variable set of
~. V, N 1, .... Ni-h J!;, N;+ 1, ... , N11 • Since the ordering of components is arbitrary, we shall
refer to the n functions as U'(§., V, Jl.h N2o···· N,J . In a similar manner, there are
(n + 2)(n + 1)/2 second Legendre transforms: one is another potential function, the
Gibbs free energy G(T, P, N 1••.., N,.); there are n of the formA'(T,IJ. 1, V.N2,•.. ,Nn) and
n of the fonn H'(P, IJ.~o ~. N'b···· N,J, and (n)(n - 1)12 of the form
U"(~ 1 .1J.2,~, V, N3, ... ,N,J. Third Legendre transforms would involve G',A".H" , and
U"' potential functions. Table 5 .2lists the important potential functions used in chemical
thermodynamics.
Again it is instructive to re-e:umine the plots of the Fundamental Equation to show
how various Legendre Transfonns are geometrically related to the USV surface.
Fundamental equations are plotted in Figures 5.1 and 5.2 for pure water and ethylene.
Multiple phase equilibrium conditions, such as the triple point of pure water and the
liquid-vapor coexistence envelope occur at specific temperatures and pressures for each
particular substance. At first glance, one might expect that the thermodynamic potential
that involves the natural variables of temperature T and pressure P would yield useful
relationships. This function is, of course, the Gibbs free energy G which is the second
Legendre transform of U and can be visualized geometrically as the intersection point
of a tangent plane to the U ~ V surface with the U axis.
Section 5.7 Graphical Aapresentations of Thermodynamic Functions and Thei' T1ansforms
151
Table S.2 Thermodyna mic Potential Functions and the Gibbs·Duhem
Relation for Simple Systems
Symbol
Function1
Ca.o.onk.al
Coordinates'
!l'"" T~-PY+L,J11 N,
v
yfO>
§., y, N1 (i=l, ..., n)
-H=U+PV
- -
H
y(J)
~~ P,
A
·l>
T, V, N. (i=:l ,... , n)
G
yfl)
T, P, ~ (i=1,... 1 .n)
Total
frr+1)
T, P, )J.i (i=l,... , n)
Type
Functional Equation~
lntemal&.ergy
•
;.J
Enthalpy
Helmbolt:z Free
Energy
G=U+Pf-~=H-~
Gibbs Free
Energy
Gibbs·Duhem
Relation
N, (i=l,..., n)
-
I
Transform
Type
Total Differential
II
Internal Energy
dU =Td~- Pdf+
L llflN,
i=l
II
Enthalpy
dH '=' Td§, + Y..dP +
L JJ.tlN;
i•l
Helmholtz Free
Energy
"
~ = -§.dT- PdY, +L ll,dN1
ial
Gibb5 Free
Energy
Gibbs-Duhem
Relation
1
y<0>::: basis function
2
y(l >=First Legendre transform,
Obtaioed by Euler integration
3
"
dG=-~dT+ Y.dP+ l:wm,
b:1
dftt+'l) =""§.tlf + 'f.dP -
•
L N,dfl, = 0
~I
yrl) =Second Legendre transfonn, ...
The canonical coordinates td'er to the set of natural variables that preserves the infonnational
content of the Fundameoral Equation, yr!J) = U.
The Calculus of Thermodynamics Chapter 5
152
For a pure component system, the intensive form of the Fundamental Equation can
be used as we did with Fi~s 5.1 and 5.2 where U is plotted as a function of S and V.
0
In this case, -y' >= U andy >= G =Jl ani points d multiple tangency with a plane rolling
over the USV surface will have the same intersection point on the U axis and thus will
2
have identical values of,< >= G = j.l. Later in Chapter 6 we will show that this geometric
condition rigorously describes the mathematical criteria for phase equilibrium. Thus, it
is no accident that the triangular plane in Figure 5.1b corresponds to the triple point
where solid ice. liquid water. and water vapor or steam coexist in equllibrium. The triple
point is equivalent to a unique condition where triple planar tangency exists on the USV
surface of Figure 5.1a. Similar common points for double tangency on the USV surface
can be used to characterize two-phase equilibria, such as the liquid-vapor coexistence
region for a pure substance.
5.8 Modifications to the Fundamental Equation for Non-simple Systems
Additional variables given in the original expression for the Fundamental Equation
[Eq. (5-l)] may be needed to describe the behavior of systems in many practical
situations in chemical thennodynamics. Some of these cases have already been
introduced in Chapters 3 and 4. For example, there is frequently a need to account for
gravitational or inertial forces- potenti al and kinetic energy contributions to the total
energy E. Systems with these effects, of course, are not simple systems as we have
defined them in this text; but nonetheless, their behavior can be represented by modifying
the Fundamental Equation given in Eqs. (5-l) or (5-2). A common approach is to
int oduce additional generalized work terms. such that
m
dE or dU =Til§. - PdV +
n
L F1 · dx1+ L Jl,d.Nt
j•l
(5-109)
i=l
where the 1ast summation involving tenns Fj · ~represents all non-PdV work effects
that may be important. A number of these are listed in Table 3.1 in Chapter 3. These
effects typicaliy include:
•
Surface defonnation work- ada
• Electric charge transport - E dq
• Electric or magnetic polarization - EdD or HdB
• Linear elastic deformation· Fx ~
The addition of these other work or energy effects does not violate the four postulates
that we have set forth to develop the laws ofthermodynamics. In particular, Postulate I
still hoJds in that two independently variable properties plus the masses of all
components present are sufficient to specify the equilibrium states of a simple system.
Non-simple systems with these non-Pd V work effects, with potential or kinetic energy
effects, or with other constraints (such as semi-penneable boundaries) can be treated.
Section 5.8 Modifications to the Fundamental Equation for Non-simple Systems
153
With sufficient information we can specify the equilibrium state of these systems or
the
path of a particular quasistatic process.
