Ideal gas partition function: Difference between revisions

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m (New page: Canonical Ensemble Partition function, $Q$, for a system of $N$ identical particles each of mass $m$ \begin{equation} Q_{NVT}=\frac{1}{N!}\frac{1}{h^{3N}}\int\int d{\bf p}^N d{\bf r}^N \ex...)
 
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Canonical Ensemble Partition function, $Q$,
[[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''
\begin{equation}
 
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>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>
\end{equation}
 
When the particles are distinguishable then the factor $N!$ disappears.
When the particles are distinguishable then the factor ''N!'' disappears.
${\sf h}({\bf p}^N, {\bf r}^N)$ is the Hamiltonian
<math>{\sf h}({\bf p}^N, {\bf r}^N)</math> is the Hamiltonian
(Sir William Rowan Hamilton 1805-1865 Ireland)
(Sir William Rowan Hamilton 1805-1865 Ireland)
corresponding to the total energy of the system.
corresponding to the total energy of the system.
${\sf h}$ is a function of the $3N$ positions and $3N$ momenta of the particles in 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
\begin{equation}
 
{\sf h}({\bf p}^N, {\bf r}^N)= \sum_{i=1}^N \frac{|{\bf p}_i |^2}{2m} + {\cal V}({\bf r}^N)
<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>
\end{equation}
 
Thus we have  
Thus we have  
\begin{equation}
 
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]
<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{\bf r}^N  \exp \left[ - \frac{{\cal V}({\bf r}^N)} {k_B T}\right]
\int  d{\bf r}^N  \exp \left[ - \frac{{\cal V}({\bf r}^N)} {k_B T}\right]</math>
\end{equation}
 
This separation is only possible if ${\cal V}({\bf r}^N)$ is independent of velocity (as is generally the case).
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:
\begin{equation}
 
\int d{\bf p}^N \exp \left[ - \frac{|{\bf p} |^2}{2mk_B T}\right]=(2 \pi m k_b T)^{3N/2}
<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>
\end{equation}
 
Thus we have  
Thus we have  
\begin{equation}
Q_{NVT}=\frac{1}{N!} \frac{1}{h^{3N}} \left( 2 \pi m k_B T\right)^{3N/2}
\int  d{\bf r}^N  \exp \left[ - \frac{{\cal V}({\bf r}^N)} {k_B T}\right]
\end{equation}


The integral over positions is known as the {\it configuration integral}, $Z_{NVT}$
<math>Q_{NVT}=\frac{1}{N!} \frac{1}{h^{3N}} \left( 2 \pi m k_B T\right)^{3N/2}
\begin{equation}
\int  d{\bf r}^N  \exp \left[ - \frac{{\cal V}({\bf r}^N)} {k_B T}\right]</math>
Z_{NVT}= \int  d{\bf r}^N  \exp \left[ - \frac{{\cal V}({\bf r}^N)} {k_B T}\right]
 
\end{equation}
 
In an ideal gas there are no interactions between particles so ${\cal V}({\bf r}^N)=0$
The integral over positions is known as the {\it configuration integral}, ''Z_{NVT}''
Thus $\exp(-{\cal V}({\bf r}^N)/k_B T)=1$ for every gas particle.
 
The integral of 1 over the coordinates of each atom is equal to the volume so for $N$ particles
<math>Z_{NVT}= \int  d{\bf r}^N  \exp \left[ - \frac{{\cal V}({\bf r}^N)} {k_B T}\right]</math>
the {\it configuration integral} is given by $V^N$ where $V$ is the volume.
 
In an ideal gas there are no interactions between particles so <math>{\cal V}({\bf r}^N)=0</math>
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 ''configuration integral'' is given by <math>V^N</math> where ''V'' is the volume.
Thus we have  
Thus we have  
\begin{equation}
 
Q_{NVT}=\frac{V^N}{N!}\left( \frac{2 \pi m k_B T}{h^2}\right)^{3N/2}
<math>Q_{NVT}=\frac{V^N}{N!}\left( \frac{2 \pi m k_B T}{h^2}\right)^{3N/2}</math>
\end{equation}
 
If we define the {\it de Broglie thermal wavelength} as $\Lambda$
If we define the [[de Broglie thermal wavelength]] as <math>\Lambda</math>
where
where
\begin{equation}
 
\Lambda = \sqrt{h^2 / 2 \pi m k_B T}
<math>\Lambda = \sqrt{h^2 / 2 \pi m k_B T}</math>
\end{equation}
 
we arrive at\\
we arrive at\\
\fbox{\parbox{\columnwidth}{
 
\begin{equation}
 
\label{eqpartitionideal}
 
Q_{NVT}=\frac{1}{N!} \left( \frac{V}{\Lambda^{3}}\right)^N
<math>Q_{NVT}=\frac{1}{N!} \left( \frac{V}{\Lambda^{3}}\right)^N</math>
\end{equation}
 
}}
 




Revision as of 19:32, 22 February 2007

Canonical ensemble partition function, Q, for a system of N identical particles each of mass m

Failed to parse (unknown function "\sf"): {\displaystyle 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]}

When the particles are distinguishable then the factor N! disappears. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\sf h}({\bf p}^N, {\bf r}^N)} is the Hamiltonian (Sir William Rowan Hamilton 1805-1865 Ireland) 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

Failed to parse (unknown function "\sf"): {\displaystyle {\sf h}({\bf p}^N, {\bf r}^N)= \sum_{i=1}^N \frac{|{\bf p}_i |^2}{2m} + {\cal V}({\bf r}^N)}

Thus we have

This separation is only possible if is independent of velocity (as is generally the case).

The momentum integral can be solved analytically:

Thus we have


The integral over positions is known as the {\it configuration integral}, Z_{NVT}

In an ideal gas there are no interactions between particles so Thus for every gas particle. 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 where V is the volume. Thus we have

If we define the de Broglie thermal wavelength as where

we arrive at\\




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 particle interactions, i.e. \begin{equation} Q_{NVT}=Q_{NVT}^{\rm ideal} Q_{NVT}^{\rm excess} \end{equation}

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