\(\Omega(U,V,N)\) 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 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 |
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.
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: \(dW_{on} = -p dV\)
\(dQ\) is a small amount of heat added to a system.
The change in internal energy is then \(dU = dQ + dW_{on}\).
Heat capacity 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 \(C_P = \frac{dU}{dT} + p \frac{dV}{dT}\)
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
\(\mu = \left( \frac{d F(T,V,N)}{dN} \right)_{T,V}\)
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 \(T,p,N\):
\(dS(U,V,N) = \frac{1}{T} dU + \frac{p}{T} dV - \frac{\mu}{N} dN\)
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