Chapter 4: General Relationships between State Variables

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CHAPTER 4
GENERAL RELATIONSHIPS BETWEEN STATE VARIABLES
OF HOMGENEOUS SUBSTANCES
INTRODUCTION
The first law of thermodynamics relates the differential change in internal energy .I of a system
to the differential amount of energy crossing the system boundary in the form of heat and work according to the
equation
.I œ $ U $ [
(4.1)
The internal energy I is a state variable (for an ideal gas it is related only to the temperature of the system),
whereas $ U and $ [ depend upon the specific process. We have discovered that for reversible processes, the
process-dependent heat and work can both be written in terms of changes in state variables. During quasi-static
(reversible) processes, we can write
$ U</@ œ X .W
and
$ [</@ œ T .Z
(4.2)
This allows us to write the first law for reversible processes solely in terms of state variables
.I œ X .W T .Z
(4.3)
The last equation is usually refered to as the combined first and second law equation, and serves as the starting
point for discussing any problem in thermodynamics.
The ability to write the first law in terms only of state variables allows us to calculate many useful
relationships between the thermodynamic properties of a system. This is because the first law, written in terms of
quasi-static processes, is actually applicable in general, since each of the variables in this equation are state
variables. This means that relationships which we can derive from this equation hold true for any process. This
is because the system will be in some unique state before some arbitrary process, and at the end of that process, it
will again be in some unique state. The actual process the system undergoes may not be quasi-static, but the end
points of the process are on the T @X surface of the =C=>/7 - and that's all that matters!
To illustrate this last point, consider the combined first and second law, where we consider Z and X as
independent variables. The changes in internal energy, then, can be expressed in terms of Z and X by the
equation
$ U œ .I T .Z œ Œ
`I
`I
.Z Œ
.X T .Z
`Z X
`X Z
`I
`I
$ U œ ”Œ
T •.Z Œ
.X
`Z X
`X Z
(4.4)
Now, if the volume were to remain constant during some process, the first term on the right-hand side is zero, and
we are left with
$UZ œ Œ
`I
.XZ
`X Z
(4.5)
Now the heat capacity at constant volume is defined to be
GZ ´ Œ
$U
`I
œŒ
.X Z
`X Z
(4.6)
This relationship is valid, not only for constant volume processes, but in general; i.e., we can always replace the
quantity `I /`X Z with GZ no matter what kind of process is being considered. This is because the internal
energy I is a function of the state of the system, not the process.
Chapter 4: General Relationships between State Variables
2
In the special case where we are dealing with an ideal gasß the internal energy of this gas is solely a
function of the temperature. This last equation, then, becomes
GZ œ 8 -Z œ
.I
.X
(4.7)
which means that we can calculate the change in the internal energy of an ideal gas using the equation
?IEF œ (
F
8 [email protected] .X œ 8 [email protected] ?XEF
(4.8)
E
since the heat capacity for the ideal gas is a constant. But the process E p F can be any process, and is not
restricted to a constant volume process!
Note: Although this equation is process independent, the value of [email protected] does depend upon the kind
of ideal gas we have (monatomic, diatomic, etc.). For polyatomic gases each type of motion
(vibration or rotation) has a “turn-on” temperature arising from quantum considerations. This
last equation, then, is valid over the whole range of temperatures only for monatomic ideal
gases. For polyatomic ideal gases it is valid only over limited temperature regions where [email protected] can
be considered constant.
THERMODYNAMIC POTENTIALS AND THE MAXWELL RELATIONSHIPS
The combined first and second law
$U œ X .W œ .I T .Z
(4.9)
provide some very useful relationships between the various system parameters of pure a substance. This equation
contains five system properties (state variables), but, as pointed out in the first chapter, only two of these are
required to completely specify the macroscopic state of the system. This means that we can choose any two of the
system parameters (state variables) as independent, and the other three must be functions of these two parametersÞ
This will allow us to derive a large number of relationships between the various system parameters. Some of
these relationships will prove to be particularly important in the development of statistical thermodynamics. In
addition, we can form combinations of some of these system parameters to define new system parameters which
are often quite useful.
In the material that follows we will be developing relationships between different system parameters.
These will be organized by examining how the change in a particular Thermodynamic potential varies with other
parameters of the system. The first of these potentials is what we have already defined as the internal energy of
the system.
Changes in The Internal Energy
Beginning with our statement of the combined first and second law, the change in the internal energy of a
system can be expressed directly in terms of a change in the entropy of the system and the change in the volume
of that system as
.I œ X .W T .Z
(4.10)
This equation is very useful when we are dealing with a constant volume process. In that case we can equate the
cnange in the internal energy directly to the change in entropy of the system.
Since the right-hand side of this last equation is written in terms of a change in entropy and a change in
volume, we will choose these two system parameters are our independent variables and write
.I œ Œ
`I
`I
.W Œ
.Z œ X .W T .Z
`W Z
`Z W
(4.11)
Chapter 4: General Relationships between State Variables
3
from which we can immediately see that
Œ
`I
œX
`W Z
(4.12)
`I
œ T
`Z W
(4.13)
and
Œ
These last two relationships are significant because they give us new insight into the meaning of the
temperature and the pressure of a gas. We see that the pressure of a gas is just a measure of the rate at which the
internal energy of an isolated system (constant entropy implies adiabatic processes) changes with a change in
volume. Obviously the internal energy decreases as the volume increases. A high pressure system is one in which
the internal energy decreases very rapidly with a volume change, whereas a low pressure system is one in which
the internal energy varies slowly with volume.
Likewise, the temperature is a measure of the rate at which the internal energy changes with entropy at
constant volume. We can also express this equation for the temperature, using the reciprocity theorem, in the
form
Œ
`W
"
œ
`I Z
X
(4.14)
This equation demonstrates that the reciprocal of the temperature is a measure of the rate at which the entropy of a
system changes with respect to internal energy when the volume is held constant. This particular equation will
turn out to be very significant in our study of statistical thermodynamics. We will see later on, in our study of
statistical thermodynamics, that it is fairly easy to define the entropy of a system relative to the internal energy
based upon a statistical model of the thermodynamic system. The definition of temperature, however, is not quite
so straight forward. This last relationship will allow us to define the temperature in a precise way when we work
with our statistical models.
The partial derivatives of the internal energy which we found in Equ. 4.12 and 4.13 can be utilized to
find yet another relationship between the system parameters by remembering that the second partials of exact
differentials must be the same, no matter what the order of differentiation. Thus
Œ
`
`I
`
`I
Œ
œŒ
Œ
`Z W `W Z
`W Z `Z W
(4.15)
which gives
Œ
`X
`T
œ Œ
`Z W
`W Z
(4.16)
This last equation is one of the so-called “Maxwell relations.”
Changes in the Enthalpy
We indicated that the combined first law in the form
.I œ X .W T .Z
(4.17)
turned out to give a very simple relationship between .I and .W for the case of constant volume processes.
However, this form of the equation is not as useful when we want to consider internal energy changes for constant
pressure processes, or if we simply want to calculate the change in internal energy for processes in which the
pressure is varying. We can, however, express this last equation in terms of a change in pressure as well an
entropy. The differential of the quantity T Z can be written as
. T Z œ T .Z Z .T Ê T .Z œ . T Z Z .T
(4.18)
Chapter 4: General Relationships between State Variables
4
enabling us to rewrite the combined first and second law in the form
Rearranging this last equation, we have
.I œ X .W c. T Z Z .T d
(4.19)
. I T Z œ X .W Z .T
(4.20)
We now define the combination of system parameters I T Z to be a new system parameter L which we call the
enthalpy. The mentioned earlier, enthalpy is one of a group of combinations of thermodynamic parameters called
thermodynamic potentials (note that they all have the dimensions of energy). These thermodynamic potentials are
often useful in special situations or for special processes. For example, writing the combined first and second law
in terms of the enthalpy
.L œ X .W Z .T
(4.21)
we see that the change in enthralpy for a constant pressure process (as is often the case in the laboratory) is just
the heat ($ U œ X .W ) added to, or taken from, the system. So just as $ U œ .I for constant volume processes,
$U œ .L for constant pressure processes. Similarly, for a constant pressure process, we have
.LT œ X .WT œ $ UT
from which we can see that
GT ´ Œ
$U
`L
œŒ
.X T
`X T
In most engineering applications, changes in enthalpy are considered, rather than changes in internal energy,
because the enthalpy is related to the heat added to a system at constant pressure, whereas the internal energy is
related to the heat added to a system at constant volume - and working in the laboratory, we typically work under
constant pressure conditions.
