Ideal gas partition function: Difference between revisions

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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>
:<math>Q_{NVT}=\frac{1}{N!}\frac{1}{h^{3N}}\int\int dp^N dr^N \exp \left[ - \frac{h(p^N, r^N)}{k_B T}\right]</math>
 
When the particles are distinguishable then the factor ''N!'' disappears. <math>h(p^N, r^N)</math> is the Hamiltonian


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)
(Sir William Rowan Hamilton 1805-1865 Ireland)
corresponding to the total energy of the system.
corresponding to the total energy of the system.
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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>
:<math>h(p^N, r^N)= \sum_{i=1}^N \frac{|p_i |^2}{2m} + V(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]
:<math>Q_{NVT}=\frac{1}{N!}\frac{1}{h^{3N}}\int dp^N \exp \left[ - \frac{|p_i |^2}{2mk_B T}\right]
\int  d{\bf r}^N  \exp \left[ - \frac{{\cal V}({\bf r}^N)} {k_B T}\right]</math>
\int  dr^N  \exp \left[ - \frac{V(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).


This separation is only possible if <math>V(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>
:<math>\int dp^N \exp \left[ - \frac{|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}
:<math>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]</math>
\int  dr^N  \exp \left[ - \frac{V(r^N)} {k_B T}\right]</math>




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


<math>Z_{NVT}= \int  dr^N  \exp \left[ - \frac{{\cal V}(r^N)} {k_B T}\right]</math>
:<math>Z_{NVT}= \int  dr^N  \exp \left[ - \frac{V(r^N)} {k_B T}\right]</math>


In an ideal gas there are no interactions between particles so <math>{\cal V}({\bf r}^N)=0</math>
In an ideal gas there are no interactions between particles so <math>V(r^N)=0</math>
Thus <math>\exp(-{\cal V}({\bf r}^N)/k_B T)=1</math> for every gas particle.
Thus <math>\exp(-V(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.

Revision as of 19:02, 22 February 2007

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

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 Q_{NVT}=\frac{1}{N!}\frac{1}{h^{3N}}\int\int dp^N dr^N \exp \left[ - \frac{h(p^N, 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 h(p^N, 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 (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 h(p^N, r^N)= \sum_{i=1}^N \frac{|p_i |^2}{2m} + V(r^N)}

Thus we have

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 Q_{NVT}=\frac{1}{N!}\frac{1}{h^{3N}}\int dp^N \exp \left[ - \frac{|p_i |^2}{2mk_B T}\right] \int dr^N \exp \left[ - \frac{V(r^N)} {k_B T}\right]}

This separation is only possible if 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 V(r^N)} is independent of velocity (as is generally the case). The momentum integral can be solved analytically:

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 \int dp^N \exp \left[ - \frac{|p |^2}{2mk_B T}\right]=(2 \pi m k_b T)^{3N/2}}

Thus we have

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 Q_{NVT}=\frac{1}{N!} \frac{1}{h^{3N}} \left( 2 \pi m k_B T\right)^{3N/2} \int dr^N \exp \left[ - \frac{V(r^N)} {k_B T}\right]}


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

In an ideal gas there are no interactions between particles so 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 V(r^N)=0} Thus 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 \exp(-V(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 the configuration integral is given by 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 V^N} where V is the volume. Thus we have

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 Q_{NVT}=\frac{V^N}{N!}\left( \frac{2 \pi m k_B T}{h^2}\right)^{3N/2}}

If we define the de Broglie thermal wavelength as where

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 \Lambda = \sqrt{h^2 / 2 \pi m k_B T}}

one arrives at

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


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.

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 Q_{NVT}=Q_{NVT}^{\rm ideal} Q_{NVT}^{\rm excess}}
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