Ideal gas partition function: Difference between revisions
Carl McBride (talk | contribs) No edit summary |
mNo edit summary |
||
Line 2: | Line 2: | ||
for a system of ''N'' identical particles each of mass ''m'' is given by | for a system of ''N'' identical particles each of mass ''m'' is given by | ||
:<math>Q_{NVT}=\frac{1}{N!}\frac{1}{h^{3N}}\int\int | :<math>Q_{NVT}=\frac{1}{N!}\frac{1}{h^{3N}}\int\int d{\mathbf p}^N d{\mathbf r}^N \exp \left[ - \frac{H({\mathbf p}^N,{\mathbf r}^N)}{k_B T}\right]</math> | ||
where ''h'' is [[Planck's constant]], ''T'' is the [[temperature]] and <math>k_B</math> is the [[Boltzmann constant]]. When the particles are distinguishable then the factor ''N!'' disappears. <math>H(p^N, r^N)</math> is the [[Hamiltonian]] | where ''h'' is [[Planck's constant]], ''T'' is the [[temperature]] and <math>k_B</math> is the [[Boltzmann constant]]. When the particles are distinguishable then the factor ''N!'' disappears. <math>H(p^N, r^N)</math> is the [[Hamiltonian]] | ||
Line 9: | Line 9: | ||
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(p^N, r^N)= \sum_{i=1}^N \frac{| | :<math>H({\mathbf p}^N, {\mathbf r}^N)= \sum_{i=1}^N \frac{|{\mathbf p}_i |^2}{2m} + {\mathcal V}({\mathbf 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 d{\mathbf p}^N \exp \left[ - \frac{|{\mathbf p}_i |^2}{2mk_B T}\right] | ||
\int | \int d{\mathbf r}^N \exp \left[ - \frac{{\mathcal V}({\mathbf r}^N)} {k_B T}\right]</math> | ||
This separation is only possible if <math>V(r^N)</math> is independent of velocity (as is generally the case). | This separation is only possible if <math>{\mathcal V}({\mathbf 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 d{\mathbf p}^N \exp \left[ - \frac{|{\mathbf 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 d{\mathbf r}^N \exp \left[ - \frac{{\mathcal V}({\mathbf r}^N)} {k_B T}\right]</math> | ||
The integral over positions is known as the ''configuration integral'', <math>Z_{NVT}</math> (from the German ''Zustandssumme'' meaning "sum over states") | The integral over positions is known as the ''configuration integral'', <math>Z_{NVT}</math> (from the German ''Zustandssumme'' meaning "sum over states") | ||
:<math>Z_{NVT}= \int | :<math>Z_{NVT}= \int d{\mathbf r}^N \exp \left[ - \frac{{\mathcal V}({\mathbf r}^N)} {k_B T}\right]</math> | ||
In an [[ideal gas]] there are no interactions between particles so <math>\ | In an [[ideal gas]] there are no interactions between particles so <math>{\mathcal V}({\mathbf r}^N)=0</math>. | ||
Thus <math>\exp(-\ | Thus <math>\exp(-{\mathcal V}({\mathbf 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 15:12, 10 July 2007
The canonical ensemble partition function, Q, for a system of N identical particles each of mass m is given by
where h is Planck's constant, T is the temperature and is the Boltzmann constant. When the particles are distinguishable then the factor N! disappears. is the Hamiltonian 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
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 configuration integral, (from the German Zustandssumme meaning "sum over states")
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
one arrives at
where
is the single particle translational partition function.
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.