Chapter 8
Maxwell relations and measurable properties
8.1
Maxwell relations
Other thermodynamic potentials emerging from Legendre transforms allow us to switch
independent variables and give rise to alternate versions of the second law at different
constant conditions:
entropy fundamental equation
maximum at equilibrium
energy fundamental equation
minimum at equilibrium
enthalpy
minimum at equilibrium
Helmholtz free energy
minimum at equilibrium
Gibbs free energy
minimum at equilibrium
Here, we discuss some of the mathematical properties of these functions, and their
consequences for relationships between thermodynamic variables. These considerations
will allow us to connect quantities that are difficult to measure directly—like the entropy
and the chemical potential—to variables that we can easily access using experiments.
To start, note that all of the potentials above have multiple independent variables. A
general feature of well-behaved multivariate functions is that the mixed partial derivatives
don’t depend on the order in which they are taken,
(8.1)
Or, writing out the derivatives explicitly,
[(
) ]
[(
(8.2)
) ]
This fact has important consequences for potentials since their derivatives involve other
thermodynamic quantities. As an example, consider the energy equation,
(8.3)
Examine the second derivative with respect to and
[(
1
)
at constant
]
[(
conditions:
)
]
(8.4)
Substituting for the inner derivatives using the fundamental equation in (8.3),
[
]
[ ]
(
)
(
(8.5)
)
Therefore, a relationship between the derivatives of and emerges simply from the fact
that these two quantities are related by a common second derivative of a thermodynamic
potential. Such equalities based on potential second derivatives are called Maxwell
relations, after James Maxwell, one of the early founders of modern thermodynamics and
electromagnetism. There are in fact many Maxwell relations, depending on which potential
is used and which pair of independent variables is examined. Here is another example,
based on the Gibbs free energy:
[(
)
]
[(
)
]
(8.6)
Substituting for the inner derivatives,
(
)
(
(8.7)
)
which shows a relationship between the pressure-dependence of the chemical potential
and the -dependence of the system volume. In fact, the right-hand derivative is equal to
. These examples illustrate a basic recipe: pick a potential and a second derivative
involving two of its independent variables, and then substitute first derivative definitions
to produce a Maxwell relation.
Maxwell relations connect two derivatives of thermodynamic variables, and
emerge due to the equivalence of potential second derivatives under a
change of operation order:
potential and
and
, where
is a thermodynamic
are two of its natural independent variables.
Maxwell relations are quite important, for two reasons. First, they show us that derivatives
of thermodynamic parameters are not completely independent. This can serve as a
consistency check in both experiments and in pen-and-paper analysis. Second, they
provide a method to express derivatives involving difficult-to-measure quantities in terms
of ones that are readily accessible experimentally, as we will now see.
8.2
Measurable quantities
How does one measure the entropy or chemical potential from experiments? These kinds
of quantities are not usually directly accessible in the lab. What we can measure, typically,
are mechanical quantities like pressure, bulk quantities like volume and density, and
2
thermal properties like temperature and heat flow (e.g., by slow heat exchange
experiments where we can measure temperature changes in a coupled reference body). Of
the thermodynamic variables that we have discussed thus far, the following are considered
measureable:
Measurable thermodynamic variables
temperature
pressure
volume
or
number of particles or mass (related by the molecular weight)
enthalpy (latent heat) of phase change
There are also several readily-measured material properties that depend on derivatives of
thermodynamic variables. Because these quantities measure the change in a system
parameter in response to an infinitesimal perturbation, they are termed response functions:
Measurable thermodynamic response functions
( )
( )
constant volume heat capacity
( )
( )
constant pressure heat capacity
( )
( )
(
(
)
isothermal compressibility
)
thermal expansivity / expansion coef.
These response functions are defined such that they are positive in the case of “normal”
systems, such as ideal gases and simple liquids. There are occasional exceptions in some of
them; for example, the thermal expansivity of liquid water below 4 °C is negative. There
are additional measurable response functions that we have not listed; some of them can be
expressed as combinations of the above, and some of them emerge in systems with other
thermodynamic variables, such as those involving electromagnetic or interfacial work.
