Maxwell Relations

Background

Maxwell’s equations are important for electric and magnetic fields, while Maxwell’s relations focus on thermodynamic quantities like temperature, pressure, volume, and entropy and are important when analyzing thermodynamic systems like internal combustion engines.

Like Maxwell’s equations, Maxwell’s relations are based on partial derivatives. The relations describe the symmetry of second-order partial derivatives of thermodynamic quantities. Maxwell’s relations directly and indirectly cover eight thermodynamic quantities including:

• pressure (P),
• volume (V),
• temperature (T),
• entropy (S) measured in Joules per Kelvin (J/K),
• internal energy (U) measured in Joules,
• enthalpy (H),
• Helmholtz free energy (F or A),
• Gibbs free energy (G).

P, V, and T are simple quantities while the other thermodynamic quantities are more complex, thus Maxwell’s relations are very important.

For example, enthalpy (H) measures a system’s total heat content is defined as the sum of its internal energy (U) plus the product of pressure (P) and volume (V): H = U + PV.

Helmholtz free energy (F) measures the useful work obtainable from a closed system at constant temperature and volume and is defined as F = U – TS.

Gibbs free energy (G) is used to predict the spontaneity of a process at a given temperature and is defined as G = H – TS, or ∂G = ∂H – T∂S:

• When ∂G < 0, the process is spontaneous (favored).
• When ∂G > 0, the process is non-spontaneous (not favored).
• If ∂G = 0, the process is at equilibrium.

Mathematical Derivations of Maxwell's Relations

To derive Maxwell’s relations, it is useful to combine the First and Second Laws into a single mathematical statement. In order to do that, one notes that since:

for a reversible process: dq = TdS

Since: dw = TdS – pdV

for a reversible expansion in which only p-V works is done, and since dU = dq + dw

dU = TdS – pdV         (1)

This differential for dU can be used to simplify the differentials for H, A (Helmholtz energy) and G. But it is even more useful due to the constraints it places on the variables T, S, p and V due to the mathematics of exact differentials. 

Equation 1 suggests that the natural variables of internal energy are S and V or simply U (S, V). Thus, the total differential (dU) in equation 1 can be expressed as follows:

By studying equations 1 and 2, we can see that:

Since dU is an exact differential, the Euler relation must hold that:

substituting equations 3 and 4 into the above expression:

This is an example of a Maxwell Relation.

A similar result can be derived based on the definition of H:

H = U + pV

Using chain rule and differentiating d(pV) results in:

dH = dU + pdV + Vdp

Making the substitution using the combined first and second laws (dU = TdS - pdV) for a reversible change involving on expansion (p-V) work:

dH = TdS – pdV + pdV + Vdp

dH = TdS + Vdp         (5)

As in the case of internal energy, this suggests that the natural variables of H are S and p. Thus, 

If we compare equation 5 and 6 we get:

We can now see that equation 3 and equation 7 are equal to T hence are equal to each other:

Furthermore, the Euler Relation must also hold true:

This is the Maxwell Relation of H. The same can be done for G and A.

Summary

Maxwell Relations are summarized below:

Fundamentals of Partial Differentiation of the Exact Differential Equation

Carrying out partial differentiation of the Exact Differential Equation of M and N with respect to z and y, respectively, results in:

dx is a perfect differential when the above equation is satisfied for any function x.

Similarly, if y = y(x, z) and z = z(x, y), then from these two relations, we have:

In terms of p, v and T, the following relation holds true:

Thermodynamic Relations

Some properties like temperature, pressure, volume, and mass can be measured directly while other properties like density and specific volume can be determined from these using some simple relations.

However, properties like internal energy, enthalpy, and entropy are not so easy to determine as they cannot be measured directly or related to easily measurable properties through some simple relations.

Thus, it is important to develop some fundamental relations between commonly encountered thermodynamic properties and express the properties that cannot be measured directly in terms of easily measurable properties.

Partial Derivatives and Associated Relations

Most basic thermodynamic relations involve differentials. Let us consider a function f that depends on a single variable x, i.e., f = f(x) as shown below:

The derivative of the function at a point defined as Slope expressed as:

Therefore, the derivative of a function f(x) with respect to x represents the rate of change of f with x.

Let us now consider a function that depends on two (or more) variables, such as z = z(x, y). This time the value of z depends on both x and y.

It is sometimes necessary to examine the dependence of z on only one of the variables by allowing one variable to change while holding the others constant and observing the change in the function.

The variation of z(x, y) with x when y is held constant is called the partial derivative of z with respect to x, and it is expressed as:

To obtain a relation for the total differential change in z(x, y) for simultaneous changes in x and y would be:

The above equation is the fundamental relation for the total differential of a dependent variable in terms of its partial derivatives with respect to the independent variables. This relation can easily be extended to include more independent variables.

An important relation for partial derivatives is used in calculus to test whether a differential dz is exact or inexact. In thermodynamics, this relation forms the basis for the development of the Maxwell relations which was discussed in a previous blog entry here where dz is replaced by dφ.

We can now develop two important relations for partial derivatives – the reciprocity and the cyclic relations.

The function z = z(x, y) can also be expressed as x = x(y, z) if y and z are taken to be the independent variables. Then the total differential of x becomes:

Eliminating dx from the above equation by combining the dz and dx equations we get:

The variables y and z are independent of each other and thus can be varied independently. For example, y can be held constant (dy = 0), and z can be varied over a range of values (dz ≠ 0). Therefore, for this equation to be valid at all times, the terms in the brackets must equal zero, regardless of the values of y and z. 

Setting the terms in each bracket of the above equation equal to zero gives two equations:

This equation is called the reciprocity relation, and it shows that the inverse of a partial derivative is equal to its reciprocal.

This equation is called the cyclic relation, and it is frequently used in thermodynamics.