Obviously the enthalpy of a system is a state variable since I , T , and Z are state variables, so we can
write
.L œ Œ
`L
`L
.W Œ
.T œ X .W Z .T
`W T
`T W
(4.22)
which allows us to identify the partial derivatives of the enthalpy in terms of measurable system parameters,
giving
Œ
`L
œX
`W T
(4.23)
Œ
`L
œZ
`T W
(4.24)
Again, we can use the fact that the second partial of L is the same, no matter what the order of differentiation,
and obtain another Maxwell relation
Œ
`X
`Z
œŒ
`T W
`W T
(4.25)
Changes in the Helmholtz Free Energy
If we wish to write the combined first law
.I œ X .W T .Z
(4.26)
in a form which is more suitable to changes in temperature and volume than changes in entropy and volume, we
can write
.I œ . X W W .X T .Z
(4.27)
Chapter 4: General Relationships between State Variables
or
. I X W œ W .X T .Z
5
(4.28)
We define a new thermodynamic potential J œ I X W called the Helmholtz free energy to give
.J œ W .X T .Z
(4.29)
from which we obtain
Œ
`J
`J
.X Œ
.Z œ W .X T .Z
`X Z
`Z X
(4.30)
and the relationships
Œ
`J
œ W
`X Z
(4.31)
Œ
`J
œ T
`Z X
(4.32)
Again you should note that .J is particularly useful when working at constant temperature, in which case
.JX œ T .Z
(4.33)
which is the amount of external work done on the system.
We can again use the fact that the Helmholtz free energy is an exact differential to obtain the Maxwell
relationship
Œ
`W
`T
œŒ
`Z X
`X Z
(4.34)
Changes in the Gibbs Free Energy
In a similar manner we define the Gibbs free energy as
K œ I XW TZ œ L XW
(4.35)
which gives
.K œ W.X Z .T
from which we obtain the relationships
Œ
`K
œ W
`X T
Œ
`K
œZ
`T X
(4.36)
(4.37)
and the final Maxwell relation
Œ
`W
`Z
œŒ
`T X
`X T
(4.38)
The Gibbs free energy is often useful when we are dealing with systems with chemically active
constituents. A differential change in the Gibbs free energy can be written as
.K œ .I X .W W.X T .Z Z .T œ .I ÐX .W T .Z Ñ W.X Z .T
(4.39)
For systems with chemically active constituents, the combined first and second law is given by
.I œ X .W T .Z ..R
(4.40)
Chapter 4: General Relationships between State Variables
6
where . is the chemical potential. The Gibbs free energy can, therefore, be expressed by the equation
.K œ . .R W .X Z .T
(4.41)
Thus, for constant pressure and temperature processes, the chemical potential is just equal to the change in the
Gibbs free energy with respect to the change in the number of particles in the system!
The Potential Memory Box (or Hints for Remembering Various Useful Relationships)
The relationships between the system parameters Z , X T , and W , and the Thermodynamic Potentials I ,
J , K, and Lß as well as the Maxwell relations turn out to be very useful relationships between the various system
parameters. One way of remembering these relationships is illustrated in the use of the “Memory Box” shown
below. This “Memory Box” can be remembered with the acronmy "Very Tall Physics Students Eat For Good
Health" Þ
J
Z
I
X
ÐÑ
K
W
L
T
ÐÑ
The bold symbols Iß J ß Kß L , represent the Thermodynamic potentials, and are ordered alphabetically
clockwise around our square “Memory Box”. The more conventional system parameters are positioned on the
four corners of the box. To determine an expression for the temperature X , for example, you take the partial of
the potential on the opposite side of the box with respect to the variable which is diagonally across from the
temperature, holding the variable on the opposite side of the potential function constant. For example:
X œŒ
`L
`I
œŒ
`W T
`W Z
The sign on the right indicates that the partials from which you obtain the temperature and the volume are
multiplied by a plus sign, whereas the partials from which you obtain the pressure and entropy require a minus
sign:
T œ Œ
`J
`I
œ Œ
`Z X
`Z W
Remembering that the order of differentiation is unimportant when taking the second partials, we can use
these relationships to quickly derive the Maxwell equations. For example, we have just shown that
Œ
`I
œ T
`Z W
and
Œ
`I
œX
`W Z
Taking the partial of the first of these equations with respect to W (holding Z constant) and the partial of the
second of these equations with respect to Z (holding W constant) we obtain the Maxwell relationship
Œ
`T
`X
œŒ
`W Z
`Z W
Now, this last equation can also be obtained rather quickly simply by using our “Memory Box”. Notice
that the system parameters T , W , and Z in the partial derivative on the left side of this equation are all on adjacent
corners of our “Memory Box”, in the same order as indicated in that partial derivative, `T /`W Z . If you go
one more step around the box in the same direction, you come to the system parameter X , and if you now reverse
Chapter 4: General Relationships between State Variables
7
directions, you have all the parameters (including X ) found in the partial derivative on the right side of the
equation in the same order as indicated in that partial derivative, `X /`Z W . Now, since your starting point for
the partial on the left is on the bottom of the box, while the starting point of the partial on the right is on the top of
the box, you must introduce a negative sign. Whereas, if your starting point for the partials on each side of the
equation were both on the top, or both on the bottom of our box, we do not have to introduce a negative sign.
This process will work, starting at any corner of the “Memory Box” and going in either direction to give all the
four of the Maxwell relations which are summarized below:
Œ
`X
`Z W
`X
Œ
`T W
`W
Œ
`Z X
`W
Œ
`T X
œ Œ
`T
`W Z
`Z
œŒ
`W T
`T
œŒ
`X Z
`Z
œ Œ
`X T
CALCULATING GENERAL PROPERTIES FOR HOMOGENEOUS SUBSTANCES
As we mentioned earlier, the combined first and second law, in its original form
X .W œ .I T .Z
contains five system properties, only two of which are independent. Typcially, we find it easier to measure things
like pressure, temperature and volume, or the gradients of these quantities than to measure the internal energy, or
entropy directly. Thus, we often write this equation and its various parameters as functions of these more easily
measured quantities. We now wish to examine some of the relationships that we can derive by considering two
special cases: 1) where X and Z are the independent variables, and 2) where X and T are in the independent
variables.