Note that the heat capacities as presented above are extensive, while more conventionally
we might see intensive versions, such as
and
.
Maxwell relations enable us to express experimentally-inaccessible quantities in
terms of the measurable ones just listed. Consider the following example derivative,
(
)
(8.8)
(
3
)
This is reminiscent of the definition for
(
, so that we can substitute:
(8.9)
)
Hence the entropy’s dependence on pressure at constant temperature is related to the
thermal expansivity. Many times, however, getting to measurable quantities isn’t so easy.
Consider the following Maxwell relation,
(
)
(8.10)
(
)
Here, we moved to something that appears even less measurable than before. Fortunately,
there are other ways to relate thermodynamic variables using the principles of multivariate
calculus. We can use the so-called “triple product rule,” which shows that,
(
)
(
)
(
)
(
)
(
(8.11)
Rearranging,
(
)
(
)
(
)
)
(8.12)
which finally gives an expression involving only measurable quantities,
(
(8.13)
)
We now turn to a more complex example that relies on additional mathematical
manipulations; let us find the relationship between
and . From the definitions of
these quantities, we can infer a good place to start: both response functions depend on a
derivative of the entropy with temperature. The difference between the two is the variable
that is held constant in the derivative. We can relate a change in the constant conditions of
a derivative in the following way. Construct the function
, assuming constant
conditions throughout. Notice that this is not a fundamental potential because is not a
function of its natural variables. We can still, however, perform the construction since is
a state function and and stem from different conjugate pairs. We choose as an
4
independent variable because it is the temperature derivative that is relevant to the heat
capacities. We now expand
in differential form as,
(
)
(
)
(8.14)
(
)
To start relating the constant volume conditions to those at constant pressure, we take the
-derivative of this expression at constant . This is operationally equivalent to dividing
by
and applying constant conditions to any complete derivative that is formed,
(
)
(
) (
We recognize that the LHS can be replaced using
to ,
(
(8.15)
and the rightmost derivative connects
) (
(
)
)
(8.16)
)
Finally, we use a Maxwell relation to address the final term in our expression, giving,
(
)
by Maxwell relation
(
)
from previous example
(8.17)
Simplifying everything,
(8.18)
As a final example, we will compute a quantity called the isentropic compressibility,
defined by,
(
)
(8.19)
Experimentally, the isentropic compressibility measures the fractional change in volume of
a system during a reversible adiabatic (
) compression. To proceed, we use the triple
product rule to remove the entropy from the constant condition,
5
(
)
[(
) (
(
)
(
)
) ]
(8.20)
Now we use a new calculus rule, called “addition of variable,” that will enable us to expand
the numerator and denominator by including temperature. We want to add temperature
because there are no measurable properties that are derivatives of at constant and vice
versa. The rule produces,
(
)
(
) (
)
(
) (
)
(
)
(
)
(8.21)
Note that the remaining derivatives all involve permutations of the same three variables.
This suggests the use of the triple product rule again,
(
)
[(
[ (
) (
) ]
(8.22)
) ]
Finally, we see that we can insert the expression for the isothermal compressibility,
(
8.3
)
(8.23)
General considerations for calculus manipulations
Many thermodynamic calculus manipulations can be derived from a relatively simple
procedure. The first step is to construct a state function involving the variables of interest.
Some examples include:
,
,
, or
. Note that, if the
state function is a potential, we do not necessarily need to use natural variables. We can
choose any independent variables that we like so long as they form a complete set of
thermodynamic information (i.e., all stem from different conjugate pairs).
The second step is to write out the full differential of the state function. As an
example, take
,
6
(
)
(
)
(
)
(8.24)
The third step is to set the differentials to zero for any terms that are constant in the
problem of interest. For example, if is constant then
and we have:
(
)
(
(8.25)
)
Note that we must keep in mind that the current equation corresponds to constant
conditions. This is important as any new derivatives that are formed will acquire as a
constant variable.
