$\displaystyle\frac{ \partial }{ \partial T }=- k_B \beta ^2\displaystyle\frac{ \partial }{ \partial \beta }$

d / dT =- k_B * beta ^2* d / dbeta

$\ln \Omega = \beta E +\ln Z $

ln( Omega ) = beta * E + ln( Z )

$ \Omega = B V ^ N E ^{3 N /2}$

Omega = B * V ^ N * E ^(3* N /2)

$\ln( Z ) =\ln( B V ^ N E ^{3 N /2})- \beta E $

ln( Z ) = ln( B * V ^ N * E ^(3* N /2) ) - beta * E

$ U = E $

U = E

$ p =\displaystyle\frac{ N k_B T }{ V }$

p = N * k_B * T / V

$ U =\displaystyle\frac{3}{2} N k_B T $

U =3* N * k_B * T / 2

$ C_V = k_B \beta ^2\displaystyle\frac{\partial^2\ln Z }{\partial \beta ^2}$

C_V = k_B * beta ^2* d2Z / dbeta2

$\displaystyle\frac{1}{ k_p }=-\displaystyle\frac{ V }{ \beta }\frac{\partial^2\ln Z }{\partial V ^2}$

k_p =-1/( V *@DIFF( ln( Z ), V , 2 ) / beta )

$ k_T = k_p k_B \left(\beta\displaystyle\frac{\partial^2\ln Z }{\partial \beta \partial V }-\displaystyle\frac{\partial \ln Z }{ \partial V }\right)$

k_T = k_p * k_B *(beta *diff(ln( Z ), V ,2) - diff(ln( Z ), V ))

$ C_p = k_B \beta ^2\displaystyle\frac{ \partial^2 \ln Z }{ \partial \beta ^2}+ k_B V \left(\displaystyle\frac{\partial \ln Z }{ \partial V }-\beta \displaystyle\frac{\partial^2 \ln Z }{\partial \beta \partial V }\right)$

C_p = k_B *beta ^2*diff( ln( Z ), beta ,2)+ k_B * V *(diff(ln( Z ), V )- beta *diff(diff(ln( Z ), beta ), V ))

$ Z = B V ^ N \left(\displaystyle\frac{2 \pi m }{ \beta }\right)^{3 N /2}$

Z = B * V ^ N *(2* pi * m / beta )^3* N /2