Maxwell Relations

Prerequisites: Thermodynamics. Full path (9 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

A fixed amount of gas in equilibrium is completely described by two quantities, for example its temperature and its volume. Everything else, like the pressure, the entropy or the energy, is then fixed. This means that the energy is a function of the state, a quantity that has one definite value at every point of a two-dimensional map of states.

Think of the energy as the height of a hilly landscape drawn on this map. At each point the hill has a slope towards the east and a slope towards the north. How fast the eastward slope changes as we walk north is always equal to how fast the northward slope changes as we walk east. If we use entropy and volume as the two directions on the map, the two slopes of the energy are the temperature and the pressure (up to a sign), so this fact about hills becomes a set of equations that link the ways in which different quantities respond to changes. These are the Maxwell relations.

They are useful because they connect things that are hard to measure with things that are easy to measure. For example, how much the entropy of a gas changes when we let it expand at constant temperature is difficult to measure directly. A Maxwell relation says that it equals how much the pressure rises when we heat the gas in a closed container of fixed volume, which only needs a thermometer and a pressure gauge. Different choices of which two quantities describe the state give different relations, one for each of the thermodynamic potentials.

Concrete

Derivation

The Maxwell relations follow directly from the fact that partial derivatives commute: $\partial_x \partial_y = \partial_y \partial_x$.

If we have some function $U(S,V)$ (called the internal energy) that depends on the entropy $S$ and the volume $V$, the total change of it is given by

$$dU = \frac{\partial U}{\partial S} \big |_V dS + \frac{\partial U}{\partial V} \big |_S dV,$$

where $\big |_V$ means that we keep $V$ fixed.

Then, we introduce two definitions:

$$\frac{\partial U}{\partial S} \big |_V \equiv T , \quad -\frac{\partial U}{\partial V} \big |_S \equiv P.$$

The minus sign here is just a convention and can be understood as follows: The internal energy usually gets smaller when we increase the volume. Thus, if we want to work with positive pressure $P$ in most situations, we need to include the minus sign here.

Using these definitions equation reads

$$dU = T dS - P dV,$$

which is the fundamental thermodynamic relation.

Next, we use that partial derivative commute:

$$\frac{\partial^2 U}{\partial V\partial S} = \frac{\partial^2 U}{\partial S\partial V}$$

and put in our two definitions from above:

$$\frac{\partial T}{\partial V} \big |_S = -\frac{\partial P}{\partial S}\big |_V$$

This is one of the Maxwell relations.

The other Maxwell relations follow completely analogous, but with different functions instead of the internal energy $U$.

For example, if we start with the Helmholtz free energy $A(T,V)$:

$$A = U -TS$$

and follow exactly the same steps ($dA= dU-d(TS) = (TdS-PdV)-(SdT+TdS)=-SdT-PdV$), we can derive

$$\frac{\partial S}{\partial V} \big |_T = \frac{\partial P}{\partial T} \big |_V$$

The other Maxwell relations follow by starting with the enthalpy $H(S,P)$ or the Gibbs free energy $G(T,P)$.

(These notions appear, since:

• a system with fixed entropy and volume will choose the state with minimum internal energy $U$,
• a system with fixed temperature and volume will choose the state with minimum Helmholtz free energy $A$,
• a system with fixed entropy and pressure will choose the state with minimum enthalpy $H$,
• a system with fixed temperature and pressure will choose the state with minimum Gibbs free energy $G$.)

For a great explanation, why the Maxwell relations are "just a sneaky way of saying that the mixed partial derivatives of the function U commute". see https://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/

(The fact that partial derivatives commute is known as Schwarz' theorem (see https://en.wikipedia.org/wiki/Maxwell_relations . Schwarz' theorem is simply a way of stating "the fact that a function S doesn’t change when we go around a parallelogram" Classical Mechanics versus Thermodynamics (Part 2) | Azimuth)

Abstract

For a simple system with fixed particle number, the equilibrium states form a two-dimensional manifold $\Sigma$, and $S$, $V$, $T$, $P$ are smooth functions on it. The first law in the form $dU = T\,dS - P\,dV$ is a statement about one-forms on $\Sigma$. Taking the exterior derivative and using $d^2 = 0$ gives

$$0 = d(dU) = dT\wedge dS - dP\wedge dV, \qquad\text{that is}\qquad dT\wedge dS = dP\wedge dV.$$

This single equation of two-forms contains all four Maxwell relations. On a two-dimensional manifold every two-form is a multiple of $dx\wedge dy$ for any local coordinates $(x,y)$, and the equation reads

$$\frac{\partial(T,S)}{\partial(x,y)} = \frac{\partial(P,V)}{\partial(x,y)}$$

in terms of Jacobians. Choosing $(x,y)$ to be one of the pairs $(S,V)$, $(T,V)$, $(S,P)$, $(T,P)$ gives the four relations. For example, with $(x,y) = (T,V)$ we have $dT\wedge dS = (\partial S/\partial V)_T\, dT\wedge dV$ and $dP\wedge dV = (\partial P/\partial T)_V\, dT\wedge dV$, hence $(\partial S/\partial V)_T = (\partial P/\partial T)_V$. The same equation also says that the map $(T,S)\mapsto(P,V)$ preserves oriented area, so any reversible cycle encloses the same area in the $T$-$S$ plane and in the $P$-$V$ plane (the net heat absorbed equals the net work done).

A more global way to state this uses contact geometry. On the five-dimensional space with coordinates $(U, S, V, T, P)$ take the contact form

$$\theta = dU - T\,dS + P\,dV.$$

The equilibrium states of a given substance form a two-dimensional Legendre submanifold $L$, meaning $\theta|_L = 0$. Projecting out $U$, the form $\omega = dT\wedge dS - dP\wedge dV$ is a symplectic form on the four-dimensional space $(T,S,P,V)$, and $\theta|_L = 0$ implies $\omega|_L = -d\theta|_L = 0$, so the projection of $L$ is a Lagrangian submanifold. The Maxwell relations are the statement that the state space is Lagrangian. The thermodynamic potentials $U(S,V)$, $A(T,V) = U - TS$, $H(S,P) = U + PV$, $G(T,P) = U - TS + PV$ are generating functions of the same Lagrangian submanifold in different coordinate charts, related by Legendre transformations. This is the same structure as in Hamiltonian mechanics, where $\omega = dp\wedge dq$ and generating functions of canonical transformations play the role of the potentials. With more variables, for example particle number $N$ and chemical potential $\mu$ with $dU = T\,dS - P\,dV + \mu\,dN$, the symplectic form acquires the term $d\mu\wedge dN$ and the same argument gives the additional Maxwell relations.

Why is it interesting?

The Maxwell relations encode useful relationships between notions of thermodynamics.

Path to this page

1. Calculus
2. Linear Algebra
3. Newton's Second Law
4. Statistics
5. Vector Calculus
6. Entropy
7. Newtonian Mechanics
8. Energy
9. Thermodynamics