Thermodynamics Statistical Basis Equation Formula Latex Code and Summary

Thermodynamics Statistical Basis

Tags: #physics #thermodynamics

Equation

$$P=N!\prod_i\frac{g_i^{n_i}}{n_i!} \\ n_i=\frac{N}{Z}g_i\exp\left(-\frac{W_i}{kT}\right) \\ Z=\sum\limits_ig_i\exp(-W_i/kT) \\ Z=\frac{V(2\pi mkT)^{3/2}}{h^3} \\ \text{Entropy in Thermodynamic Equilibrium} \\ S=\frac{U}{T}+kN\ln\left(\frac{Z}{N}\right)+kN\approx\frac{U}{T}+k\ln\left(\frac{Z^N}{N!}\right) \\ \text{Ideal gas} \\ S=kN+kN\ln\left(\frac{V(2\pi mkT)^{3/2}}{Nh^3}\right)$$

Latex Code

                                 P=N!\prod_i\frac{g_i^{n_i}}{n_i!} \\
            n_i=\frac{N}{Z}g_i\exp\left(-\frac{W_i}{kT}\right) \\
            Z=\sum\limits_ig_i\exp(-W_i/kT) \\
            Z=\frac{V(2\pi mkT)^{3/2}}{h^3} \\
            \text{Entropy in Thermodynamic Equilibrium} \\
            S=\frac{U}{T}+kN\ln\left(\frac{Z}{N}\right)+kN\approx\frac{U}{T}+k\ln\left(\frac{Z^N}{N!}\right) \\
            \text{Ideal gas} \\
            S=kN+kN\ln\left(\frac{V(2\pi mkT)^{3/2}}{Nh^3}\right)

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Introduction

Equation

$P=N!\prod_i\frac{g_i^{n_i}}{n_i!} \\ n_i=\frac{N}{Z}g_i\exp\left(-\frac{W_i}{kT}\right) \\ Z=\sum\limits_ig_i\exp(-W_i/kT) \\ Z=\frac{V(2\pi mkT)^{3/2}}{h^3} \\ \text{Entropy in Thermodynamic Equilibrium} \\ S=\frac{U}{T}+kN\ln\left(\frac{Z}{N}\right)+kN\approx\frac{U}{T}+k\ln\left(\frac{Z^N}{N!}\right) \\ \text{Ideal gas} \\ S=kN+kN\ln\left(\frac{V(2\pi mkT)^{3/2}}{Nh^3}\right)$

Latex Code

            P=N!\prod_i\frac{g_i^{n_i}}{n_i!} \\
            n_i=\frac{N}{Z}g_i\exp\left(-\frac{W_i}{kT}\right) \\
            Z=\sum\limits_ig_i\exp(-W_i/kT) \\
            Z=\frac{V(2\pi mkT)^{3/2}}{h^3} \\
            \text{Entropy in Thermodynamic Equilibrium} \\
            S=\frac{U}{T}+kN\ln\left(\frac{Z}{N}\right)+kN\approx\frac{U}{T}+k\ln\left(\frac{Z^N}{N!}\right) \\
            \text{Ideal gas} \\
            S=kN+kN\ln\left(\frac{V(2\pi mkT)^{3/2}}{Nh^3}\right)

Explanation

Latex code for the Thermodynamics Statistical Basis. I will briefly introduce the notations in this formulation.

$P$ : number of possibilities
$N$ : number of particles
$n$ : number of possible energy levels
$g$ : g-fold degeneracy
$n_i$ : The occupation numbers in equilibrium(with the maximum value for P)
$Z$ : State sum Z is a normalization constant
$\Delta T_{\rm k}=\frac{RT_{\rm k}^2}{r_{\beta\alpha}}x_2~~,~~ \Delta T_{\rm s}=-\frac{RT_{\rm s}^2}{r_{\gamma\beta}}x_2$ : one component in a second gives rise to decrease of the freezing point $\Delta T_{\rm s}$

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