Independent Variables X ß Z . The first case we wish to consider is one in which assume that the
independent parameters for our system are the temperature X and the volume Z . This means that we can express
the entropy W and the internal energy I as functions of temperature and volume, i.e., W œ W X ß Z and
I œ I X ß Z , and we can also express the pressure as a function of temperature and volume, T œ T X ß Z . The
change in any one of these dependent parameters can be expressed in a form similar to the equation
.W œ Œ
`W
`W
.X Œ
.Z
`X Z
`Z X
(4.42)
Thus, the combined first and second law can be written
X ”Œ
`W
`W
`I
`I
.X Œ
.Z • œ ”Œ
.X Œ
.Z • T .Z
`X Z
`Z X
`X Z
`Z X
`W
`W
`I
`I
X ”Œ
.X Œ
.Z • œ Œ
.X ”Œ
T •.Z
`X Z
`Z X
`X Z
`Z X
(4.43)
from which we obtain
Œ
`W
" `I
œ Œ
`X Z
X `X Z
`W
"
`I
Œ
œ ”Œ
T•
`Z X
X
`Z X
(4.44)
Chapter 4: General Relationships between State Variables
8
The heat $U added to a system during a reversible process can be written
$U œ X .W œ X ”Œ
`W
`W
.X Œ
.Z •
`X Z
`Z X
(4.45)
If this process is a constant volume process, then .Z œ ! and we have
$UZ œ X .WZ œ X Œ
`W
.X
`X Z
(4.46)
We define the heat capacity at constant volume as the heat added to the system at constant volume divided by the
temperature change, or
GZ ´
$ UZ
`W
`I
œ XŒ
œŒ
.X
`X Z
`X Z
(4.47)
where we have used the relationship between the partials derived above. As mentioned earlier in the chapter, this
equation relates partial derivatives which are independent of any given processes to a process dependent variable,
GZ , which is a measurable quantity. The partials with respect to temperature, holding volume constant are,
therefore,
Œ
`I
œ GZ
`X Z
(4.48)
Œ
`W
GZ
œ
`X Z
X
(4.49)
and
Now look back at the relationships between the partial derivatives of the energy and the entropy. Since
the entropy is a state variable, the second partial of the entropy will have the same value no matter what order the
partials are taken! (This last statement is true for any state variable.) This means that
Œ
Œ
`
`W
`
`W
Œ
œŒ
Œ
`Z X `X Z
`X Z `Z X
(4.50)
`
" `I
`
"
`I
” Œ
•œŒ
œ ”Œ
T •
`Z X X `X Z
`X Z X
`Z X
(4.51)
" `
`I
"
`I
"
`
`I
`T
Œ
Œ
œ # ”Œ
T • ”Œ
Œ
Œ
•
X `Z X `X Z
X
`Z X
X
`X Z `Z X
`X Z
(4.52)
But the internal energy is also a state variable, so that the second partial of the energy will have the same value no
matter what order the partials are taken. This means that this last equation simplifies to give
"
`I
`T
”Œ
T• œ Œ
X
`Z X
`X Z
(4.53)
or
Œ
`I
`T
œ XŒ
T
`Z X
`X Z
(4.54)
(Notice that the quantity on the left hand-side of Equ. 4.53 is just equal to the ` S/` VX . We will come back to
this later.)
The relationship in Equ. 4.54 tells us how the internal energy of the system changes with respect to
volume if the temperature is held constant. You may be tempted to say that this partial has to be zero, and this is
Chapter 4: General Relationships between State Variables
9
indeed true if the system is an ideal gas, where T Z œ 8VX . However, the equation which we derived from the
combined first and second law is valid for any system no matter what the equation of state! This means that a
measurement of the change in pressure with respect to temperature in a constant volume process can be used,
along with the measurement of the temperature and pressure of the system to determine how the internal energy
changes with volume.
You will remember from the first chapter that there are two typical quantities that are measured for many
systems - the coefficient of isothermal compressibility 5 and the coefficient of thermal expansion " . We pointed
out in that chapter that we can often rewrite the partials in terms of other partials using the cyclic and reciprocity
relationships. Using these in this case, we find that
Œ
`T
`X
`Z
Œ
Œ
œ "
`X Z `Z T `T X
(4.55)
"
`T
`Z /`X T
œ
œ
,
`X Z
`Z /`T X
(4.56)
or
Œ
so that our expression for the change in internal energy with respect to volume can be expressed in terms of
experimentally measurable quantities as
Œ
`I
X"
T
œ
`Z X
,
(4.57)
Using this relationship for the change in internal energy with respect to volume, we can also obtain an expression
of the change in entropy with respect to volume from the equation
Œ
`W
"
`I
œ ”Œ
T•
`Z X
X
`Z X
(4.58)
`W
"
X"
T T • œ " Î,
œ ”œ
`Z X
X
,
(4.59)
which gives
Œ
We have been able to express the partial derivatives of W , I with respect to Z and X in terms of the
experimentally measurable quantities GZ , ,, " , T , and X . This means that we can write an expression for the
heat added to the system,
$ U œ X .W œ X Œ
`W
`W
X"
.Z
.X X Œ
.Z œ GZ .X ,
`X Z
`Z X
(4.60)
the entropy change in the system,
.W œ Œ
`W
`W
GZ
"
.X .Z
.X Œ
.Z œ
`X Z
`Z X
X
,
(4.61)
and the change in internal energy of the system
.I œ Œ
`I
`I
X"
T •.Z
.X Œ
.Z œ GZ .X ”
`X Z
`Z X
,
(4.62)
in terms of experimentally measurable quantities.
Evaluation of Partials for Real and Ideal Gases. For an ideal gas " œ "ÎX and , œ "ÎT , and we
obtain the expected result
Œ
`I
œ!
`Z X
ideal gas
(4.63)
Chapter 4: General Relationships between State Variables
10
that the internal energy of an ideal gas is independent of the volume. On the other hand, the partial derivative of
entropy with volume for an ideal gas is expressed as
Œ
`W
T
8V
V
œ
œ
œ
`Z X
X
Z
@
ideal gas
(4.64)
Notice that the change in entropy with volume (the slope of the entropy-volume curve) depends only upon the
specific volume of an ideal gas, and decreases as the volume becomes larger, going to zero as Z p ∞. Also
notice that this partial is an intensive quantity, independent of the size of the system. Since the volume of the gas
is an extensive variable, the entropy must also be an extensive variable - depending upon the size of the system.
For a real gas, however, where there is an interaction between the molecules, we expect that the internal
energy will be a function of the pressure and volume. As an example of a real gas, let's consider a gas which
obeys the Van der Waal equation. We found an expression for the coefficient of thermal expansion, " , in the first
chapter for a Van der Waal's gas expressed in terms of the specific volume
" `@
V @# @ , Œ
œ
@ `X T
V X @$ # + @ , #
(4.65)
" `Z
VZ # Z 8, Œ
œ
V X Z $ # 8+ Z 8, # ‘
Z `X T
(4.66)
"œ
In terms of the absolute volume, this is
"œ
The isothermal compressibility , can be expressed as
" `@
" `Z
"
"
,œ Œ
œ Œ
œ
@ `T X
Z `T X
Z `T Î`Z X
(4.67)
For a Van der Waal's gas, the pressure can be expressed as
T œ
VX
+
8VX
+8#
# œ
#
@ , @
Z 8, Z
(4.68)
The partial is easily evaluated to give
Œ
`T
8VX
#+8#
#+8# Z 8, # 8VX Z $
œ
œ
# `Z X
Z$
Z 8, Z $ Z 8, #
(4.69)
from which we can derive the isothermal compressibility
5œ
"
Z $ Z 8, #
Z # Z 8, #
œ
Z #+8# Z 8, # 8VX Z $
8VX Z $ #+8# Z 8, #
(4.70)
We can now determine how the internal energy of a Van der Waal gas varies with volume
Œ
`I
X"
T
œ
`Z X
,
`I
"
Œ
œ X †"† T
`Z X
,
(4.71)
`I
VZ # Z 8, 8VX Z $ #+8# Z 8, #
8VX
+8#
œ
X
†
†
”
#•
Œ
#
#
$
#
V X Z # 8+ Z 8, ‘
`Z X
Z
Z 8, Z Z 8, #
#
8VX
+8
+8
+
`I
8VX
#•œ # œ #
”
Œ
œ
Z 8, Z
Z
@
`Z X
Z 8, Thus, for a “real” gas the internal energy does depend upon the volume of the gas. Remember that the constant +
in the Van der Waal gas is related to the intermolecular forces acting on the particles. If this intermolecular force
is zero, then the internal energy is not a function of the volume of the gas. But even for a real gas where there is
Chapter 4: General Relationships between State Variables
11
an intermolecular attraction, the dependence of the internal energy on the volume gets smaller as the specific
volume of the gas becomes larger, i.e., as the average distance between the molecules becomes larger. Thus, for
large specific volumes, the Van der Waal gas approaches an ideal gas.
Similarly, the partial of the entropy with respect to volume can also be evaluated for a Van der Waal gas
from the relation
Œ
`W
"
`I
" +8#
8VX
+8#
8V
V
#•œ
œ
œ ”Œ
T• œ ” # `Z X
X
`Z X
X Z
Z 8, Z
Z 8, @,
(4.72)
Notice that the entropy does not depend upon the intermolecular forces. The change in entropy with volume is
only a function of the specific volume accessible to the gas.