Assuming that Eq. (5-109) can be integrated using Euler's theorem, we can write:
E or U = fC§., V, N1•.•• , Nn, z 1, •••, !m)
(5-110)
which should be a well-behaved continuous function. Then all of the relationships
developed earlier in this chapter can be utilized for manipulating partial derivatives
and
the like. For example, Maxw ell reciprocity relationships can be applied:
(a:S•l =(:~I
l - -·
l,y,
Nl' %i
(5-lll l
Nl ' xi [k)
Legendre transfonnations can also be developed to interrelate thermodynamic
properties. The following example problem illustrates the use of Legendre Transforms
in manipulating partial derivatives.
Example 5.4
5.4{a) Express (d~laP>r. N. in terms of PYTN properties and/or their derivatives.
•
Solution
The variables held constant (T. N;) and variable involved in lhe differentiation (P) sugges
t
that y(J.) = G should be used
dG=-§.dT+ VdP +
l:" JJ.•.dN;
i=l
and we see that a Mu.well relationship for y~>: yJfl gives the desired result
y<?l
=a
==11
~
)]
5.4(b) Express [ d{g_/T
d(l/7)
(a§_l
(avl
aP
ar
=
·~
-
(5-112)
.~
in terms of a Legendre transform of U.
P.N;
Solution
=
Because Q .,<2>is involved with its natural coordinates of P. T, N , first try to simplif
y by
1
expanding the derivative, recognizing that Gr == (OGioT)p, N. =~ and
that
TS.
-G=y(2)=U-T
- -S+P'V
- ~H--
d(l / 1))]
[d(QIT
I
(aol
=G- T oT
P, N1
,
= G + T§.=H - T§. +T§.= H
(5-113)
N1
Eq. (S-113) is the famous Gibbs-Helmholtz relationship which will be used frequently
to
show the temperature dependendes of various derived properties.
The Cslculus of Thermodynamics Chapter 5
154
5.4(c) Express (ac,,aV)T in terms of PVTproperties and/or their derivatives.
Solution
We start by using the definition of Cv and the basis function of y<0>= U in intensive form.
c."'(~~l
and
dU=T&-PdV=Cft+(~~ldV
Therefore. with (aulaV),-::; T(i:)SiiJV)T-P and a Maxwell relation
(~;1 =
[
a[i~) -PJ
a;
Expanding.
acv[ =(as! (accas;av)r)l-(aPI
( av )r av )r + T
ar
ar}
With the V. T variable set, use ytl) =A to see if a suitable Maxwell relation can be
developed to express (aS!oV>r differently
dA=-SO-PdV and
A1v={~~1 ={~~l=Avr
and we can simplify the equation for (aC/dV>r to
(~~1 ~+T(~l
(5-114)
5.4(d) For systems where surface forces are important. the Fundamental Equation can be
redefined by adding a term + a
to account for the reversible work due to surface
deformation, where a--surface tension in J/m2 and a= area in m2 • Develop an expression
for (a¥oa)r, y, Ni in terms of properties that can be measured experimentally.
a
Solution
The modified Fundamental Equation in differeotiaJ and integrated fonn is:
n
dll. =Ttl§.- PdV +ada+
L ~idN;
Fl
and
IJ
u = T§.- PV + a a + I, J.1.1 N1
i=.
Section S.9 Relationships Between Partial Derivatives of Legendre Transforms
155
Again by inspecting the variable set 0', T, V, and Ni involved in the derivative, a yC2>
transform is suggested that yields that set, so two lransformatjons from S to T and a to a
coordinates are needed:
n
ttyC2l ~ -§.dT- ada - PdV +
L, J.Lj tiN;
i=J
A Maxwell relation on the fU"st two tenns gives the desired result:
y~=-(;~1.fr. =-[~l ~.
!• N;
,
Ni
(5-115)
=f:
5.9 Relationships Between Partial Derivatives of Legendre Transforms
Although not obvious at this stage of our theoretical development, it is extremely
helpful to be able to express derivatives of Legendre transforms. For example, later in
Chapter 7 we will introduce criteria that wilJ determine the limits of phase stability for
single and multicomponent systems. Often the functional forms cf various PVTN
equations of state and other models of non-ideal behavior, such as activity coefficient
models, require that certain variables be used to facilitate calculation of the phase
stability criteria. This situation leads directly to the manipulation of partial derivatives
of Legendre transformations.
The presentation below follows earlier work by Beegle, Modell, and Reid (1974)
which has been updated by Kumar and Reid (1986) who applied Jacobian
transformations to calculate partial derivatives of Legendre transforms.
Single tJarUlbk transforms. Starting with Eq. (5-87), we wish to investigate the
relations between derivatives of yCOl and the first transform y(l>, where
y
(I)
~
(5-116)
= /(..., 1, ~·· ••, xm)
~~=
a
(~l
!X)
(5-117)
2'"""' XIII
There are several ways to obtain the desired results, but the most expeditious involves
the use of the derivative operators
(fori> 1)
(_£._1
o.l'; ,.,..... lx~ •...,x,..
and
(_l__l
a~.
lx, ,....
Y)!:J (aa
r
Ytt
and
oz·· ··· :em
(i-;t: 1)
x
"'
(5-118)
The Calculus of Thermodyntmics Chapter 5
156
?
1.. . ~ Y~~> (a:1. . ..
(a~
(5-119)
The terms ft~> and >{ are second-o rder derivatives:
0
co>,. a2 y<o>
(5-120)
Yti - dXI dXi
co>- a2fo>
a.J
Yu -
(5-121)
1
Equation s (5-118} and (5-119) are of little value to obtain first derivatives since, in
view of Eqs. (5-94) and (5-95).
all)l
(
x.
= ct) __
-Y, -
a~~
,....x.
(~1
=y~t) = y~O) = ,_.
dX·
'
1
l
(5-122)
(i > 1)
-,;
,.xl,..., lx;], .. •, x.
