Editing Ideal gas partition function
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[[Canonical ensemble]] partition function, ''Q'', | |||
for a system of ''N'' identical particles each of mass ''m'' | for a system of ''N'' identical particles each of mass ''m'' | ||
<math>Q_{NVT}=\frac{1}{N!}\frac{1}{h^{3N}}\int\int d{\bf p}^N d{\bf r}^N \exp \left[ - \frac{{\sf h}({\bf p}^N, {\bf r}^N)}{k_B T}\right]</math> | |||
When the particles are distinguishable then the factor ''N!'' disappears. | |||
<math>{\sf h}({\bf p}^N, {\bf r}^N)</math> is the Hamiltonian | |||
(Sir William Rowan Hamilton 1805-1865 Ireland) | |||
corresponding to the total energy of the system. | corresponding to the total energy of the system. | ||
'' | ''h'' is a function of the ''3N'' positions and ''3N'' momenta of the particles in the system. | ||
The Hamiltonian can be written as the sum of the kinetic and the potential energies of the system as follows | The Hamiltonian can be written as the sum of the kinetic and the potential energies of the system as follows | ||
<math>{\sf h}({\bf p}^N, {\bf r}^N)= \sum_{i=1}^N \frac{|{\bf p}_i |^2}{2m} + {\cal V}({\bf r}^N)</math> | |||
Thus we have | Thus we have | ||
<math>Q_{NVT}=\frac{1}{N!}\frac{1}{h^{3N}}\int d{\bf p}^N \exp \left[ - \frac{|{\bf p}_i |^2}{2mk_B T}\right] | |||
\int d{\ | \int d{\bf r}^N \exp \left[ - \frac{{\cal V}({\bf r}^N)} {k_B T}\right]</math> | ||
This separation is only possible if <math>{\cal V}({\bf r}^N)</math> is independent of velocity (as is generally the case). | |||
The momentum integral can be solved analytically: | The momentum integral can be solved analytically: | ||
<math>\int d{\bf p}^N \exp \left[ - \frac{|{\bf p} |^2}{2mk_B T}\right]=(2 \pi m k_b T)^{3N/2}</math> | |||
Thus we have | Thus we have | ||
<math>Q_{NVT}=\frac{1}{N!} \frac{1}{h^{3N}} \left( 2 \pi m k_B T\right)^{3N/2} | |||
\int d{\ | \int d{\bf r}^N \exp \left[ - \frac{{\cal V}({\bf r}^N)} {k_B T}\right]</math> | ||
The integral over positions is known as the | The integral over positions is known as the {\it configuration integral}, ''Z_{NVT}'' | ||
<math>Z_{NVT}= \int d{\bf r}^N \exp \left[ - \frac{{\cal V}({\bf r}^N)} {k_B T}\right]</math> | |||
In an | In an ideal gas there are no interactions between particles so <math>{\cal V}({\bf r}^N)=0</math> | ||
Thus <math>\exp(-{\ | Thus <math>\exp(-{\cal V}({\bf r}^N)/k_B T)=1</math> for every gas particle. | ||
The integral of 1 over the coordinates of each atom is equal to the volume so for ''N'' particles | The integral of 1 over the coordinates of each atom is equal to the volume so for ''N'' particles | ||
the ''configuration integral'' is given by <math>V^N</math> where ''V'' is the volume. | the ''configuration integral'' is given by <math>V^N</math> where ''V'' is the volume. | ||
Thus we have | Thus we have | ||
<math>Q_{NVT}=\frac{V^N}{N!}\left( \frac{2 \pi m k_B T}{h^2}\right)^{3N/2}</math> | |||
If we define the [[de Broglie thermal wavelength]] as <math>\Lambda</math> | If we define the [[de Broglie thermal wavelength]] as <math>\Lambda</math> | ||
where | where | ||
<math>\Lambda = \sqrt{h^2 / 2 \pi m k_B T}</math> | |||
we arrive at\\ | |||
<math>Q_{NVT}=\frac{1}{N!} \left( \frac{V}{\Lambda^{3}}\right)^N</math> | |||
Thus one can now write the partition function for a real system can be built up from | Thus one can now write the partition function for a real system can be built up from | ||
the contribution of the ideal system (the momenta) and a contribution due to | the contribution of the ideal system (the momenta) and a contribution due to | ||
particle interactions, | particle interactions, i.e. | ||
\begin{equation} | |||
Q_{NVT}=Q_{NVT}^{\rm ideal} Q_{NVT}^{\rm excess} | |||
\end{equation} | |||