Independent Variables X ß T . Choosing X and T as independent variables, we write the combined
first and second law in the form:
X ”Œ
`W
`W
`I
`I
`Z
`Z
.X Œ
.T • œ Œ
.X Œ
.T T ”Œ
.X Œ
.T • (4.73)
`X T
`T X
`X T
`T X
`X T
`T X
`W
`W
`I
`Z
`I
`Z
X ”Œ
.X Œ
.T • œ ”Œ
TŒ
•.X ”Œ
TŒ
•. T
`X T
`T X
`X T
`X T
`T X
`T X
or
Œ
`W
"
`I
`Z
œ ”Œ
TŒ
•
`X T
X
`X T
`X T
`W
"
`I
`Z
Œ
œ ”Œ
TŒ
•
`T X
X
`T X
`T X
(4.74)
Just as in the previous case, the second partials of the entropy and the internal energy are equal, no
matter what the order of the partials. Taking the partial of the first equation above with respect to pressure and
the second equation with respect to temperature, we have
Œ
`
`W
"
`
`I
`Z
`
`Z
Œ
œ ”Œ
Œ
Œ
TŒ
Œ
•
`T X `X T
X
`T X `X T
`X T
`T X `X T
(4.75)
and
Œ
`
`W
"
`I
`Z
Œ
œ # ”Œ
TŒ
•
`X T `T X
X
`T X
`T X
"
`
`I
`
`Z
”Œ
Œ
TŒ
Œ
•
X
`X T `T X
`X T `T X
(4.76)
If we subtract the second equation from the first, we obtain
!œ
"
`I
`Z
" `Z
”Œ
TŒ
• Œ
X#
`T X
`T X
X `X T
(4.77)
Solving for the change in internal energy with respect to pressure (holding X constant), we obtain
Œ
`I
`Z
`Z
œ X Œ
TŒ
`T X
`X T
`T X
(4.78)
And this can be written in terms of , and " to give
Œ
`I
œ X " Z T ,Z œ T ,Z X " Z
`T X
This equation tells us how the internal energy varies with pressure.
(4.79)
Chapter 4: General Relationships between State Variables
12
An alternate way of determining `I /`T X is to realize that when the temperature is held constant, we
can choose the independent variables to be the volume and the pressure, and a change in one of these will
necessarily imply a change in the other. Thus, from Equ. 4Þ57
Œ
`I
`I
`Z
`I
œŒ
Œ
œŒ
† ,Z `T X
`Z X `T X
`Z X
X"
œŒ
T † ,Z œ X " Z T ,Z
,
(4.80)
The partial of the entropy with respect to the pressure can now be evaluated, using this last relationship to
give
Œ
`W
"
`I
`Z
"
`I
œ ”Œ
TŒ
• œ ”Œ
T , Z • œ " Z
`T X
X
`T X
`T X
X
`T X
(4.81)
The heat $U added to a system during a reversible process can be written
$U œ X .W œ X ”Œ
`W
`W
.X Œ
.T •
`X T
`T X
(4.82)
If this process is a constant pressure process, then .T œ ! and we have
$UT œ X .WT œ X Œ
`W
.X
`X T
(4.83)
We define the heat capacity at constant pressure as the heat added to the system at constant pressure divided by
the temperature change, or
GT ´
$ UT
`W
`I
`Z
œ XŒ
œ ”Œ
TŒ
•
.X
`X T
`X T
`X T
(4.84)
where we have used the relationship between the partials derived above. As mentioned earlier in the chapter, this
equation relates partial derivatives which are independent of any given processes to a process dependent variable,
GT , which is a measurable quantity. The partials with respect to temperature, holding volume constant are,
therefore,
Œ
Œ
`W
GT
œ
`X T
X
(4.85)
`I
`Z
œ GT T Œ
œ GT T " Z
`X T
`X T
(4.86)
Again we have been able to express the partial derivatives of W , I with respect to T and X in terms of
the experimentally measurable quantities GT , ,, " , T , Z and X . This means that we can write an expression for
the heat added to the system,
$ U œ X .W œ X Œ
`W
`W
.X X Œ
.T œ GT .X X " Z .T
`X T
`T X
(4.87)
the entropy change in the system,
.W œ Œ
`W
`W
GT
.X " Z .T
.X Œ
.T œ
`X T
`T X
X
and the change in internal energy of the system
(4.88)
Chapter 4: General Relationships between State Variables
.I œ Œ
13
`I
`I
.X Œ
.T œ cGT T " Z d .X cT ,Z X " Z d .T
`X T
`T X
(4.89)
in terms of experimentally measurable quantities.
Evaluation of Partials for Real and Ideal Gases. For an ideal gas " œ "ÎX and , œ "ÎT , whicn
gives the expected result
Œ
`I
œ!
`T X
ideal gas
(4.90)
The change in entropy with respect to pressure for an ideal gas is given by
Œ
`W
Z
8V
œ " Z œ œ `T X
X
T
ideal gas
(4.91)
Notice that this partial is an extensive quantity, since the pressure is an intensive variable and the entropy is an
extensive variable.
To see how the internal energy of a real gas varies with pressure, we again use the relationships " and ,
derived for the Van der Wall gas. To obtain the desired result, let's use Equ. 4Þ80
`I
`I
+8#
Z # Z 8, #
†Z
œŒ
† ,Z œ # † `T X
`Z X
Z
8VX Z $ #+8# Z 8, #
(4.92)
`I
+8# Z Z 8, #
œ
`T X
8VX Z $ #+8# Z 8, #
(4.93)
`W
VZ # Z 8, VZ $ Z 8, †
Z
œ
œ " Z œ V X Z $ # 8+ Z 8, # ‘
V X Z $ # 8+ Z 8, # ‘
`T X
(4.94)
Œ
Œ
and
Œ
for the Van der Waal gasÞ
The Difference in Heat Capacities for Homogeneous Substances. In this section we will start again
with the combined first and second law written in terms of the independent variables Z and X :
$U œ .I T .Z œ Œ
`I
`I
.X Œ
.Z T .Z
`X Z
`Z X
(4.95)
We have identified the first partial as the heat capacity at constant volume, GZ , so we can write this last equation
as
$U œ GZ .X ”Œ
`I
T •.Z
`Z X
(4.96)
We have already found an expression for `IÎ`Z X , which gives us
Œ
`I
X"
T
œ
`Z X
,
(4.97)
so we obtain
$ U œ GZ .X X"
.Z
,
(4.98)
To find an expression for the heat capacity at constant pressure, we will write the change in volume in terms of a
change in temperature and a change in pressure to obtion
Chapter 4: General Relationships between State Variables
14
X"
`Z
`Z
”Œ
.X Œ
.T •
,
`X T
`T X
$ U œ GZ .X (4.99)
or
$ U œ ”GZ X "#Z
X"
X "#Z
c,Z d .T œ ”GZ •.X •.X X " Z .T
,
,
,
(4.100)
During a constant pressure process, .T œ !, giving
X "#Z
•.X
,
(4.101)
X "#Z
$ UT
œ GZ .X
,
(4.102)
$ UT œ ”GZ But the heat capacity at constant pressure is just
GT œ
so the difference in the heat capacities is given by
GT GZ œ ”Œ
X"
X "#Z
T T •" Z œ
,
,
(4.103)
Now, X , Z , " # , and , are all positive quantities, so we see that for all homogeneous substances GT / GZ ! For an
ideal gas, this difference is just given by
TZ
œ 8V
X
(4.104)
8# V # X Z $
8VX Z $ #+8# Z 8, #
(4.105)
GT GZ œ
but for a real gas (a Van der Waal gas) we find
GT GZ œ
which reduces to the ideal gas result if + œ !.