YW
to second-order derivatives of
Howeve r, to relate, for example ,
employ the operator equation (Eq. (S-119)] on y~ll. Thus for i# 1:
ayp)l
(d~l
- co - _!_ (ay~ul
"""Y1i - Y\of 'd.xl
I" • o
.(Ill
-y~~) [ay~o'l -Y~~>y\0(
I
-
-
a.%t
2'"" "I .x.
I" • ..
ycm, one can
(5-123)
.t,..
where Eq. ( 5-96) was used to simplify the third step. In a similar manner, other secondand third-order derivatives may be readily transformed. A list of these is presente d in
Table 5.3.
Example 5.5
Assume that the basis function is U
derivatives of this basis function.
=f (§, y, N1, .••, N ). Determine yW2 in teems of
11
Solution
0
Withfl) =f(T, V, N 1, ... ,N11) and -/ ) =/(§_, y, N 1, ..•, N11),
yt l) =A = U- TS
- -
-
a is the Helmholtz energy. From Table 5.3,
3
iPsA
a
- -A
yCI) -A
112 - - TTY- rrv- afl av - - arav
(5-124)
Section 5.9 Rela1ionstips Between Partial Derivatives of Legendre Transforms
157
(O)
YU2
- (y~~~2
_ Ussv
- u2 -
Usss Usv
ss
u3
ss
where the underbars have been omitted for simplicity. Now we can write
-
2
(J
s
arav
= -
(a Pl
2
ar
~N
=
n;asav- (riTirJ§?)v.JdTI'dYJs N
(fJTI~~ N
~
('dTifJ$l
(5-125)
~N
Table 5.3 Second- and Third-Order Derivadves ofy(l) in Terms ofy<O>
Derint:fve
ln _yCI)
Quantity
operated
upon:
~II
:Equadoo
Used
(5-119)
(S-119)
(S-118)
.111
Yu1
Yu
.lll
ylll
YII
. JI)
YuJ
•.II>
(5-ll9)
(t)
(S·Il9)
(S-119)
(5-118)
The Calculus ol Thermodynamics Chapter 5
158
Multipk Varillble Transforms. The development shown above was limited to the
case where only a single variable was transformed. Should one wish to transform more
than a single variable~ it is always possible to proceed a step at a time and transform each
separately as was shown above. It is also possible to develop a more general technique
to allow one to express the partial derivatives of a Legendre transform
-f)(~1 ,... , ~i' -Xj+1, ... , Xm~ tenns of the basis function y<o) or, in general, some other
Legendre transform y ~. where q = 1, 2, ...,j. The equations to obtain these secondorder derivatives are shown in Table 5.4 for the basis function and in Table 5.5 for other
functions. In Table 5.4 there are three cases for~: that is,j > i~ k; lc ~j > i; andj s; i. k.
Various cases are illustrated in Examples 5.6 and 5.7. Note that we can make use of
Maxwell's reciprocity relation -/N. = y<fj to simplify the calculations.
Table S.4 Relations Between Second-Order Derivatives of thejth
Legendre Transfonn and the Basis Function
D~O)
B
D
F
A
0
~-
=.1_ -
0
YW = :...___AD~O):....____
j > i. k
y~ =- - - - J_>< ,k.
D~O)
J
1
J-
D
F
n<o>
~
/J)_
ik-
D(O)
C1E G
"<"
. <k
1-
) _ l
D~O)
J
0.11: Kronecker delta, = 1 if i = j, = 0 if i ~ j.
A: j terms; each with a value of (-Ow·). where m = 1. 2, ... ,j.
B: j terms; each with a value of (~,4-)~ where m 1, 2, ... , j .
C: (j- l) terms; each with a value of (1- ~;)Y;. where m = 1, 2, ... , (j- 1).
D: (j- l) terms; each with a value of (1- SiJym! where m = 1, 2,... , (j- 1).
E: One term; (1 - 51
8j;·
F: One term; (1- Sfllk - 1"'
=
;).Yt-
G: (1- oi;X1- B,>y:>.
o
and
,co)11
,co>
12
(0)
YJi
,co)
21
D~O)=
J
(0)
Yjl
(0)
. Y··
}}
Section 5.9 Relationships Between Partial Derivatives of Legendre Transforms
159
Table S.S Relationships Between Second-Order Derivatives of thejth
Legendre Ttansform and the (j- q) Transfonn"
B'
n~> Ir'
Y&'
A'IA" H'
---:-:-~j > i, k
D(j-q)
=
q
n<r>
D'
p
A1 IA" f
j >i
Y(Jik~ = -~~-(j-q)
•
1Dq
<"
J"<.
_l
"<k
1and
U-q)
{j-q)
Y(i-q+l)(j-q+l) Y<f-qf-l)(i-qt2)
nN>
=
q
U-q)
· · · Y(j-q+I)j
.li-4)
(j-q)
Y~l)(;--q+l) >'(i-q+2)~2)
(J-q)
Yj(j-q+l)
=
a Zn 0 if r S s
:: 1 if r> s
A': (q-l)terms;each with avalueof[Z(j-q+l)iY~~-8cf.t,)i]wherep=(q-1),(q-2),... ,1
A": Z(}-q+-l)iY~-q)
.
B': (q- 1) ~s. each with a value of [Z(;-q+I)k y~~k- Su-p)k] where p = (q-1), (q-2), ... , 1
B'': Z(j-.q+l}JtYjtil>
.
H': ZU-q+I);Z{j-q+J)l;)'~)
.
D': (q- 1) ten:nsS each with a value of (1 - Ojk)Y~~ where p = (q-1), (q-2), ...~ 1
P: (1 - Oj.t)Y~-q - Ojft
f: (1- Ojk}Z(j-q+t)iY}k-q)
J': (q- 1) te~s. each with a value of (1- Oji)~t· where p =(q-1), (q-2)..... 1
K~ o- Oji)yJtti>- s-.
