Open systems, free energy and spontaneity

An open system tracks its own measure, not the whole universe

Intuition

The second law is about the entropy of the whole universe. That is awkward to track. If you only watch your own room, you can still tell whether things are going the right way, as long as you also know what the outside is doing. Free energy is the bookkeeping for “my room only”.

An isolated system has fixed energy, volume and particle number: $U, V, N$. An open system trades one of these with its surroundings. The lecture assumes the surroundings are so much larger than the system that their state stays unchanged: an infinite reservoir.

• If the system exchanges heat, $U$ is no longer fixed. At equilibrium the system’s temperature $T$ equals the surroundings’.
• If it can expand or contract, $V$ is no longer fixed. At equilibrium its pressure $p$ equals the surroundings’.
• If it can swap particles, $N$ is no longer fixed. At equilibrium its chemical potential $\mu$ equals the surroundings’.

The same argument works for any pair of generalised force and displacement.

At constant T, the Helmholtz free energy A decides

Put a system in a big bath at fixed $T$. The total entropy cannot fall:

$dS_{\text{total}} = dS + dS_{\text{bath}} \ge 0$

The big bath is always close to equilibrium, so it gains the heat the system loses: $dS_{\text{bath}} = -dQ/T$. Substitute and rearrange:

$dS \ge \frac{dQ}{T}$

This is the Clausius version of the second law. Now use the first-law balance $dQ = dU - dW$ (it is $\Delta U = Q + W$ rearranged):

$dS \ge \frac{dU}{T} - \frac{dW}{T} \quad\Rightarrow\quad dU - T\,dS \le dW$

Define the Helmholtz free energy

$A \equiv U - TS$

At constant $T$, $dA = dU - T\,dS$, so the inequality reads

$dA \le dW$

In words, $A$ is the maximum work that can be extracted from the system at constant $T$. The $A$ is for Arbeit, German for work. Think of $U$ as your pay, $TS$ as tax that disorder takes, and $A$ as take-home pay available for useful work.

If the volume is constant and there is no non-$pV$ work, then $dW = -p\,dV = 0$ and

$dA \le 0$

So at constant $T$ and $V$ a spontaneous change lowers $A$, and the system tries to minimise its free energy. The lecture stresses that we are still maximising the entropy of the universe. We are just tracking properties of the system.

Enthalpy and Gibbs free energy handle constant pressure

State functions depend only on the equilibrium state, not on history. For pV work only, the internal energy obeys

$dU = T\,dS - p\,dV$

At constant volume, $dU = dQ_{\text{rev}}$. And $\left(\dfrac{dU}{dS}\right)_V = T$.

Most chemistry runs at constant pressure, not constant volume, so we want a partner for $U$ there. Define the enthalpy

$H \equiv U + pV$

Differentiate (the slide shows each step):

$dH = d(U+pV) = T\,dS - p\,dV + p\,dV + V\,dp = T\,dS + V\,dp$

At constant $p$ with no non-$pV$ work, $dH = dQ_{\text{rev}}$. When a reaction is called exothermic or endothermic, the lecture says we usually mean $\Delta H$, not $\Delta U$.

The Helmholtz partner at constant pressure is the Gibbs free energy

$G \equiv U - TS + pV = H - TS = A + pV$

From $dA \le dW$ the slide gets $dG \le dW + p\,dV + V\,dp$. At constant $T$ and $p$ this becomes $dG \le dW + p\,dV = dW_{\text{non-expansion}}$. With no non-$pV$ work, the test for spontaneity at constant $T$ and $p$ is

$\Delta G \le 0$

One table gathers every case

Rows are the volume or pressure condition. Columns are adiabatic ($dQ = 0$) or isothermal ($dT = 0$).

	$dQ = 0$ (adiabatic)	$dT = 0$ (isothermal)
$dV = 0$ (isochoric)	$dS \ge 0$	$dA \le 0$
$dp = 0$ (isobaric)	$dH \le 0$	$dG \le 0$

Each quantity is a system-based test of whether the entropy of the universe is rising. All state functions can also be written in terms of the partition function, for example

$G = -NkT\ln\frac{q}{N}$

At constant $T$ and $N$, $G$ is proportional to $-\ln q$. The lecture reads this as: equilibrium maximises the number of accessible energy states.

Pitfall

Do not use $\Delta G \le 0$ when $T$ or $p$ is not held constant. Look up the row and column first. Adiabatic and isochoric is $dS \ge 0$, not $dA$ or $dG$.

Exam detail

Lecture 16 summary: open systems shift constraints, for example from constant $U$ to constant $T$, or from constant $V$ to constant $p$. $A$, $G$ and $H$ govern spontaneity in different kinds of open system. They are system-based measures of whether the entropy of the universe is increasing. Statistical mechanics lets every state function be written from the partition function.

Recall

Which free-energy-type quantity decides spontaneity at constant T and p, and what is the condition?

The Gibbs free energy $G = H - TS$. With no non-$pV$ work, a spontaneous change has $\Delta G \le 0$.