Notes

General notes

Ideal gas law

• P = pressure of gas
• V = volume of gas
• n = number of moles
• R = gas constant, $8.314 \frac{J}{K\text{mol}}$
• T = temperature in kelvin
• N = number of atoms
• K = Boltzman's constant, $1.381\times 10^{-23}J/K$

$PV=nRT$

$PV=NKT$

During adiabatic compression

1. $PV^{\gamma} = const$, $\gamma = (f+2)/f$
2. $VT^{\frac{f}{2}} = const$

Entropy of an ideal gas

Given by the famous Sackur-Tetrode equation

$S = Nk\left(\ln\frac{VU^{f/2}}{N^{(f/2)+1}} + c \right)$

Multiplicy

Multiplicity of paramagnet

${N \choose q} = \frac{N!}{(N-q)!q!}$

Multiplicity of Einstein solid

${N+q-1 \choose q} = \frac{(N+q-1)!}{(N-1)!q!}$

Sirling approximation

Stirling's approximation is a good approximation for factorials

$\ln N! \approx N\ln(N) - N$

Chapter 5

Thermodynamic identities

$dG = -SdT + VdP + \mu dN$
$dU = TdS - PdV + \mu dN$
$dF = dU - d(ST) = -SdT - PdV + \mu dN$
$dG = dH - d(ST) = -SdT + vdP + \mu dN$

Maxwell relations

$\frac{\partial}{\partial N} (\frac{\partial U}{\partial S} |_{V,N})|_{V,S} = \frac{\partial}{\partial S} (\frac{\partial U}{\partial N} |_{V,S})|_{V,N}$

Clausius-Clapeyron equation

According to wikipedia: This equation a way of characterizing a discontinuous phase transition between two phases of matter of a single constituent.

$\frac{dP}{dT} = \frac{L}{T\Delta V}$

Gibbs free energy and chemical potential

$\mu = \left(\frac{\partial G}{\partial N}\right)_{T,P}$

$G = N\mu$

An easy way to interpret this result is that $\mu$ is just the gibbs free energy per particle