L': ( 1 - OJ;)(l -
Ojk)y~>
Example 5.6
Rerate the derivative yfR to second-order derivative ofy!0 >for four cases:
(a) j= 3. i = l, k:=::2
{b) j=3, i= l,k=3
The Calculus of Thermodynamics Chapter 5
160
=2, i =3, k =4
(c) j
(d) j = 3, i = 3, k ;::: 3
Solulion
Table 5.4 is employed. The results are:
(a) j=3, i= L,k=2
--S12
- 022
D(30)
- -- - -
y<GJ
12
- 032
3
--S,, --812 --813 0
y( ) - ------:-::-;---- - =
(b) j
y~
D~O)
12-
y<O>
13
yCO)
33
D{O)
3
=3, i = L. k =3
i
0
(l--S33)Y\
( 1--SJ3)y~~
<1-o33)Y ~~ (3) _ _ -_s_. _l
Y13 -
(c) j
((Jj
Y12
o33
_--S_2:....:.•_~_3_1~--o___ == _
D~O)
13
y~~ y~
D~O)
=2, i =3, k = 4
(1 - ~24)y~~
( 1 - ~2,)}'i~ -
riP>
(0)
2
(0)
(d) j =3, i = 3. k = 3
rf.O)
2
Y33 =- D(O)
(3)
3
Bxtlmple 5.7
Using Table 5.5, express:
Y'ti in terms of derivatives of yCll.
(b) ~~in tenns of derivatives ofy<2>.
y<O)
fO)
y~
y<O)
y(O)
'3
y<O>
y<O)
II
024
21
24
=
DiO)
y<O}
y<O)
(1 - ~23)y n (1 - 023)Y23 - 023 ( 1- On)(l - o )f~
(a)
y(O)
12
23
D(O)
2
14
24
34
Section 5.9 Relationships Between Partial Derivatives of Legendre Transfoons
161
Solution
(a) In this case j = 4, i =2, k =3, q = 3. We will use the case where j > i, k. Notice
that Z(;-q+t)i = ZC4-3+l)i = ~i· Since 2 =i, then Zu = 0. Also, Z(;-q+t)t: = 0.
Dt''=D~ )
1
where
(1)
Y22
nc•>3 -
(l)
Y23
y<i)
y<f] y~
y<l) y(O)
33
34
yO> y<i}
34
Then
-&23
-~33
DCO
y(4}_
0
--~n --032 0 0
23-
y~ y~
-
D<•>
3
y!}j
lll
3
D(ll
3
(b) Here
j>i
j<k
j = 5, i
=2, k =6, q =3,
yC2)
y<2)
33
~2)
34
yf2)
3S
y(S)-
26-
34
y<}l
y(2)
-45
y<2)
35
~2)
4S
y<fj
z(i-q+t); = ~z
y<l)
y(2)
y<2>
46
y(2)
56
y!l) yfl) y<2> y~
42
32
52
n<2>
3
yc;;
y<l>
33
yC2)
24
y(2)
34
iii
-).2>
35
yc;)
y(2)
45
Y46
y(2)
45
y(2)
y(2)
23
36
(2)
Y35
==-
=I
55
y(2)
26
yC2)
36
(2)
56
v<2>
3
Let us now consider derivatives in intensive variables only and show that they can be
reduced to expressions involving only S, T. P, V. In general, a partial derivative of
intensive variables may involve U, H. A, G, S, T, P, V. We have excluded N because it
is extensive;~ is also excluded because Jl is equal to Gfor a pure material
The first step in reducing the general derivative (CJb/()c)d is to eliminate any of the
four potential functions U, H. A. or G, which may appear as b, c. or d. The three
possibilities are treated as follows:
1. If b is U, H, A, or G, eliminate it by using the differential form of the transform. For
example, with Eq. (5-101), for a closed, non-reacting system,
162
The Calculus of Thermodynamics Chapter 5
s
(5-126)
2. If c is U, H. A, or G, invert the derivative using Eq. (5-30):
1
(~
= (db)dc)4
(5-127)
and then proceed by step 1.
3. If dis U, H, A. or G, bring the potential function into the brackets by using the triple
product relation of Eq. (5-32):
(abl
ac
= .... (ad/Clc)b
(5-128)
(od/ob)c
and then proceed by step 1.
Using these three steps, we can reduce any partial derivative involving only S, V. T,
and P to a second-order derivative c:i one ri tbe four fonns of rhe Fundamental Equation.
This is illustrated in Example 5.8 below.
EmmpkS.B
For a pure material there are 24 partial derivatives involving S, T, P. and V. Suggest a
techoique to relate any such derivative to second-order derivatives of the Gibbs energy
and, therefore, to CP' 1C7' and a, where
~,.-~(~:1
and
a,s~(~l
Solution
Of the 24 derivatives, 12 are simply inverses. The others are as foJlows: ( 1) (oS/ilT)p• (2)
cas;anv. (3) (iJSioP>']\ C4> cas;ap)\h C5) cas;aV)7\ (6) cas;aV)J?, (7) cavloP)p (8)
(dV/dP)s. (9) (aVIa1)P' (10) (oV/C)T)St (11) (aP/oT)s- (12) (dPidT)v. Number (3) is
equal to the negative of (9) from the Maxwell reciprocity theorem; similarly for (4) and
(10). Number (S) is equal to (12) and (6) to (11). Thus, there remajn (1) through (8). Three
of these are second-order derivatives of the Gibbs energy:
(1) (oSidT)p == -o.,~
c,1r
(3) (oSidP)r==-GrP==-Gn=-(oV/CJ1), =-VtxP
<7) <avldP)1 = GPI' =-Vte.,
In addition, derivatives (2), (5), and (lO)arerelated by theX-Y-Z.l rule, as are (4), (6), and
(8). [The sets (1)~ (3), (11) and (71 (9). (12) are also so related, but chese derivatives have
already been eliminated.] Thus, it is only necessary to relate (2), (41 and (6) to derivatives
a
Section 5.9 Relationships Betweert Partial Derivatives of Legendre Transf
arms
163
in G to allow all derivatives to be so expressed. The reduction of (2)
has been inlroduced
as Problem 5.12. For (4) and (6), we define om basis function G
=y(O) =f(T , P, N )
1
where thex~ values arc shown below. For (6}, we define y<t)::: H =!~.
