Lecture 2 Typed

The fundamental thermodynamic relation

First Law

$$dE=\delta Q+\delta W$$ $$\delta W=pdV$$ $$\delta Q=TdS\text{ ( Quasistatic Reversible Process)}$$ $$dE=TdS-pdV=\text{"heat" + "work"}$$

If particle number is not held constant:

$$dE=TdS-pdV+\mu dN$$ $$dS=\frac{1}{T}dE+\frac{P}{T}dV-\frac{\mu }{T}dN⟹S(E,V,N)$$ $$\frac{1}{T}=(\partial S)$$ $$\frac{1}{T}={\left(\frac{\partial S}{\partial E}\right)}_{\text{V,N}},\frac{P}{T}={\left(\frac{\partial S}{\partial R}\right)}_{\text{E,N}},\frac{\mu }{T}=-{\left(\frac{\partial \sqrt{y}}{\partial N}\right)}_{E,V}$$

Example: Ideal gas (assuming fixed N)

$$E=\frac{3}{2}NkT;PV=NkT$$ $$\frac{1}{T}=\frac{3}{2}\frac{Nk}{E}+Nk\frac{dV}{V}$$

S is entropy? yes!

$$dS=\frac{3Nk}{2}\frac{dE}{E}+Nk\frac{dV}{V}$$ $$dS=\frac{3Nk}{2}\frac{dE}{E}+Nk\frac{dV}{V}$$ $$S(E,V)\frac{3}{2}Nk\ln \frac{E}{E_0}+Nk\ln \frac{V}{V_0}$$

Where $E_0\&V_0$ are arbitrary

Often Needed:

$$S(T,V)=\frac{3}{2}Nk\ln \frac{T}{T_0}+Nk\ln \frac{V}{V_0}$$

and

$$\Delta S(1\to 2)=\frac{3}{2}Nk\ln \left(\frac{T_2}{T_1}\right)+Nk\ln \left(\frac{V_2}{V_1}\right)$$

This approximation works only at high temp and low density. Falls apart at low temps.

Gibbs - Duharm Relation

$$S=S(E,V,N)$$

Maximum at fixed E,V,N

If you double the size of the system , you will ideally have double the entropy.

$$dS={\left(\frac{\partial S}{\partial E}\right)}_{\text{V,N}}dE+{\left(\frac{\partial S}{\partial V}\right)}_{\text{E,N}}dV+{\left(\frac{\partial S}{\partial \mu }\right)}_{\text{ E,V}}dN$$

Extensive State Function: (?)

$$E=E(S,V,N)$$

$\lambda$ is the size of the system?

extensibility?

$$\frac{\partial E}{\partial (\lambda S)}\frac{\partial \lambda S}{\partial \lambda }+\frac{\partial E}{\partial (\lambda V)}+\frac{\partial E}{\partial (\lambda V)}+\frac{\partial E}{\partial (\lambda N)}\frac{\partial (\lambda N)}{\partial \lambda }=E(S,V,N)$$ $$\frac{\partial E}{\partial (\lambda S)}S+\frac{\partial E}{\partial (\lambda V)}V+\frac{\partial E}{\partial (\lambda N)N}=E(S,V,N)$$

Then set $\lambda =1$

$$E(S,V,N)=TS-PV+\mu N$$ $$dE=TdS+SdT-pdV-Vdp+\mu dN+Nd\mu$$

Compare with first law:

$$SdT-Vdp+Nd\mu =0$$

This is the Gibbs-Duham relation

The third law of thermodynamics

$$\underset{T\to 0}{\lim }S=0$$

(With exceptions)

Quasi-static case:

$$\frac{\delta Q}{T}=dS$$

@ constant volume

$$\delta Q=C_V(T)dT$$

The heat capacity isn’t always a constant, it may depend on temperature.

$${\int }_{T_1}^{T_2}dS={\int }_{T_1}^{T_2}\frac{\delta Q}{T}={\int }_{T_1}^{T_2}\frac{C_v(T)}{T}dT$$ $$S(T_2,V)-S(T_1,V)={\int }_{T_1}^{T_2}\frac{C_V(T)}{T}dT$$ $$S(T_1,V)=\underset{T_1\to 0}{\lim }{\int }_{T_1}^{T_2}\frac{C_V(T)}{T}dT⟹\underset{T\to 0}{\lim }{C}_V(T)=0$$

Similarly, you can repeat the same for constant pressure P

$$\underset{T\to 0}{\lim }{C}_p(T)=0$$

Generalized Thermodynamic Potentials

Equilibrium with Gradients

System small compared to a large reservoir.

Subcase: Hemholtz Free energy

• Fixed volume

Maxwell Relation and Applications

Why does this Maxwell dude do literally everything