Classical Thermodynamics

Entropy

$\Omega(U,V,N)$ is the number of ways that one can get certain values of $U,V, N$ .

$S(U,V,N) = k\ln \Omega(U,V,N)$

For two unconnected systems, $S_{tot} (U_1,U_2,V_1,V_2,N_1,N_2) = S_1(U_1,V_1,N_1) + S_2(U_2,V_2,N_2)$

Reversible processes: $\Delta S_{tot} = 0$

Irreversible processes $\Delta S_{tot} > 0$

State functions

State functions are quantities that describe the system and its relationship with the environment.

Properties of the system	Relationship to the environment
$U$ : internal energy	$T$ : temperature
$V$ : volume	$p$ : pressure
$N$ : number of particles	$\mu$ : chemical potential

Property	What it means
Temperature	How much the system can give energy to the environment
Pressure	How much the system can take volume from the environment
Chemical potential	How much the system can give particles to the environment

Equilibrium

When a system and its environment have settled down and their state functions no longer change, we say that they are in equilibrium. Equilibrium occurs when the total entropy is maximized, subject to any external constraints.

Thermodynamic functions

The total entropy is always maximized in equilibrium. In equilibrium, the entropy is a function of the internal energy, volume and number of particles.

Often, we want to consider a system connected to something much larger. The system can exchange something with the environment to equilibrate both. In that case, we can use the free energy of the system, which is minimized. In equilibrium, the free energy is now a function of the environmental state functions.

Function	Exchanges	Formula	Name	Equilibrium condition
$S(U,V,N)$	Nothing	$k\ln\Omega(U,V,N)$	Entropy	Maximized
$F(T,V,N)$	Energy $U$	$U-TS$	Helmholtz free energy	Minimized
$G(T,p,N)$	Energy $U$ and volume $V$	$U-TS+pV$	Gibbs free energy	Minimized

Work and heat

Work: $dW_{on} = -p dV$

$dQ$ is a small amount of heat added to a system.

The change in internal energy is then $dU = dQ + dW_{on}$ .

Heat Capacity

Heat capacity is the amount of energy it takes to change the temperature. $C = \frac{dQ}{dT} = \frac{dU + pdV}{dT}$

If the system is heated/cooled at constant volume, $\frac{dV}{dT}=0$ , so $C_V = \frac{dU}{dT}$

If the system is heated/cooled at constant pressure, then $C_P = \frac{dU}{dT} + p \frac{dV}{dT}$

Relationship between environment and system variables.

These relationships define $T, p, \mu$

$\frac{1}{T} \equiv \left( \frac{d S(U,V,N)}{dU} \right)_{V,N} = \frac{\partial S(U,V,N)}{\partial U}$

$\frac{p}{T} \equiv \left( \frac{d S(U,V,N)}{dV} \right)_{U,N}$

$\frac{-\mu}{T} \equiv \left( \frac{d S(U,V,N)}{dN} \right)_{U,V}$

There are many other relationships like this for $F$ and $G$ . In this class, we will mainly also use

$\mu = \left( \frac{d F(T,V,N)}{dN} \right)_{T,V}$

Fundamental relation of thermodynamics

From the chain rule,

$dS(U,V,N) = \frac{\partial S}{\partial U} dU + \frac{\partial S}{\partial V} dV + \frac{\partial S}{\partial N} dN$

When the system is in equilibrium, we can fill in the definitions of $T,p,N$ :

$dS(U,V,N) = \frac{1}{T} dU + \frac{p}{T} dV - \frac{\mu}{N} dN$

Thermodynamic processes

Quasistatic: the system is in equilibrium

Reversible: the total entropy change is zero

Isothermal: the temperature of the system does not change

Isobaric: the pressure does not change

Isochoric: the volume does not change

Adiabatic: no heat comes in or out of the system