P, N1, ...) and the
values are also shown below.
x--s
,,
y<ll
.X
~
1.
T
-s
2.
p
3.
N•
.X
;
J.
-s
-T
v
2.
p
v
Jl,, etc.
3.
Na
. _v== Y2(1)•
Thus, st.nce
(ava~)
-
'P,N
==
(as1
av
=
~·etc.
<-Yo>)-1
12
But, by Table 5.3.
and (dSidV)p =Ciet.PVT.
For derivative (4), using the same basis function. we need to use a doubl
e transform; thus,
y<2>= U -f~, ~. N 1....) with
y(lj
X
2..
-s
v
-P
3.
N,
J,1.1, etc.
1.
·T
and (iJ§.IdP)v,N:N(as/CJP)v:N(y~~rt. s1nceP =-y~l. From Table
5.4,
(0
YJl
-
(2)Y12 - .lOL(O ) _ 1,,(0)) 2YttY2 2
v t2
and
GTP
G G
TI' PP
_ G2
TP
The Calculus of Thennodynamics Chapter 5
164
Although the procedure described above is always valid and unequivocal, there are
shortcut methods such as those introduced earlier in Sections 5.3 and 5.4 which, when
applicable, are usually less tedious.
BXIlmple 5.9
Evaluate the following partial derivatives as functions of P. V, T, their partial derivataves,
and CP: {a) (aS;aP)0 ; (b) (dA/aG)T
Solution
(a) Using Eqs. (5-126) and (5-128). we find that
as! __ (oG/CJP)s
(()p _1- (oG/c)S)11
-S(CJTioP)s+ v
S(()T/dS)p
(5-129)
Using Eq. (5-128) to eliminate (CJTiaP)s,
(}TJ = _ (()SI()P)r = (dVId1)p
( iJP
(iJSioT)p
C,IT
(5-130)
asj =~-(av(
(()p} TS dT),
(5-131)
1
so that
The entropy, S, io Eq. (5-131) can be expressed as a function of the desired variables, as
demonstrated jn Section 8.1 [see Eq. (8.23)].
(b) This partial deTivativecan be reduced by the following shortcut procedure
aA I
(aa
caAiiJV)r
P
)r = caG!dV)r =- V(oPioV)r
in which Eq. (5-34) was employed. The insertion of V was a convenient but not an arbitrary
choice. Although P could have been used in place of Vwithequal simplicity, S would have
been less convenjenL The choice was guided by the fact that A =/(T, V) and G =f(Tl P);
since Twas the constant in the differentiation, either T, V or T. P would be a convenient
set of independent variables. As an alternative, Jacobian transforms could be used.
Section 5.10 SUmmary
165
5.10 Summary
Starting with the Fundamental Bquation U = f(S. V. N) as a basis function, this
chapter introduces a number ofmathematical techniques formanipulating variables. "''he
important methods discussed were:
l. Euler integration of the Fundamental Equation
2. Maxwe1rs reciprocity theorem for relating 2nd and higher order derivatives
3. Derivative inversion, triple product_ and chain rule expansion for manipulating
variables
4. Jacobian transformations for derivative manipulation
S. Legendre transformations 10 express the U-/<$... V. N) Fundamental Equation in
terms of different variables without losing any iof'onnation content
6. Modifications to the Fundamental Equation to account for non-PV work effects
7. Derivation of relationships involving partial derivatives of Legendre transforms
In addition. graptUcal techniques were used to illustrate the geometric relationships
of the USV surface and various Legendre transfomled functions.
References and Suggested Readings
Alberty, R.A. (1994), "Lecendre transforms in c:bemical thermodynamics." CMmical
Revi~s. 94(6). p 1457-1482 (Chemist's pcnpcctivcon Legendrelransforms]
Aria, R. and N.R. Amwtdson, (1973). Ma~Mmarical Mtlhods in Cht~mi&al Engineering,
Prentice HaJJ, Eng~wood Cliffs, NJ. Vol2, p 197-201. {Iattoduetion to Legendre
transformations}
Beegle. B.L., M. ModelL and R.C. Reid (l97.ot). "Legendre t.ransfonns and their application in
thermodynamics." .AICitE J. 20 (2). p 1194-1200.
Cafteu. H.B. (1985), Tlumnodynamics and an /IIU'odlltfion to TMrmo.rtatistict, 2nd cd,
Wiley. New York. [Legendre transfonnation.s)
Clark, A.L and L. Katz ( 1939). 'Thermodynamic surfaces of lb.O," TJWU Royal Soc.
CtJit/JdD. 3rd series, Section III. 33. p. 59-72 [Figure S.la taken from p 72].
Coy, D.C. (1993), "Visualizing lhennodynalnic stabilily and phase equilibrium through
computer graphics. PhD dissertation. Dept. of Otem. Bng., Iowa State Univ., Ames, Iowa
[3--0 images of variously ttansformtd and scaJed fundaDWIC.alsurface.s of pure. binary, and
ternary aystems]
DebenedeU.i, P.G. (1986), "Generalized Massieu-Planck functions.: Geometric representation
exttema and uni~tueness properties," J. Clumt. Plrys. 85(4), p 2131-2139.