The fourth step is to take the derivative of interest, essentially dividing by the
appropriate differential. All new derivatives acquire any previously applied constant
conditions. Moreover, we must specify additional constant conditions as necessary to meet
one less than the number of independent variables. In our example, we can also specify
to be constant when taking the derivative:
(
)
(
)
(
)
(
)
(8.26)
In conjunction with this procedure, one can then begin to substitute various fundamental
definitions, expressions for measurable quantities, and Maxwell relations.
⁄
As another example, consider the expression
. When derivatives involve
a thermodynamic potential, we often begin with its differential form:
(8.27)
Constant
conditions implies,
(8.28)
Taking the temperature derivative at constant volume,
(
)
(
)
(
)
(8.29)
in which we can begin to see opportunities to substitute response functions.
There are a number of common calculus manipulation techniques that are useful in
thermodynamic analysis. These are briefly summarized below.
7
Rules of calculus manipulations
inversion
( )
⁄( )
triple product rule
( ) ( ) ( )
addition of variable
( )
(
) ⁄(
nonnatural derivative
( )
)
( ) ( )
( )
As we have seen earlier, there are also rules that apply specifically to thermodynamic
potentials:
Rules for thermodynamic potentials
Maxwell relations
potential transformation
(
)
(
)
( )
( )
⁄
The combination of all of these techniques enables us to relate virtually any
thermodynamic quantity to the set of measurable variables described earlier.
8
Problems
prove the triple product rule
1. Fundamentals problem. Given only the ideal gas equation of state,
, prove
the following properties using fundamental equations and Maxwell relations. In the first
two parts, you are proving that several ideal-gas properties are volume independent.
a) ( )
b) (
)
and (
)
c)
2. Fundamentals problem. Express the following in terms of measurable quantities for a
single component system at constant conditions:
a) ( )
b) ( )
c) ( )
d) ( )
e) ( )
f) ( )
g) ( )
h) ( )
3. Conceptual problem.
general:
a) ( )
Indicate whether each of the following relations is true in
( )
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b) ( )
(
)
c) ( )
(
)
1. Prove that all of the response functions,
and
are in fact measurable. For the
former two, consider the first law in conjunction with heating at different constant
conditions, where can be quantified by coupling to a reference body.
3. Conceptual problem. A quasi-static adiabatic process is performed on a system at
starting at state 1. Indicate the direction the system will proceed in the P-T plane (i.e., find
), in terms of measurable quantities.
4. Fundamentals problem. The Van der Waals equation of state for a gas is:
where
energy,
and and are positive constants. Find the change in molar internal
, upon isothermal expansion from molar volume to .
6. Applied problem. A Joule-Thompson process is one by which a gas is throttled
(expanded) with no heat exchanged. Therefore, any change in internal energy is due to
internal changes in pressure-volume work, and total enthalpy is conserved. Often during
such an expansion, the temperature of the gas will change. The Joule-Thomson coefficient
measures how much the temperature changes for a given pressure drop:
Here
( )
.
does not indicate a relationship to the chemical potential, but instead simply comes
from convention. Show that the Joule-Thomson coefficient is given by
6. Applied problem. Let the fundamental equation for an elastic band be
10
.
where
is the length and the tension.
a) An elastic band heats up when it is slowly stretched adiabatically. When it is cooled at
constant tension, will the same band expand or contract?
b) The same amount of heat flows into two identical elastic bands (not necessarily the same
as in part a) while one is held at constant tension and the other at constant length. Which
experiences the largest temperature rise? Prove this.
7. Applied problem. Liquid water exhibits a wealth of properties that are considered
anomalous when compared to simple liquid models like the van der Waals fluid. One wellknown anomaly is that, at constant pressure, water exhibits a temperature of maximum
density (TMD). At atmospheric pressure, this temperature is around 4C. In general,
though, we can construct a line in the
plane that gives the TMD for any pressure.
Consider the properties of this line:
a) Along the TMD line, what must be true about the thermal expansion coefficient? Prove
⁄
also that
along this line. Be careful to address all of the terms in your proof.
b) Show that the slope of this line is given by:
(
)
(
) ⁄(
)
c) Water’s isothermal compressibility increases upon cooling. Can anything be said about
the slope of the TMD line?
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