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 | :<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]] | |||
(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. | ''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>H( | :<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 | :<math>Q_{NVT}=\frac{1}{N!}\frac{1}{h^{3N}}\int dp^N \exp \left[ - \frac{|p_i |^2}{2mk_B T}\right] | ||
\int | \int dr^N \exp \left[ - \frac{V(r^N)} {k_B T}\right]</math> | ||
This separation is only possible if <math> | 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 | :<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 | \int dr^N \exp \left[ - \frac{V(r^N)} {k_B T}\right]</math> | ||
The integral over positions is known as the | The integral over positions is known as the ''configuration integral'', <math>Z_{NVT}</math> | ||
<math>Z_{NVT}</math> | |||
:<math>Z_{NVT}= \int | :<math>Z_{NVT}= \int dr^N \exp \left[ - \frac{V(r^N)} {k_B T}\right]</math> | ||
In an | In an ideal gas there are no interactions between particles so <math>V(r^N)=0</math> | ||
Thus <math>\exp(- | 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. | ||
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:<math>\Lambda = \sqrt{h^2 / 2 \pi m k_B T}</math> | :<math>\Lambda = \sqrt{h^2 / 2 \pi m k_B T}</math> | ||
one arrives at | one arrives 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 | ||
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particle interactions, ''i.e.'' | particle interactions, ''i.e.'' | ||
:<math>Q_{NVT}=Q_{NVT}^{\rm ideal} | :<math>Q_{NVT}=Q_{NVT}^{\rm ideal} Q_{NVT}^{\rm excess}</math> | ||
[[Category:Ideal gas]] | [[Category:Ideal gas]] | ||
[[Category:Statistical mechanics]] | [[Category:Statistical mechanics]] |