ENTROPY CHANGES FOR AN IDEAL GAS
One of the most obvious uses of the combined first and second law
$U œ X .W œ .I T .Z
(4.106)
is to find an equation for the change in entropy of a systemÞ Solving for .W we obtain
.W œ
.I
T
.Z
X
X
(4.107)
Thus, the change in entropy of a system can be determined simply by integrating from the initial to the final states
of the system
?WEF œ (
F
E
.W œ (
F
E
F
.I
T
(
.Z
X
E X
(4.108)
For an ideal gas, where .I œ [email protected] .X and T Z œ 8VX , and where we can assume the [email protected] is a constant, we can
write this as
?WEF œ (
F
E
.W œ [email protected] (
F
E
F
.X
.Z
8V(
X
E Z
(4.109)
Chapter 4: General Relationships between State Variables
15
Thus, the change in entropy of an ideal gas as it goes from state E to state F (characterized by TE , ZE , XE and
TF , ZF , XF , respectively) is given by
?WEF œ [email protected] lnŒ
XF
ZF
8V lnΠXE
ZE
(4.110)
This same equation can be expressed in terms of changes in pressure by using the ideal gas law to obtain
?WEF œ [email protected] lnŒ
XF
XF TE
XF
XF
TE
8V lnŒ
œ [email protected] lnŒ 8V lnŒ 8V lnŒ XE
XE TF
XE
XE
TF
(4.111)
or
?WEF œ [email protected] V lnŒ
XF
TE
XF
TE
8V lnŒ œ 8-: lnŒ 8V lnŒ XE
TF
XE
TF
(4.112)
Likewise, one can use the ideal gas law to elliminate the temperature and write the change in entropy in terms of
the volume and pressure. Thus, we can easily calculate the changes in the entropy of an ideal gas, provided we
know the temperature and volume, the temperature and pressure, or the volume and pressure for the initial and
final states:
?WEF œ [email protected] lnŒ
XF
ZF
8V lnΠXE
ZE
XF
TE
œ 8-: lnŒ 8V lnŒ XE
TF
TF
ZF
œ [email protected] lnŒ 8-: lnŒ TE
ZE
(4.113)
In what follows we will find expressions from which we can calculate the changes in entropy for any system, not
just an ideal gas.
Appendix 4.1
The $U and X .W Equations:
A Systematic Derivation of Relationships
Between Parameters of Arbitrary Systems
In this appendix we will follow a very systematic approach to develop many equations which relate various
partial derivatives to experimentally measurable quantities. To simplify the expressions and the math, we
consider only closed systems where .R œ 0. The first law simplifies to
.I œ X .W T .Z
(A4.1.1)
and I is a function only of Wß X ß T ß and Z , where only two of these variables are independent. Since the entropy
cannot usually be measured directly, we wish to express the energy in terms of the more easily measured variables
T ß Z ß and X . This means that we have a choice of expressing the energy as
I = I ( Z ,X )
I = I (T ,X )
or
or
I = I (T ,Z ).
(A4.1.2)
Since we can often measure the amount of heat which is added to a system during various processes, we will
rearrange the energy equation to give
$U = X .W œ .I + T .Z
(A4.1.3)
By considering the energy I as a function of any two of the three easily measurable parameters T ß Z ß and X , we
can derive the three so-called $U equations.
I. I œ Z ß X Here we have
.I œ Œ
`I
`I
.X Œ
.Z
`X Z
`Z X
(A4.1.4)
`I
`I
.X Œ
.Z T .Z
`X Z
`Z X
(A4.1.5)
which gives for the $U equation
$U œ X .W œ Œ
or
$U œ X .W œ Œ
`I
`I
.X ”Œ
T •.Z
`X Z
`Z X
(A4.1.6)
which is the first $U equation.
II. I œ IÐT ß X Ñ
The second $U equation arises from the assumption that I = I (T ,X ), giving
$U œ X .W œ Œ
`I
`I
.X Œ
.T T .Z
`X T
`T X
(A4.1.7)
but we must remember that dZ is also an exact differential, with Z = Z (T ,X ), so that this equation becomes
$U œ X .W œ Œ
`I
`I
`Z
`Z
.X Œ
.T T ”Œ
.X Œ
.T •
`X T
`T X
`X T
`T X
(A4.1.8)
Chapter 4: General Relationships between State Variables
17
or
$U œ X .W œ ”Œ
`I
`Z
`I
`Z
T Œ
•.X ”Œ
T Œ
•.T
`X T
`X T
`T X
`T X
(A4.1.9)
which is the second $U equation.
III. I œ IÐT ß Z Ñ
The last $U equation is found by taking I = I (T ,Z ) so that
$U œ X .W œ Œ
`I
`I
.T Œ
.Z T .Z
`T Z
`Z T
(A4.1.10)
or
$U œ X .W œ Œ
`I
`I
.T ”Œ
T •.Z
`T Z
`Z T
(A4.1.11)
which is the third $U equation.
In this appendix we will carefully examine some of the consequences of these three equations. In
particular we want to determine expressions for the change in the internal energy and the entropy of a system in
terms of experimentally measureable quantities. Once we determine expressions for these partial derivatives, we
will be able to write out expressions for the heat flowing into a system or the change in entropy of the system, or
the change in internal energy of the system in terms of experimentally measureable quantities. Some of the
relationships derived in the main part of the chapter will be repeated here for completeness, but there are many
more relationships derived in this appendix than in the chapter itself.
I. Consequences of the First $U equation: I œ IÐX ß Z Ñ and W œ WÐX ß Z Ñ
The First $U equation is
$U œ X .W œ Œ
`I
`I
.X ”Œ
T •.Z
`X Z
`Z X
(A4.1.12)
In addition, since the entropy can also be written as a function of X and Z we can write
X .W œ X Œ
`W
`W
.X X Œ
.Z
`X Z
`Z X
(A4.1.13)
Examining these two equations, we immediately find some relationships between the energy partials and the
entropy partials:
XŒ
`W
`I
œŒ
`X Z
`X Z
(A4.1.14)
`W
`I
œ ”Œ
T•
`Z X
`Z X
(A4.1.15)
and
XŒ
This last equation is very useful in studying the variation of the internal energy of a gas as a function of volume.
Rearranging this equation, one can write
Œ
`I
`W
œ XŒ
T
`Z X
`Z X
(A4.1.16)
Chapter 4: General Relationships between State Variables
18
Now we can use the Maxwell relation (derived in the chapter)
Œ
`W
`T
"
œŒ
œ
`Z X
`X Z
,
(A4.1.17)
to write
Œ
`I
X"
T
œ
,
`Z X
(A4.1.18)
We will now use the $U equation to find expressions for these partial derivatives in terms of measurable
quantities, such as the heat capacities GZ and GT , the isothermal compressibility , and the coefficient to thermal
expansion " , and the easily measureable parameters T ß Z ß and X .
A. Isochoric Processes. We will first consider a system undergoing an isochoric process (.Z = 0), so that the
first $U equation simplifies to give
$UZ œ X .WZ œ Œ
`I
.XZ
`X Z
(A4.1.19)
where the subscripts indicate the variable which must remain constant in the process. We define the heat capacity
at constant volume GZ as the ratio of the amount of heat added to the system, $ UZ , to the rise in temperature,
.XZ , of the system for a given process.
GZ œ
`W
`I
$ UZ
œX Œ
œŒ
.XZ
`X Z
`X Z
(A4.1.20)
[Note: Remember that the ratio of $ UZ to .XZ is simply a ratio since $ U is not an exact differential, while the
ratio of .WZ to .XZ is equivalent to the partial derivative of W with respect to X when Z is held constant, because
the entropy is a state variable.]
We see from this equation that the heat capacity at constant volume is directly related to the how the energy
of the system changes with temperature if we hold the volume constant. Likewise, it is directly related to the
change in entropy of the system during this isochoric change in temperature. These partials can therefore be
written in terms of an experimentally measureable quantity according to:
Œ
`I
œ GZ
`X Z
and
Œ
`W
GZ
œ
`X Z
X
(A4.1.21)
[Note: We have written these equations in terms of extensive variables, which means that these
relationships depend upon the size of the system. The heat capacity, therefore, also depends upon
the size of the system. To get the equations in terms of intensive variables, we must divide by the
mass or the number of moles in the system. But this must be done for all the variables. For
example, the combined first and second law would have to be written as
$ ; = X .= œ . % + T [email protected]
(A4.1.22)
where ; œ U/8, = œ W /8, etc. Under these circumstances the specific heat capacity - is given by
-œ
G
$;
œ
8
.X
(A4.1.23)
All the equations which we derive can easily be changed to intensive form by dividing the
extensive variables by the number of moles. Notice that " and , are defined in terms of ratios of
the volume so that their definition is independent of size!]
