2022: SklogWiki celebrates 15 years on-line

Difference between revisions of "Ideal gas Helmholtz energy function"

From SklogWiki
Jump to: navigation, search
m (New page: From equations :<math>Q_{NVT}=\frac{1}{N!} \left( \frac{V}{\Lambda^{3}}\right)^N</math> and :<math>A=-k_B T \ln Q_{NVT}</math> one has :<math>A=-k_BT\left(\ln \frac{1}{N!} + N\ln\frac{V}...)
 
m
Line 2: Line 2:
 
:<math>Q_{NVT}=\frac{1}{N!} \left( \frac{V}{\Lambda^{3}}\right)^N</math>
 
:<math>Q_{NVT}=\frac{1}{N!} \left( \frac{V}{\Lambda^{3}}\right)^N</math>
 
and  
 
and  
:<math>A=-k_B T \ln Q_{NVT}</math>
+
:<math>\left.A\right.=-k_B T \ln Q_{NVT}</math>
 
one has
 
one has
 
:<math>A=-k_BT\left(\ln \frac{1}{N!} + N\ln\frac{V}{\Lambda^{3}}\right)</math>
 
:<math>A=-k_BT\left(\ln \frac{1}{N!} + N\ln\frac{V}{\Lambda^{3}}\right)</math>

Revision as of 15:20, 21 February 2007

From equations

Q_{NVT}=\frac{1}{N!} \left( \frac{V}{\Lambda^{3}}\right)^N

and

\left.A\right.=-k_B T \ln Q_{NVT}

one has

A=-k_BT\left(\ln \frac{1}{N!} + N\ln\frac{V}{\Lambda^{3}}\right)
=-k_BT\left(-\ln N! + N\ln\frac{VN}{\Lambda^3N}\right)
=-k_BT\left(-\ln N! + N\ln\frac{N}{\Lambda^3 \rho}\right)

using Stirling's approximation

=-k_BT\left( -N\ln N +N + N\ln N - N\ln \Lambda^3 \rho \right)

one arrives at

A=Nk_BT\left(\ln \Lambda^3 \rho -1 \right)