Gyftopoulos, E.P. and O.P. BereUa (1991), ~s: F~ andAppliaUions,
MaeMillanf New Yort. {Mechanical engineering porspec:tive]
The Calculus of Thennodynamics Chapter 5
lolls. K.R. ( 1990). "Gibbs and the art of thermodynamics", Proceedings ofthe Gibbs
Symposium, published in G.D. Mostow and D.G. Caldi (eds.) Amer. Math. Soc. and
presented at Yale University, New Haven, CT(May 15-17, 1989) p 293-321. [computer
generated fundamental surfaces for a pure fluid]
Jolls, K.R., M.C. Schmitz, and D.C. Coy (1991)J "A new look. at an old subject." The
C~mical. Engineer, [nstitution of Engineers (UK), No. 497, May 30, p 42.
lolls, K.R. and D.C. Coy (1992), ..Gibbs's models visualized," Physics Today. March, p 96.
Kumar, S.K. and R.C. Reid. (1986), "Derivation of the relationships between partial
derivatives of Legendre transforms'', AIChE 1. 32 (7). p 1224-1226.
Tisza, L. (1966), Generalized Themwdynmnics, Mrr Press, Cambridge, MA, Ch. 2 in general
and specifically, p 61, 136, 236. [Transformation methods]
Problems
5.1.
(a)
Demonstrate the utility of Eq . (5-96) fer a system with six components and
where i 4. Order the Fundamental Equation as
=
U == [email protected]_, V, N 1, ... , N6)
(b)
If one were to write the Fundamental Equation as
~=f(U, V, N1, ... , N")
(c)
What would the derivative of the tota) Legendre transform be?
Suppose that one were write the Fundamental Equation as
yCO) = G =J(T, P~ N 1, •••~ N,)
(d)
Prepare a table showing the various conjugate coordinates, ~;. X;. Next, write the third
Legendre transform of the basis function yC0>shown above. Prepare a~~. x 1 table
for this transform. What generalization can you infer from this exercise?
Choose the basis function for the Fundamental Equation to be
/
0
) =A= f(J_,
N 1, ...• N11, T)
Obtain an expression for y1}d in terms of derivatives y(O}.
S.l.
Given y<O> =f(§., Y, N1N2••.• , Nn) obtain the Legendre transform and its differential if one
wished to work in the fonowing coordinate systems:
(a)
(b)
5.3.
f(T. y, ~~~· .., J.tJ
f[{l/1), V, N1, N1 , .. ., N,J
Express the following partial derivatives in an equivalent form using the Fundamental
Equation
Problems
167
and Maxwell's relations. Indicate, if possible, how they might be measured
experimentally.
(a)
ayC2ll
( av-
S'
SA.
(b)
(c)
(d)
(e)
(f)
S.6.
N
(b)
laQ;'
(c)
,N
l(~l
dV
soN
(d)
(a~
l
S
(1)
•
N
Carry out the following transformations.
(a)
s.s.
(.al&.l
Express (CJ~/()_!07, N,,.•., N11 as a function of P, ~. T. and their derivatives.
Express @H/dP)T, N...h. N as a function of G and its derivatives and show how these
may be given in tenns o(P, ~. T, N, and c,.
Express [()~/7)/0(1 /1)]y, N...... N,. as a function of U and its derivatives.
Express (dHiafh. N••..., N as a function of G and its derivatives and show how these
may be given in terms o(P, f, T, N, and CP.
Express ((}T/(}NA)¥... !· p,N" ... as a function of U and is independent derivatives.
Express (aTlaNA)~. P, "• Nco ... as a function of u and its independent derivatives.
For a one-component system, show:
(a)
v ~
-~JN av
()N .f-
(b)
v (Jp
--N -av
~~L=
The fa) lowing discussion is limited to one-dimensional motion along the x coordinate and
for a constant-mass system. Newtonian mechanics relates the force on a particle to the
mass and acceleration; that. is,
Another way in which to study the dynamics of motion is with Lagrangian mechanics. In
this case, a function Lis defined as L = g - fl), where ~ =the kinetic energy = ~l/2
and Cl> is the potential energy= j{x). Newton's Law in Lagrangian mechanics is given as
l (:t)(~tl
(~:
L=f(x.X.)
=
. dx
x= dt
.
=vel octty
Another branch of mechanics uses a function (-H). which is a function of x and the
momentum of a particle, p; that is,
-H=f(p,x)
with
The Calculus of Thermcxlynamics Chapter 5
168
- (aLl_(asl_
ax - a.x - mx·
P_
(a)
(b)
x.
Using the concepts of Legendre transfonns, define -H in terms of L. p, and
In the latter branch of mechanics. H is called the Hamiltonian; with your definition
of H complete the following equations:
(~~l =
5.7.
and
(~l =
Express the following in terms of CP, P, V, T and derivatives of these variables.
(a)
(b)
(dSiona. N
(()AiaGh,N
5.8.
Prove that for an n-component mixture there are (n + 2)(n + l)n independent
second-order derivatives of the Fundament.al Equation.
5.9.
A spherical tank contains l mol of heliwn gas at 10 bar and 300 K (see Figure P5.9). We
would like to carry out an experiment in which helium is released to the atmosphere, but
at the same time, the remaining contents of the sphere maintain a constant tot.al energy.
U. Heating or cooling coils may be used to keep U constant during venting. Helium
behaves as ao ideal gas with a constant value of Cv 12.6 J/mol K. Choose a base state
where the specific enthalpy is zero at 300 K.
=
(a)
When one-half of the helium has been vented, what is the temperature and pressure
of the residual helium? What is the heat interaction?
(b)
(c)
Detennine (oT/dP)Q.~ at the instant when venting begins.
Repeat parts (a) and (b) if the base state were chosen so that the specific internal
energy,
were zero at 300 K.
For part (a), what would be the residual helium temperature when 60% of the gas
had been removed?
(d)
u.
