Maxwell Relations

$\displaystyle dU = TdS - pdV$    
Exterior derivative,    
$\displaystyle ddU = 0 = dT \wedge dS - dp \wedge dV$    
$\displaystyle \boxed{dT \wedge dS = dp \wedge dV}$    

Pick two variables to relate: for example, U(S,V).

$\displaystyle dT = \frac{\partial T}{\partial S}dV \wedge dS + \frac{\partial T}{\partial V} dV$ (159)
$\displaystyle \frac{\partial T}{\partial V}dV \wedge dS = \frac{\partial p}{\partial S} dS \wedge dV$ (160)
$\displaystyle \rightarrow \boxed{\frac{\partial T}{\partial V} = - \frac{\partial p}{\partial S}}$ (161)


Classical Ideal Gas

$\displaystyle \mathcal{Z} = \frac{V}{h^3} \int d^3 p \bigg[e^{-\beta p^2/2m} \bigg] = \frac{V}{h^3}\sqrt{\frac{2\pi m}{\beta}}^3$    
$\displaystyle \mathcal{Z} = \frac{V^N}{h^{3N}}\bigg(\frac{2\pi m}{\beta}\bigg)^{3N/2} = V^N\bigg( \frac{2 \pi m}{h^2 \beta} \bigg)^{3N/2}$    

$\displaystyle \mathcal{Z} = \frac{V^N}{\lambda_{th}^{3N}}$ (162)
$\displaystyle A = -kT \ln{\mathcal{Z}}$ (163)
$\displaystyle S = -\frac{\partial A}{\partial T}$ (164)


$C_p$, $C_v$, $\alpha_p$, $k_T$ definitions

$C_p$, $C_v$, $\alpha_p$, $k_T$ relations

$\displaystyle C_v = C_p - \frac{v\alpha^2_p}{k_T}$ (165)
$\displaystyle k_T = -\frac{1}{v}\frac{\partial v}{\partial p}\big\vert _p$ (166)
$\displaystyle \alpha_p = -\frac{1}{v}\frac{\partial v}{\partial T}\big\vert _p$ (167)


$\displaystyle \nonumber$   Expand at constant T    
$\displaystyle dW = -P dV$    
$\displaystyle W = - \int_{v_i}^{v_f} P dV$    
$\displaystyle = - \int_{v_i}^{v_f} \frac{NT}{V}dV$    
$\displaystyle \boxed{W = NT \ln{\frac{v_i}{v_f}}}$    

Adiabats Adiabatic lines connect lines of different isotherms. No heat is allowed in or out of the system. Work on a gas with no heat escape means the total energy increases.

$\displaystyle U = \frac{f}{2}NT \rightarrow dU = \frac{f}{2} NdT$    
$\displaystyle \frac{f}{2} N dT = - P dV \rightarrow \frac{f}{2}N dT = -\frac{NT}{V}dV$    
$\displaystyle \frac{f}{2}\frac{dT}{T} = - \frac{dV}{V} \rightarrow VT^{f/2}=V_0T_0^{f/2}$    
$\displaystyle \boxed{ P V^{\gamma} = \text{ const; } \gamma = \frac{f+2}{f}}$