Final exam review material

Boltzman factor

$e^{-\beta E} = e^{-\frac{E_i}{kT}}$

Probability a system will occur in a given state is proportional to the Boltzman factor

The Partition function

$Z(T) = \sum_{i=1}^{k} e^{-\beta E}$

Can be written as a sum over all distinct energies with $\Omega(E)$ being the number of states with energy $E$. Known as multiplicity

$Z(T) = \sum_{\text{Energies E}} \Omega(E)e^{-\beta E}$

Average Energy

$\bar{E} = \sum_{i=1}^{k}P(i)E_i = \sum_{i=1}^{k} \frac{e^{-\beta E_i}}{Z} E_i = \frac{1}{Z}\sum_{i=1}^{k}E_i e^{\beta E_i}$

Can also be written as:

$\bar{E} = \sum_{E}^{k}P(E)E = \sum_{E} \frac{\Omega(E)e^{-\beta E}}{Z}E = \frac{1}{Z}\sum_{E}\Omega(E)e^{-\beta E}E$

or

\bar{E} = -\frac{1}{Z}\frac{\partial Z}{\partial \beta}|_{V,N} = -\frac{\partial (\ln{Z})}{\partial \beta}|_{V,N}$

where

$Z = \sum_{\text{energies E}}\Omega(E)e^{-\beta E}$

Microcannonical / Cannonical approach

Hold at constant volume $V$ and number of particles $N$

Microcannonical

• fix internal energy $U$
• find multiplicity $\Omega(U,V,N)$
• $S = k \ln{\Omega}$
• $T^{-1} = \frac{\partial S}{\partial U}|_{V,N}$

Cannonical

• fix temperature $T$
• find partition function $Z(T,V,N)$
• $F = -kT\ln{Z}$
• $U = -\frac{\partial \ln{Z}}{\partial \beta}|_{V,N}$

Partition function of N indistinquishable particles

$Z_N = (Z_1)^N$

Dilute limit

A system is in the dilute limit if the number of thermodynamically accessible states is much, much larger than the number of particles. In this limit, the probability of any 2 particles being in the same state is negligibly small.

Bosons vs Fermions

Important note: Bosons can share states whereas fermions cannot share states due to the Pauli exclusion principal.

Gibbs factor

Thermodynamic identity for $U$ implies then,

$\Delta U_R = T\Delta S_R - P\Delta V_R + \mu\Delta N_R$

then assuming volume doesn't change, let $\Delta U_R = -E$ and $\Delta N_R = -N$

$-E = T\Delta S_R - \mu N$

$\rightarrow \boxed{\Delta S_R = -\frac{E}{T} + \frac{\mu N}{T}}$

The probability of this state, also known as the Gibb's factor, is proportional to

$\boxed{P(N,E)\approx e^{\Delta S_R/k} = e^{-\beta(E-\mu N)}}$

Distributions

Knowing the distribution $n(E)$, one can calculate the average number of particles in a system

$N = \sum_{\text{states }i} n(E_i)$

and the average internal energy of a system

$U = \sum_{\text{states }i} n(E_i)E_i$

Fermi-Dirac distribution

$Z = 1 + e^{-\beta(\epsilon - \mu)}$

Average number of particles in state of energy $\epsilon$:

$\bar{n_{\text{FD}}} = \frac{1}{e^{\beta(\epsilon - \mu)} + 1}$

Bose-Einstein distribution

$Z = \frac{1}{ 1 + e^{-\beta(\epsilon - \mu)}}$

Average number of particles in state of energy $\epsilon$:

$\bar{n_{\text{FD}}} = \frac{1}{e^{\beta(\epsilon - \mu)} - 1}$

Boltzman distribution

$\bar{n}_{\text{boltzman}} = \frac{1}{e^{\beta(\epsilon - \mu)}} = e^{-\beta(\epsilon - \mu)}$

Density of states

Density of states, $g(E)$ is the number of 1-particle states per unit of energy.

$g(E) = \frac{\Delta k}{\Delta E}$

Number of states between 2 energies:

$\boxed{k(E_1,E_2) = \int_{E_1}^{E_2}dk = \int_{E_1}^{E_2}{g(E)} dE}$

Knowing the density of state $g(E)$ and how many particles occupy each state $n(E)$ allow us to determine both the total number of particles in the system and what the internal energy of the system is.

$\text{\# of particles: }N = \sum_{\text{states }i}n(E_i) = \int n(E)dk = \int n(E) g(E) dE$

$\text{internal energy: } U = \sum_{\text{states} i}E_in(E_i) = \int En(E)dk = \int En(E)g(E)dE$

Helmholtz free energy

Helmholtz free energy measures the useful work obtainable from a closed thermodynamic system at constant temeprature and volume.

$\boxed{F = U - TS}$

Alternate forms:

$\Delta F = \Delta U - T\Delta S$

$\Delta F = Q + W - T\Delta S$

$F = -kT\ln Z$

• $F$ must stay the same if a process is reversible, $Q = T\Delta S$
• $F$ must decrease if a process is irreversible, $Q < T\Delta S$

Gibbs free energy

Gibbs free energy is a thermodynamic potential used to calculate maximum of reversible work performed by a thermodynamic system at constant temperature and pressure.

$\boxed{\Delta G = \Delta H - T\Delta S}$

Alternate forms:

$G = U + PV - TS$

$G = F + PV$

• Spontaneous process will have $\Delta G < 0$
• Non-Spontaneous process will have $\Delta G > 0$
• Phase with lower Gibbs free energy is more stable

Entropy

Entropy is an extensive property (depends on size of system) of a thermodynamic system.

$\boxed{S = k_B\ln{\Omega}}$

• Spontaneous process will have $\Delta S > 0$ at constant volume and energy

Batteries

Heat generated

$\boxed{Q = -T\Delta S = \Delta G - \Delta H}$

Voltage of battery

$\boxed{\text{voltage} = \frac{work}{2\times N_A \times e^-}}$

Entropy in universe

$\Delta S = \frac{-Q}{T} = \frac{\Delta H - \Delta G}{T}$

Power emitted by hot surfaces

How fast energy is leaking out

$\dot{U} = \frac{\text{energy}}{\text{time}} = (\text{intensity})(\text{area})$

Intensity

General formula

$I = \text{Intensity} = \frac{\text{energy}}{(\text{time})(\text{area})}$

For spheres

$I \propto T^4R^2$

Chemical potential

Chemical potential is an intensive quantity (independant on size of system) that is the same when two systems are in diffusive equilibrium

$\boxed{\mu \equiv -T\left.\frac{\partial S}{\partial N}\right\vert_{U,V}}$

Alternate forms:

$\mu = \left. \frac{\partial (-kT \ln Z)}{\partial N}\right\vert_{T,V,N}$

• particles flow from high potential to low potential
• relates to free energy by $\mu_i = \left.(\frac{\partial F}{\partial N_i})\right\vert_{T,V,N_{j\neq i}}$ where $F = -kT\ln Z$

Stirling's approximation

$\boxed{\ln n! = n\ln n - n + O(\ln n)}$