Thermodynamics

Equalibrium

• Properties (measurements) do not change over obeservation time
• e.g.

0th Law

• AB, BC AB
• As the consequence:
  
  
  We can define as Temperature. And is Equation of state
• $T=\frac{(PV){sys}}{(PV){tri}}\times 273.16K$

1st Law

1. Adiabatic wall: no heat exchange
   diathermic wall: can heat exchange
   isolated system

2. State1State2: “Inner Energy”:

   State1State2: “Heat”: or

3. Quasi-static process:
   
   is the generalize force (Intensive quantity), and is the generalized displacement (Extensive quantity)

4. Heat capacity:

   1. Const. V:

   $$
   C_{V} = \left( \frac{\mathrm{d} Q}{\mathrm{d} T} \right){V}=(\frac{\partial U}{\mathrm{\partial} T}){V}
   $$

   2. Const. P:

   $$
   C_{P}=\left( \frac{\mathrm{d}Q}{\mathrm{d}T} \right){P}=\left( \frac{\partial U}{\partial T} \right){P}+P\left( \frac{\partial V}{\partial T} \right){P}=\left( \frac{\partial H}{\partial T} \right){P}
   $$

   : Enthalpy

e.g. Ideal Gas: Jouler’s Expension:


$$C_{P}-C_{V} =P \left( \frac{\partial V} {\partial T} \right){P}=Nk{B}$$

Heat Engine

2nd Law

1. Kevin: No process is possible whose sole effect is complete conversion of heat to work
   Clausius: No process is possible whose sole effect is transfer of heat from colder to hotter body
2. Carrot Engine:
   • Cycle, reversible
   • Operates between two heat baths at temperatue and
   • Carrot Thereom: Carrot is the best
3. Clausius Inequality
   • Carrot:
   • Other cycles: (: temperature of heat bath)

Entropy

1. Reversible:
   potential: entropy
   
   Infinitesimal:
   
2. Inreversible:
   
   Infinitesimal:
   
3. Adiabatic process: ,

   • e.g.1. Jouler’s Expension:
     

   • e.g.2. ![[1.Thermodynamics 2023-10-30 18.35.49.excalidraw]]
     
     Consider a const volumn process:
     
     

     Choose a reversible process to calculate

   • 
     : natural variable S, V.

Systems of variable particles

• Can happen: open system; phase transition
• : Extensivity,
  1. Thermodynamic limit: fixed.
  2. Short-ranged interaction
  3. Consequence of extentivity:
     
     : 1st order homogeneous function.
     : scale factor.
     
     Differential and
     __Gibbs-Duhem relation: __

Thermodynamic potentials

1. , we are at liberty to choose other variables.
2. Enthalpy:
   
3. Helmholtz Free Energy:
   
4. Gibbs Free Energy:
   
5. Grand potential:
   (Gibbs-Duhem relation)

Towards Equal

1. fixed,
2. fixed,
3. fixed,

Maxwell’s Relation

$$\left( \frac{\partial U}{\partial S} \right){V}=T,\ \left( \frac{\partial U}{\partial V} \right){S}=-p,\ \therefore \frac{\partial ^{2}U}{\partial S \partial V}=\left( \frac{\partial T}{\partial V}\right){S}=-\left( \frac{\partial P}{\partial S} \right){V}$$
Relate quantities to experimentally measurable quantities.

• e.g. Heat Capacity
  $$C_{V}=\left( \frac{\partial U}{\partial T} \right){V}=T\left( \frac{\partial S}{\partial T} \right){V},C_{p}=\left( \frac{\partial H}{\partial T} \right){p}=T\left( \frac{\partial S}{\partial T} \right){p}\text{And }\left( \frac{\partial S}{\partial T} \right){p}=\left( \frac{\partial S}{\partial T} \right){V}+\left( \frac{\partial S}{\partial V} \right){T}\left( \frac{\partial V}{\partial T} \right){p}\therefore C_{p}-C_{V}=T\left( \frac{\partial S}{\partial V} \right){T}\left( \frac{\partial V}{\partial T} \right){p}=T\left( \frac{\partial p}{\partial T} \right){V}\left( \frac{\partial V}{\partial T} \right){p}$$
• Response Function
  1. Extensivity:
  2. Conpressibility: $\kappa_{T}=- \frac{1}{V}\left( \frac{\partial V}{\partial p} \right){T}$\therefore C{p}-C_{V}=\frac{TV\alpha^{2}}{\kappa_{T}}$$

Phase Equalibrium

• Partition into uniform subsystem.
  
  Maximize , subject to constraint:
  

• Lagrange Multiphier:
  
  
  
  

• Stability condition: is maximiazed.
  
  
  
  
  
  e.g.

  Choose as independed variables: :
  $$\text{So: } \delta S=\left( \frac{\partial S}{\partial T} \right){V}\delta T+\left( \frac{\partial S}{\partial V} \right){T}\delta V,\ \delta p=\left( \frac{\partial p}{\partial T} \right){V}\delta T+\left( \frac{\partial p}{\partial V}\right){T}\delta V \Rightarrow\left( \frac{\partial S}{\partial T} \right){V}(\delta T)^{2}+\left[ \left( \frac{\partial S}{\partial V} \right){T}-\left( \frac{\partial P}{\partial T} \right){V} \right]\delta T\delta V-\left( \frac{\partial p}{\partial V} \right){V}(\delta V)^{2}\geq 0$$

  $$\text{Maxwell Relation: }\left( \frac{\partial S}{\partial V} \right){T}=\left( \frac{\partial P}{\partial T} \right){V},\left( \frac{\partial S}{\partial V} \right){T}=\frac{C{V}}{T},-\left( \frac{\partial p}{\partial V} \right){T}=\frac{1}{V\kappa{T}}\therefore C_{V}\geq 0,\ \kappa_{T}\geq 0$$