!·Heating
coils
Ot"
•Cooling
coils
as
clesired
Figure P5.9
Problems
169
5.10. Choose as a basis function
YW
in terms of derivatives of the basis function. Discuss how experiments
and obtain
could be designed and conducted to obuun numerical values of the / 0) derivatives.
Using your result, consider the foHowing problem. We have a system containing, initially,
N; motes of a material. We wish to remove N/2 moles under conditions where the total
entropy and the pressure remain constant. If the initial temperature is 400 K, what is the
final temperature? Asswne that the base state for entropy is such that at 400 K and the
system pressure, the specific entropy is 10 J/mol K.. Also assume that the heat capacity is
10 J/mol K. independent of temperature.
5.11. If the basis function is chosen as
show that
by performing two single-step transforms from / 2) to /I) and then to / 0 >.
5.12. Assume that the basis Legendretransfonn is/0>= f(P, T, N1, N2,1 ... ). Express y~in terms
of derivatives of y<O) and interpret tbe results on a physical basjs.
5.13. Express y£{ in terms of derivatives of /
/
0
}
1
)
for a ternary mixture when the basis function
=U-=- f(~. V, NA• N8 , Nc)
5.14. Express y~ in terms of derivatives of the basis function
y!O) = U =f(§_,
V, NA, ... )
5.15. Show that
5.16.
y<m- 2} is a Legendre transform of a basis function y<0), where
y(0) -j(
- x 1, x2, ... , xm)
/m-2) = f(~l' ~2'"'' ~m-2' Xm-1• xm)
The Calculus of Thennodynamics Chapter 5
170
Derive a general equation to express the derivative y~==~~m-l) in terms of derivatives of
a Legendre transform y<rl, where 0 ~ r< m -2. Define any derivatives and show what
variables are held constant.
5.17. Show that "==
S.
=[av] [()P] and check to see if the relation holds for an ideal gas.
c.., aPT av s
5.18. (a) Derive Eq. (5-118), for the derivative operator;
( l.,. .~J.~. l L-~)-~.
(b)
(c)
a-
a-
dXi
= ax;
(0) ( aYli
yft>
-
OxJ
=
L.~.
Show that y~1l y<f:l /y~o; fori~ 1 as given in Table 5.1.
Express the derivative (CJPidns.tA Jl. .-~ in lenns of derivatives of the basis function
(0)
-11
y =G.
Show that the result given in part (c) above reduces to
jl
(d)
(~~t :(~~t fcupure~rW
5.19. (a) Develop a suitable expression for ('iJTioP)ni(ClT/ClP)s in terms of PVI'
properties and their derivatives.
(b)
Starting wi1h the basis function:
y<O) =
rf =f<§., V, a, N1, N2
p .. ,
N;J
wheTe new conjugate coordinates {x;. ~}=(a, cr}have been introduced to account
for surface effects. a for area and a for surface tension, develop a suitable
expression for: (a~lda>r. v, N in terms of the appropriate 2nd derivative of a 1st
order Legendre transfonn and its derivatives and show that it can be equated to a
temperature derivarive of 0'.
5.20. The thermodynamics of rubber subjected to uniaxial tension is described by a fundamental
equation given in differential form with all extensive quantities reduced by mass as,
dU= TdS + V0 'td£
=
e (L- LJ!Ln == Unear tensile strain
"t
=FI A0 == uniaxial tensile stress
L == length in the direction of tension
F
=tensile force exerted on the rubber
a
L o> _o• Vo =length. area, and volume in the unstressed state.
Problems
(a)
(b)
(c)
171
Starting from this fundamental equation, express the stress t as a linear combination
of the isothermal derivatives of U and S with respect to strain.
For understanding rubber elasticity, it is
important to determine the relative
contributions of internal energy and entropy
to 't. To achieve this, we perform the
following experiment: Keeping the ends of a
stretched rubber band fix.ed, we measure the
stress 't as a function of temperature T as
indicated in the figure. Show how the internal
energy and enttopy contributions to stress can
be separately evaluated on the basis of these
T
measurements. Give a graphical interpretation
Figure PS.20
on the plot.
A 1935 experiment by Meyer and Ferri indicated that, for a typical rubber extended
to 4.5 times its original length at temperatures T> 210 K, the function depicted in
the figure is adequately deseribed by the equation t = TflOO~ with Tin Kandt in
MPa. What conclusion can you draw about entropic contributions to the elastic
response?
S.lt. Rocky and Rochelle Jones, while backpacking near the Presidential Range of the White
Mountains of New Hampshire, were trapped in a severe thunderstorm. Rocky was very
worried that they would be struck by lightning. As thunder and lightning bolts crashed
along the mountain ridges, he remembered that one ought to be able to estimate the
distance to a lightning strike by counting the seconds between when you first see the strike
and when you first hear the thunder. Rochelle recalled her 10.40 training and remembered
that the speed of sound could be related to certain thermodynamic properties. Deep jn her
photographic memory she recalled that:
v~ = (ClP/ap)swhere vc is the sound speed, Pis the pressure) p is the fluid density~ and Sis the entropy.
But neither Rochelle nor Rocky can quite figure out how to evaluate the isentropic
derivative (dP ;ap)s in terms of measurable properties. Please help them out! Rocky just
timed the last strike and it was only 2 seconds away. State and justify all assumptions in
your analysis.
N.B. Rocky's portable solid-state weather computer which fits into his backpack has
provided some va1uable meteorological data:
air temperature = 0°C (32op)
relative humidity = 80%
barometric pressure =0.99 bar and falling rapidly!
wind speed= 50 mph (80 kmph) from theSE
In addition. Rochelle had a copy of Perry•s Handbook with her which provided some
other {hopefully useful) thennophysical parameters for air.
