Thermodynamics

Maxwell Relations

$$dU = TdS - pdV$$ Exterior derivative,

$$ddU = 0 = dT \wedge dS - dp \wedge dV$$

$$\boxed{dT \wedge dS = dp \wedge dV}$$

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

$$dT = \frac{\partial T}{\partial S}dV \wedge dS + \frac{\partial T}{\partial V} dV$$ (159)

$$\frac{\partial T}{\partial V}dV \wedge dS = \frac{\partial p}{\partial S} dS \wedge dV$$ (160)

$$\rightarrow \boxed{\frac{\partial T}{\partial V} = - \frac{\partial p}{\partial S}}$$ (161)

Etc.

Classical Ideal Gas

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

$$\mathcal{Z} = \frac{V^N}{h^{3N}}\bigg(\frac{2\pi m}{\beta}\bigg)^{3N/2} = V^N\bigg( \frac{2 \pi m}{h^2 \beta} \bigg)^{3N/2}$$

$$\mathcal{Z} = \frac{V^N}{\lambda_{th}^{3N}}$$ (162)

$$A = -kT \ln{\mathcal{Z}}$$ (163)

$$S = -\frac{\partial A}{\partial T}$$ (164)

Etc.

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

• Isothermal Compressibility:
  $k_T = - \frac{1}{V} \big( \frac{\partial V}{\partial P} \big)_T$
• Thermal Expansion Coefficient:
  $\alpha_P = - \frac{1}{V} \big( \frac{\partial V}{\partial T} \big)_P$
• Heat Capacity at constant pressure:
  $C_P = - T \big( \frac{\partial S}{\partial T} \big)_P$
• Heat Capacity at constant volume:
  $C_V = - T \big( \frac{\partial S}{\partial T} \big)_V$

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

$$C_v = C_p - \frac{v\alpha^2_p}{k_T}$$ (165)

$$k_T = -\frac{1}{v}\frac{\partial v}{\partial p}\big\vert _p$$ (166)

$$\alpha_p = -\frac{1}{v}\frac{\partial v}{\partial T}\big\vert _p$$ (167)

Isotherms

$$\nonumber$$

$$dW = -P dV$$

$$W = - \int_{v_i}^{v_f} P dV$$

$$= - \int_{v_i}^{v_f} \frac{NT}{V}dV$$

$$\boxed{W = NT \ln{\frac{v_i}{v_f}}}$$

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.

$$U = \frac{f}{2}NT \rightarrow dU = \frac{f}{2} NdT$$

$$\frac{f}{2} N dT = - P dV \rightarrow \frac{f}{2}N dT = -\frac{NT}{V}dV$$

$$\frac{f}{2}\frac{dT}{T} = - \frac{dV}{V} \rightarrow VT^{f/2}=V_0T_0^{f/2}$$

$$\boxed{ P V^{\gamma} = \text{ const; } \gamma = \frac{f+2}{f}}$$