B. Isobaric Processes. Next we will consider the system undergoing an isobaric process (.T = 0). In this case
the first $U equation gives us
Chapter 4: General Relationships between State Variables
$UT œ X .WT œ Œ
19
`I
`I
.XT ”Œ
T •.ZT
`X Z
`Z X
(A4.1.24)
If we now divide this equation by .XT (a change in temperature at constant pressure), we get
$UT
.WT
`I
`I
.ZT
œX
œŒ
”Œ
T•
.XT
.XT
`X Z
`Z X
.XT
(A4.1.25)
The first term in this equation is, by definition, the heat capacity at constant pressure, GT . Again, it is not a
partial derivative because $U is not an exact differential. However, the other ratios which appear in this equation
are partial derivatives, so that this equation should be written
GT œ
$ UT
`W
`I
`I
`Z
œ XŒ
œŒ
”Œ
T• Œ
.XT
`X T
`X Z
`Z X
`X T
(A4.1.26)
This last equation can be further simplified by using the identities:
GZ œ Œ
`I
`X Z
" `Z
Œ
Z `X T
(A4.1.27)
`W
`I
œ GZ ”Œ
T • "Z
`X T
`Z X
(A4.1.28)
and
"œ
to give
GT œ X Œ
This equation can now be used to determine the isothermal change in internal energy as the volume of the system
changes. We simply use the experimentally determined values of GT ß GZ ß T ß " ß and Z and obtain
Œ
`I
GT GZ
T
œ
`Z X
"Z
(A4.1.29)
Likewise, the entropy partial is
Œ
`W
GT
œ
`X T
X
(A4.1.30)
C. Isothermal Processes. For an isothermal process (.X = 0) the first $ U equation gives
$UX œ X .WX œ ”Œ
`I
T •.Z
`Z X
X
(A4.1.31)
This expression is interesting. It is a measure of the amount of heat which is added to a system during an
isothermal process, i.e., it is the amount of heat added to the system during a process in which the temperature of
the system does not change! Now the heat capactity is defined as G = $ U/.X , so that for an isothermal process,
the heat capacity of the system is essentially infinite! Thus the concept of heat capacity in the usual sense is not
very useful in this case. However, somewhat analogous to the definition of heat capacity, we define the latent
heat capacity (or latent heat) of the system as the amount of heat added to a system (or taken away) as the volume
of the system changes even when the temperature remains constant. We use the symbol PZ to stand for the latent
heat capacity of the system during a volume change. [Note that this is an exception to the way we normally view
subscriptsÞ Usually a subscript will mean a variable is held fixed, but when we deal with latent heats, the
subscript will stand for the quantity which changes.] The latent heat for a volume change is therefore given by
the equation
PZ œ
.WX
`W
`I
$ UX
œX
œ XŒ
œ ”Œ
T•
.ZX
.ZX
`Z X
`Z X
From Equ. (13.25) we can write PZ as
(A4.1.32)
Chapter 4: General Relationships between State Variables
PZ œ
20
GT GZ
"Z
(A4.1.33)
We have, however, introduced a new partial derivative of the entropy which can now be expressed in terms of
measureable parameters:
Œ
`W
PZ
GT GZ
œ
œ
`Z X
X
X "Z
(A4.1.34)
Using the definition of the latent heat, the first $U equation can be expressed in the simple form:
$U œ X .W œ CZ dX + LZ dZ
(A4.1.35)
D. Adiabatic ProcessesÞ For an adiabatic process ($ U œ X .W = 0) we can write the first $ U equation
$UW œ X .W œ GZ .XW PZ .ZW œ !
$ UW œ X .W œ GZ .XW GT GZ
.ZW œ !
"Z
(A4.1.36)
(A4.1.37)
Here we use the subscript W to denote an adiabatic (isentropic) process. Dividing by .ZW we obtain
.XW
`X
PZ
GT GZ
GZ GT
œŒ
œ œ
œ " Z GZ
" Z GZ
.ZW
`Z W
GZ
(A4.1.38)
which gives the partial
Œ
`X
GZ GT
œ
" Z GZ
`Z W
(A4.1.39)
However, this last equation could have been derived from previous results in this section by using the cyclic and
reciprocity rules:
`X
`W /`Z X
œ Œ
œ `Z W
`X /`Z W
G: GZ
G: GZ
GZ G:
X "Z
œ œ
GZ
GZ " Z
" Z GZ
X
(A4.1.40)
This last equation can also be written
Œ
`X
"#
œ
`Z W
"Z
(A4.1.41)
where # is the ratio of specific heats (# œ -: /[email protected] ).
The last equation above is perfectly general, for any real substance. For an ideal gas, # is a constant and
" œ "ÎX , so that this last equation can be integrated to obtain the equations relating the pressure, volume and
temperature for adiabatic processes in ideal gases.
Chapter 4: General Relationships between State Variables
21
Summary of Results for the First $U equation: I œ IÐZ ß X Ñ
Summarizing the expressions which we have developed for the energy and entropy partials, we have, from
the first $U equation:
Œ
`W
GZ
œ
`X Z
X
`I
Œ
œ GZ
`X Z
`X
GZ GT
Œ
œ
`Z W
" Z GZ
Œ
`W
GT
œ
`X T
X
Œ
`W
PZ
GT GZ
œ
(A4.1.42)
œ
`Z X
X
X "Z
`I
GT GZ
X"
T œ
:
Œ
œ
`Z X
"Z
,
`X
:
G: GZ
Œ
œ
`Z I
GZ
" Z GZ
where the last expression on the right is derived from the energy partials, using the cyclic and reciprocity rules.
The Maxwell relationships also gives us some additional information. For example
Œ
`W
`T
"
œŒ
œ
`Z X
`X Z
,
(A4.1.43)
can be used to gain some additional information concerning the heat capacities. We obtain
Œ
"
`W
PZ
GT GZ
œ
œ
œ
,
`Z X
X
X "Z
(A4.1.44)
which gives
GT GZ œ
"#X Z
,
(A4.1.45)
for the difference in the heat capacities. This equation tells us that GT is always greater than GZ since " # , X ß Z ß
and 5 are always greater than zero! For an ideal gas, where " œ 1/X and , = 1/:, this reduces to
GT GZ œ 8V
Ðideal gas)
(A4.1.46)
-T -Z œ V
Ðideal gas)
(A4.1.47)
or in terms of molar specific heat capacities
We can use the first $U equation to calculate the amount of heat added to the system for a given temperature and
volume change in terms of measureable quantities
$ U œ X .W œ GZ .X ”
GT GZ
•.Z
"Z
(A4.1.48)
X"
.Z
,
(A4.1.49)
or, using the Maxwell relation,
$ U œ X .W œ GZ .X II. Consequences of the Second $U equation: I œ IÐX ß T Ñ and W œ WÐX ß T Ñ
The second $U equation is given by
$U œ X .W œ ”Œ
`I
`Z
`I
`Z
T Œ
•.X ”Œ
T Œ
•.T
`X T
`X T
`T X
`T X
(A4.1.50)
Likewise, in terms of the differential change in the entropy
X .W œ X Œ
giving
`W
`W
.X X Œ
.T
`X T
`T X
(A4.1.51)
Chapter 4: General Relationships between State Variables
22
XŒ
`W
`I
`Z
œŒ
TŒ
`X T
`X T
`X T
(A4.1.52)
XŒ
`W
`I
`Z
œŒ
TŒ
`T X
`T X
`T X
(A4.1.53)
and
Again, we want to determine expressions for the change in the internal energy and the entropy of a system in
terms of experimentally measureable quantities. This will enable us to write out expressions for the heat flowing
into a system or the change in entropy of the system, or the change in internal energy of the system in terms of
experimentally measureable quantities.
