Thermodynamic potentials

We’ve been working with systems that have either been isolated from the surroundings or have tightly controlled heat exchange with the surroundings.

Said another way, we’ve been told exactly what the heat exchange is for each of the systems we’ve studied.

Real life doesn’t work that way: often we wish to study systems where we have constant temperature or pressure or volume rather than constant (or well-described) energy.

Internal Energy

Let’s start by summarizing what we know about internal energy. We found that three things can contribute to internal energy: mechanical work, heat, and addition of particles:

$$dU = đW + đQ + \mu dN$$

This is called the “fundamental equation of thermodynamics.” For reversible processes, we found in Lecture 5 that $đW$ and $đQ$ can be rewritten so that

$$dU = TdS + Jdx + \mu dN.$$

Natural variables

We say that $S,x, N$ are the natural variables of $U$. Why are $S, x, N$ so important that they get their own name? Because all the other thermodynamic quanitites can be expressed as partial derivatives of $U$ with respect to the natural variables!!

$T = \frac{\partial U}{\partial S}\bigg)_{N,x}$ $J = \frac{\partial U}{\partial x}\bigg)_{N,S}$ $\mu = \frac{\partial U}{\partial N}\bigg)_{x,S}$

Internal energy is minimized when it’s natural variables are held constant

But you might be working with a system where the variables $S, x, N$ are not at all natural. Instead, you might be working with a system where $T,x,N$ are constant. In that case, internal energy is not minimized at equilbrium and is therefore not a meaningful thermodynamic potential. What thermodynamic potential is minimized under constant $T,x,N$?

Helmholtz Free Energy: constant (N,x,T)

Let’s define a function $F = U - TS$. Then

$dF = dU - TdS - SdT$ $\Rightarrow dF = Jdx - SdT + \mu dN.$

The natural variables of Helmholtz Free Energy are $N, x, T$ because all other thermodynamic quantities can be found in terms of partials of $F$ with respect to the natural variables:

$J = \frac{\partial F}{\partial x}\bigg)_{N,T}$ $S = -\frac{\partial F}{\partial T}\bigg)_{N,x}$ $\mu = -\frac{\partial F}{\partial N}\bigg)_{x, T}$

The Helmholtz Free Energy is minimized when it’s natural variables $N, x, T$ are held constant.

Enthalpy (S, N, J)

Let’s define a new function $H = U - Jx$. Then

$dH = dU - Jdx - xdJ$ $\Rightarrow dH = TdS - xdJ + \mu dN.$

The natural variables of Enthalpy are $S, J, N$ because all other thermodynamic quantities can be found in terms of partials of $H$ with respect to the natural variables:

$x = \frac{\partial H}{\partial J}\bigg)_{N,S}$ $T = -\frac{\partial H}{\partial S}\bigg)_{N,x}$ $\mu = -\frac{\partial H}{\partial N}\bigg)_{x, S}$

The Enthalpy is minimized when it’s natural variables $S, N, J$ are held constant.

Gibbs Free Energy: constant (N, J, T)

We can do exactly the same thing for a new thermodynamic potential, Gibbs Free Energy.

$$G = U - TS - Jx$$

I’ll spare you the algebra and just write the final differential.

$$dG = -S dT - x dJ + \mu dN$$

Again, the natural variables are $N, J, T$ and we can write $S, x, \mu$ in terms of these variables. Give it a shot!

Maxwell Relations

We can also write various partial derivatives of thermodynamic quantities in terms of other partial derivatives using what we know about exact differentials. Let’s consider the mixed partials of the Helmholtz Free Energy:

$$\frac{\partial^2 F}{\partial T\partial x} = \frac{\partial^2 F}{\partial x\partial T}$$

We know from above that $J = \frac{\partial F}{\partial x}\bigg)_{N,T}$ and $S = \frac{\partial F}{\partial T}\bigg)_{N,x}$. Therefore,

$$\frac{\partial J}{\partial T}\bigg)_{N,x} = \frac{\partial S}{\partial x}\bigg)_{N,T}$$