The C8Jculus of Thetmodynamlcs Chapter 5
cP = 29.3 J/rnol K
molecular wt = 29 gfmol
Tc::: -140 .7°C
Pc= 37.2 bar
3
Pc = 1IVc = 350 kg/m
And at 25°C, l bar
p = 1.17 kg/m3
r
Joule-Thompson coef4ficient= a.H = (OT /aP)H =0.23°Ciba
viscosity :: 0.19 X l 0" Pa-s
Prandtl number = Pr :: 0. 70
2
thermal conductivity= 1= 2.6 x 10" W/m K
of CP and C.,:
5.22. (a) Stllrtin.g with the following ahernative definition
CP =T(()S/iJ1),. C.,= T(CJS!iJ1)v
(b)
erties only. Using lhe
show that CP- C'll can be expressed as a function of PVI prop
result of part (a) show that CP- C.,= R for an ideal gas.
ible to express the ratio
Given that c,- C.,= f(P. V, 1) from part (a). is it poss
erties and theirdmvatives? Explain
K = C,IC"' as an explicit function of PVT prop
your answer.
5.23. A basis function is defined as follows:
O=. f(T, V, N 1, N2, ~1)
With conjugate coordinate variable pairs given as
.XI
~
T
-s
v
-P
Jll
Nt
N2
fJ-2
Ft
~I
order in mass) and F 1 is a
where z1 is a generalized extensive displacement (first
other terms have their usual
generalized intensive force (zero order in mass) and the
mean.ing. Provide expressions for the following:
(a) dyco>
(b)
yeo>
(c)
dyCSl
(d)
( e)
[()Fl/dJ.L1 1r.t.N1'~ in tenns of a derivative of N1
.
(I )
(I)
1n terms o
Ysss
Ysss=
f
d
r..
~an
You may want to use Table 5.3.
.
.
. denvatiVes
1ts
Problems
5.14.
173
Ify<0l ~ U= /~. V, N1,... , N,J, are y<n+2) and dyCn+2) always equal to zero? Answer yes
or no and e:JC.plain briefly.
lf Y(l) = j[t11, P. T, N2, N3,•.. , Nn], what are yCO) and dy(0)?
What derived thermodynamic property does yC0>correspond to?
yjf)
Give one example of y~> =
for any lah Legendre transform ofy!0> ~ U and any two
independent variables z and J. Express your answer as an equality of partial derivatives
involving derived properties (U, ~. !:f., G, ~and/or primitive variables (T, P, V, N1).
5.25. A basis function for a modified Fundamental Equation that includes reversible
electromagnetic work effects in a two-component system is given as:
y!O'J = lf =/[§., V, N1, N2• B] -j(x 1• X2..... x 5)
where the conjugate pair of variables added for electromagnetic work effects is
{x5, ~s} == {y B, H}
and
H = magnetic field strength (amp/m)
V = tota1 volume subject to the magnetic field, includes the system volume
and any free space outside the boundaries of the system (m3)
B =magnetic induction (Volt·slm)
For all cases of interest, you can assume that the volumetric lll8gnetic permeability is
constanl
(a)
What are yes> and t!f>?
(b)
Derive an expression for (CJJ.L1/oBh:x. M.N2
At constant T, P, and Bcould you calculate J1l as a function of mole fraction of
component (xi) from knowledge of how JJ.1 varies with x1 ? If so. explain your
answer.
(c)
5.26. According to a recent anicle in the March. 1989 issue ofScientific American. harbor sea)s
are believed to have sophisticated acoustic transmission and detection systems that are
similar to sonar. For example, mother seals locate their pups by measuring the difference
in reflected acoustic wave arrival times in air and in water.
Given that the speed of sound (vc) can be related to an isentropic derivative of pressure
with respect to density:
v:=(dPJi)p)s
The Calculus of Thermodynamics Chapter 5
174
estimate about bow far a mother seal js from her pup if the time difference is 3 seconds.
Air can be assumed to behave as an ideal gas with CP = 5/2 R. The properties of liquid
water can be obtained from the steam tables. The air and water temperatures are about
2°C.
S.27. A non-ideal gas of constant heat capacity Cv = 12.56 J/mol K undergo a reversible
adumatic expansion. The gas is described by the van derWaals equation of state
(P + a/V2)(V- b)
=RT
where a= 0.1362 J m3/mol2 and b = 3.215 x 10-s m3/mol. Derive an expression
for the temperarure variation of the gas internal energy. and calculate its value when the
gas volume of 400 moles is 0.1 m3 and its temperature is 2.94 K.
S.28. The basic thermodynamic relationships for an axially stressed bar can be written as:
dQ,,.v = Td§., dWte\1 =- 'td§, ~ =N£
where 't is the stress and £ is the srmin. Derive the Fundamental Equation for a
one-component bar and show that
(~I~.N--{a~l
oT
(JN
.t
5.29. A very useful property is the partial molar volume of one component in a mixn.are. It is
defined fer an n-component case as
=
~ (dV/CJ~)T.PJJ, (J1
f
If we have a basis function 0l,
y<O> = f(§_. V, NA• N 8 ,... /'I,J
we can see that, if we desired V8 , then
-
(2)
VB =-y24
where
"J•
However, to evaluate
we normally only have property relations which express P as a
function of the independent variable set Vl T, NA• N8,.... N11
Use this variable set to define a new basis function y<O>, and express
~. T, NA• N 8 ,•••, N,. using the foil owing equation of state.
P=
NRT
V- NAbA -NfiJ8
-NdJc·-· -N.,./JN
f/l in zenns of
•
Prablams
17S
where bA• b8...., bNare conatants and N =VIP with i =a. b,... .N
S.30. The extensive variabJ~ that characterizes magnetic behavior is the magnetic dipole
momentlln this situation, the fundamental equation is rewritten as
U= V(§_, V,[,N)
with the added conjugate variable set £r;1, ~~a[!, H) where H is the magnetic field
strength. For a pure component system ck.vclop exptessions for;
U
(a)
(b)
f-4> and d:,'..,
(e)
(~/aB>gt, N
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