A. Isobaric Processes. Consider the isobaric process (.T = 0)
$UT œ X .WT œ ”Œ
`I
`Z
TŒ
•.XT
`X T
`X T
(A4.1.54)
Dividing by the temperature change at constant pressure, we obtain the heat capacity at constant pressure, GT ,
GT œ
`W
`I
`Z
`I
$ UT
œ XŒ
œŒ
TŒ
œŒ
T "Z
.XT
`X T
`X T
`X T
`X T
(A4.1.55)
This equation is not the same one we obtained earlier, we have a different energy partial. Here we determine how
the energy of the system varies with temperature during an isobaric process in terms of experimentally measurable
parameters from the partial
Œ
`I
œ GT T " Z
`X T
(A4.1.56)
We do obtain again the previous result,
Œ
`W
GT
œ
`X T
X
(A4.1.57)
Notice that the equation for GT can be written
GT œ
`W
`I
` [T Z ]
` [I T Z ]
`L
$ UT
œ XŒ
œŒ
Œ
œŒ
œŒ
.XT
`X T
`X T
`X
`X
`X T
T
T
(A4.1.58)
where the quantity L œ I T Z is the enthalpy of the system, as defined earlier. Here we see that the enthalpy
of the system under constant pressure processes is similar to the internal energy of the system under constant
volume processes, since
GT œ Œ
`L
`X T
(A4.1.59)
GZ œ Œ
`I
`X Z
(A4.1.60)
and
B. Isothermal Processes. Now consider an isothermal process (.X = 0)
$UX œ X .WX œ ”Œ
`I
`Z
TŒ
•.TX
`T X
`T X
(A4.1.61)
In the same way that we defined the latent heat before, we again define the latent heat of pressure change PT and
find from the equation above
Chapter 4: General Relationships between State Variables
PT œ
`W
`I
`Z
$ UX
œ XŒ
œ ”Œ
TŒ
•
.TX
`T X
`T X
`T X
23
(A4.1.62)
Remember that the isothermal compressibility is given by
" `Z
Œ
Z `T X
(A4.1.63)
$ UX
`W
`I
œ XŒ
œ ”Œ
T ,v•
.TX
`T X
`T X
(A4.1.64)
,œ so that we can write
PT œ
The quantity PT is an experimentally measureable quantity, but it can also be determined in terms of other more
familiar quantities. We will show in what follows that
PT œ PZ ,Z œ Œ
,
GT GZ
,Z œ GZ GT "Z
"
(A4.1.65)
This will mean that the energy and entropy partials can be expressed as
Œ
`I
œ PT T ,Z
`T X
(A4.1.66)
,
`I
œ GZ GT T ,Z
"
`T X
(A4.1.67)
or
Œ
and
Œ
`W
PT
œ
`T X
X
(A4.1.68)
in terms of measurable quantities.
To see the relationship between PZ and PT , we will consider the first and second $ U equations
$U œ GZ .X PZ .Z
(A4.1.69)
$U œ GT .X PT .T
(A4.1.70)
and
There are actually two different ways we can understand the relationship between PT and PZ , and we will
examine both. First, consider three different processes which can be diagramed on a T ,Z graph: 1) .T = 0; 2)
.Z = 0; 3) .X = 0, as shown below.
For the case where .T = 0 we immediately have $ U = CT .X and we can calculate the amount of heat
added to the system during the temperature increase .X . Likewise for the case where .Z = 0 we can write $ U =
CZ .X . Now for the case where .X = 0, we have two expressions which we can use
$U = LZ .Z
(A4.1.71)
$U = LT .T
(A4.1.72)
or
24
8
6
dV = 0
Presure (Pascals)
Chapter 4: General Relationships between State Variables
4
dP = 0
A
dT
2
=0
T(A)
T(C)
0
2
4
6
B
T(B)
8
10
Volume (cubic meters)
Now for an isothermal change (.X = 0) the pressure and volume are related, so that if T changes by .T then we
know exactly how much Z changes. Thus if X is constant and T changes from state E p F we can calculate $ U
from the second equation above. But during this process, Z changes also from the value it has for state E to that
for state F, so that we can in essence calculate the heat added to, or removed from, the system using Equ.
(A4.1.71) above. Now the process is the same in both cases and the end points are the same so that we must have
PZ .ZX = PT .TX
(A4.1.73)
or
PZ
`T
`Z
œŒ
œ ”Œ
•
PT
`Z X
`T X
"
œ "
,Z
(A4.1.74)
Now since we have already found an expression for PZ in terms of easily measured parameters of the system, this
equation allows us to determine PT :
PT œ PZ ,Z œ Œ
,
GT GZ
,Z œ GZ GT "Z
"
(A4.1.75)
This same result can be obtained directly from the equations
$U = CZ .X + PZ .Z
(A4.1.76)
$U = CT .X + PT .T
(A4.1.77)
and
Since Z is a function of T and X , we can write the first equation as
$U = CZ .X + LZ ”Œ
`Z
`Z
.T Œ
.X •
`T X
`X T
(A4.1.78)
`Z
`Z
• .X PZ Œ
.T
`X T
`T X
(A4.1.79)
or
$U œ ”GZ PZ Œ
Chapter 4: General Relationships between State Variables
25
Now if we compare this equation with the second $U equation
$U = CT .X + LT .T
(A4.1.80)
GT œ GZ PZ " Z
(A4.1.81)
we see that
and
PT œ PZ ,Z œ PZ ,Z œ
,
GZ GT "
(A4.1.82)
which is the result we obtained earlier.
C. Adiabatic Processes. We now examine adiabatic processes (.W œ !), for which the second $ U equation gives
$U œ X .W œ ! œ ”Œ
`I
`Z
`I
`Z
T Œ
•.X ”Œ
T Œ
•.T
`X T
`X T
`T X
`T X
(A4.1.83)
$U œ X .W œ ! œ GT .XW PT .TW
(A4.1.84)
or
from which we obtain
Œ
`X
PT
œ œ `T W
GT
,
"
GZ GT GT
œ , GZ GT " GT
(A4.1.85)
As we demonstrated last time this equation can be obtained directly from the entropy partials by using the
reciprocal and cyclical properties of partials.
We can also use this last equation to derive a useful relationship between the pressure and the temperature
of an ideal gas which undergoes an adiabatic process. Deviding the top and bottom of the equation by GZ and
substituting for , and " , this last equation can be written
Œ
`X
X " # , GZ GT œ œ `T W
:#
" GT
(A4.1.86)
Since # is constant for an ideal gas, we can integrate this equation to obtain
X:
"#
#
œ const.
Ðadiabatic process for ideal gas)A4.1.87)
(
Summary of I œ IÐX ß T Ñ and W œ WÐX ß T Ñ
Summarizing the expressions which we have developed for the energy and entropy partials using the second
$U equation, we have:
Œ
`I
œ GT T " Z
`X T
`W
GT
Œ
œ
`X T
X
, GZ GT `X
PT
œ Œ
œ " GT
`T W
GT
,
`I
œ PT T ,Z œ GZ GT T ,Z
"
`T X
,
`W
PT
GZ GT œ
Œ
œ
"X
`T X
X
Œ
The heat added to the system, from the second $U equation is given by
$U œ X .W œ ”Œ
`I
`Z
`I
`Z
T Œ
•.X ”Œ
T Œ
•.T
`X T
`X T
`T X
`T X
(A4.1.88)
Chapter 4: General Relationships between State Variables
26
or
$ U œ X .W œ GT .X ,
GZ GT .T
"
(A4.1.89)
which can be further simplified to give
$ U œ X .W œ GT .X X " Z .T
(A4.1.90)
III. Consequences of the third $U equation: I œ IÐT ß Z Ñ and W œ WÐT ß Z Ñ
The third $U equation is
$U œ X .W œ Œ
`I
`I
.T ”Œ
T •.Z
`T Z
`Z T
(A4.1.91)
and X times the change in entropy is given by
X .W œ X Œ
`W
`W
.T X Œ
.Z
`T Z
`Z T
(A4.1.92)
`W
`I
œŒ
`T Z
`T Z
(A4.1.93)
`S
`I
œŒ
T
`Z T
`Z T
(A4.1.94)
so that we have
XŒ
and
XŒ
Now we take the partial with respect to Z holding T constant in the first equation and the partial with respect to
T holding Z constant in the second equation, and make use of the fact that I and W are exact differentials to
obtain:
Œ
`W
`W
œ ,Z Œ
"Z
`T Z
`Z T
(A4.1.95)
We now examine the third $U equation under certain processes to obtain expressions for the partials of energy
and entropy in terms of experimentally measurable quantities.
A. Isochoric Processes For an isochoric process (dZ = 0) the third $ U equation gives
$UZ œ X .WZ œ Œ
`I
.TZ
`T Z
(A4.1.96)
Dividing by a small temperature change at constant volume we obtain
GZ œ
.WZ
`I
.TZ
$ UZ
œX
œŒ
.XZ
.XZ
`T Z .XZ
(A4.1.97)
which is the heat capacity at constant volume. The ratios of the exact differentials must be partial derivatives, so
we can write
GZ œ
`W
`I
`T
$ UZ
œ XŒ
œŒ
Œ
.XZ
`X Z
`T Z `X Z
(A4.1.98)
We can use the cyclic relationship to evaluate `T Î`X Z :
Œ
"Z
"
`T
`Z /`X T
œ œ
œ ,
`X Z
`Z /`T X
,Z
(A4.1.99)
Chapter 4: General Relationships between State Variables
27
so that we obtain for the heat capacity at constant volume:
GZ œ
`W
`I
$ UZ
"
œ XŒ
œŒ
.XZ
`X Z
`T Z ,
(A4.1.100)
This gives us another energy partial which we can evaluate experimentally
Œ
`I
,GZ
œ
`T Z
"
(A4.1.101)
B. Isobaric Processes For an isobaric process, the third $U equation becomes
$UT œ X .WT œ ”Œ
`I
T •.ZT
`Z T
(A4.1.102)
and we can again divide by a small temperature change at constant pressure .XT to give
GT œ
$ UT
`W
`I
`Z
œ XŒ
œ ”Œ
T• Œ
.XT
`X T
`Z T
`X T
(A4.1.103)
$ UT
`W
`I
œ XŒ
œ ”Œ
T •" Z
.XT
`X T
`Z T
(A4.1.104)
or
GT œ
which gives us another energy partial
Œ
`I
GT
T
œ
`Z T
"Z
(A4.1.105)
C. Adiabatic ProcessesÞ Now for an adiabatic process we can write
$U œ X .W œ ! œ Œ
`I
`I
.TW ”Œ
T •.ZW
`T Z
`Z T
(A4.1.106)
where we have again used the subscript W to stand for adiabatic (isentropic) changes. Now by using the energy
partials derived above
Œ
`I
GT
T
œ
`Z T
"Z
Œ
`I
,GZ
œ
`T Z
"
(A4.1.107)
we can write
,GZ
GT
.TW .ZW
"
"Z
(A4.1.108)
"
`T
GT
GT
‚
œ œ "Z
,GZ
,Z GZ
`Z W
(A4.1.109)
$ U œ X .W œ ! œ
from which we can derive
Œ
Again, we can use this last equation to derive a useful relationship between the pressure and the volume of
an ideal gas which undergoes an adiabatic process. Writing the ratio of the specific heats as # and substituting the
value of , for an ideal gas, we have
Œ
`T
:#
œ `Z W
Z
(A4.1.110)
Chapter 4: General Relationships between State Variables
28
which can be integrated to give
:Z # œ constant
(A4.1.111)
We could also examine an isothermal process, but we would get no new results.
Summary of I œ IÐZ ß T Ñ and W œ WÐZ ß T Ñ
We now summarize the partial derivative relationships which we have found in this subsection:
Œ
,GZ
`I
œ
"
`T Z
"
`T
GT
GT
‚
œ Œ
œ "Z
,GZ
,Z GZ
`Z W
`W
`W
Œ
œ ,Z Œ
"Z
`T Z
`Z T
Œ
`I
GT
T
œ
"Z
`Z T
(A4.1.112)
The heat added to the system using the third $U equation is given by
$U œ X .W œ Œ
`I
`I
.T ”Œ
T •.Z
`T Z
`Z T
(A4.1.113)
or
$ U œ X .W œ
,GZ
GT
.T .Z
"
"Z
(A4.1.114)
ADDITIONAL APPLICATION
The Internal Energy of an Ideal Gas (The Free Expansion of a Gas):
When we first introduced the concept of an ideal gas, we started with a model for that gas which assumed
that their was no interaction potential energy between the molecules making up the system, that is, the internal
energy of the system could not depend upon the location of the particles or their relative separation. This implies
that the internal energy of an ideal gas should not depend upon the volume of the gas. In fact, we showed that the
internal energy of an ideal gas is a function of the temperature only, not the volume or the pressure. From these
fundamental assumptions, we derived the equation of the ideal gas
T Z œ R 5X œ 8VX
(A4.1.115)
Let us approach this problem from a somewhat different perspective. Let's assume that the equation of state
has been determined by empirical observations (which, in fact, is the case historically). We can now use the
relationships which we obtained in the last chapter to investigate the internal energy of such a gas and how it
depends upon various parameters. Rather than start with arbitrary relationships to prove this point, however, we
will actually develop the equations we need.
If we assume a closed system, the internal energy can be a function of at most two independent parameters.
Let's make the choice I œ IÐZ ß X Ñ, so that we can write
.I œ Œ
`I
`I
.Z Œ
.X
`Z X
`X Z
(A4.1.116)
Œ
`I
œ GZ
`X Z
(A4.1.117)
`I
X"
T
œ
,
`Z X
(A4.1.118)
`I
X "ÎX T œ!
œ
`Z X
"ÎT (A4.1.119)
From the results we obtained above, we find
and
Œ
Now for an ideal gas this equation becomes
Œ
which tells us that any gas which obeys the ideal gas equation is one for which the internal energy of the gas
cannot depend upon the volume! We can use the chain rule
Œ
`I
`I
`Z
œŒ
Œ
œ0
`T X
`Z X `T X
(A4.1.120)
to see that the internal energy of an ideal gas must be independent of both volume and pressure changes, and thus
must be a function of the temperature of the gas only!
The fact that an ideal gas is not a function of the pressure or the volume of the gas implies that if we were to
cause an ideal gas to expand freely into a larger volume, the internal energy (and thus the temperature of the gas)
would not change. Such an experiment is called a free-expansion, and is performed by placing a gas in a chamber
with two sections separated by a valve. One side is evacuated and the other side is filled with a gas. Once the gas
reaches equilibrium, the valve is opened and the gas is allowed to freely expand into the adjacent volume. From
the first law we have
.I œ $ U $ [ œ $ U
(A4.1.121)
since no work is done in a free expansion (there are no walls which move). Thus, any change in internal energy
would cause heat to flow into or out of the system. Now if the system is isolated, so that the walls of our special
box are adiabatic, then the change in internal energy must be zero! Thus, for an adiabatic free-expansion we have
Chapter 4: General Relationships between State Variables
30
that .I œ !. Now let's invision a process for which .I œ !, but for which the process if reversible, or “quasistatic". In this case, we would have
.I œ ! œ X .W :.Z
(A4.1.122)
or
X ”Œ
`W
`W
.X Œ
.Z • œ :.Z
`X Z
`Z X
(A4.1.123)
Now
GZ œ
`W
$ UZ
œ XŒ
.XZ
`X Z
(A4.1.124)
and
Œ
`W
`T
GT GZ
œŒ
œ
`Z X
`X Z
"Z X
(A4.1.125)
GZ .X œ ”T X Œ
`T
•.Z
`X Z
(A4.1.126)
"
`T
”T X Œ
•.Z
GZ
`X Z
(A4.1.127)
so that we have
or
.X œ
For an ideal gas, the quantity in brackets is zero, so that the temperature of the gas would not change for a
small volume change. This is reasonable if the internal energy of the gas does not depend upon the relative
distances between the particles. However, for a Van der Waal's gas, the change in temperature is not zero. This
is because Van der Waal's model attempts to treat the gas from a more realistic point of view. As the relative
distance between molecule increases, the potential energy of the system changes - effectively changing the
location of the “bottom of the potential well". If the internal energy cannot change, and the effective potential
energy of the system “rises", then the thermal energy of the gas (the difference between the total energy and the
potential energy) must decrease. Thus the temperature of the gas must decrease. Notice that we can, in effect,
determine .X for a given change in volume for any gas (even if we do not know the equation of state) provided
we know GZ ß GT ß " ß Z ß T , and X for the gas.
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