4.2: Conditions under which Integrals Approximate Partition Functions
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The Boltzmann equation gives the equilibrium fraction of particles in the \(i^{th}\) energy level, \(\epsilon_i\), as
\[\frac{N^{\textrm{⦁}}_i}{N}=\frac{g_i}{z}\mathrm{exp}\left(\frac{-\epsilon_i}{kT}\right) \nonumber \]
so the fraction of particles in energy levels less than \(\epsilon_n\) is
\[f\left(\epsilon_n\right)=z^{-1}\sum^{n-1}_{i=1}{g_i}\mathrm{exp}\left(\frac{-\epsilon_i}{kT}\right) \nonumber \]
where \(z=\sum^{\infty }_{i=1}{g_i}\mathrm{exp}\left({\epsilon_i}/{kT}\right)\). We can represent either of these sums as the area under a bar graph, where the height and width of each bar are \(g_i\mathrm{exp}\left({\epsilon_i}/{kT}\right)\) and unity, respectively. If \(g_i\) and \(\epsilon_i\) can be approximated as continuous functions, this area can be approximated as the area under the continuous function \(y\left(i\right)=g_i\mathrm{exp}\left({\epsilon_i}/{kT}\right)\). That is,
\[\sum^{n-1}_{i=1}g_i\mathrm{exp}\left(\frac{-\epsilon_i}{kT}\right)\approx \int^n_{i=0}{g_i\mathrm{exp}\left(\frac{-\epsilon_i}{kT}\right)}di \nonumber \]
To evaluate this integral, we must know how both \(g_i\) and \(\epsilon_i\) depend on the quantum number, \(i\).
Let us consider the case in which \(g_i=1\) and look at the constraints that the \(\epsilon_i\) must satisfy in order to make the integral a good approximation to the sum. The graphical description of this case is sketched in Figure 1. Since \(\epsilon_i>\epsilon_{i-1}>0\), we have
\[e^{-\epsilon_{i-1}/kT}-e^{-\epsilon_i/kT}>0 \nonumber \]
For the integral to be a good approximation, we must have
\[e^{-\epsilon_{i-1}/kT}\gg e^{-\epsilon_{i-1}/kT}-e^{-\epsilon_i/kT}>0, \nonumber \]
which means that
\[1\gg 1-e^{-\Delta \epsilon /kT}>0 \nonumber \]
where \(\Delta \epsilon =\epsilon_i-\epsilon_{i-1}\). Now,
\[e^x\approx 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots \nonumber \]
so that the approximation will be good if
\[1\gg 1-\left(1-\frac{\Delta \epsilon }{kT}+\dots \right) \nonumber \] or \[1\gg \frac{\Delta \epsilon }{kT} \nonumber \] or \[kT\gg \Delta \epsilon \nonumber \]
We can be confident that the integral is a good approximation to the exact sum whenever there are many pairs of energy levels, \(\epsilon_i\) and \(\epsilon_{i-1}\), that satisfy the condition
\[\Delta \epsilon =\epsilon_i-\epsilon_{i-1}\ll kT. \nonumber \]
If there are many energy levels that satisfy \(\epsilon_i\ll kT\), there are necessarily many intervals, \(\Delta \epsilon\), that satisfy \(\Delta \epsilon \ll kT\). In short, if a large number of the energy levels of a system satisfy the criterion \(\epsilon \ll kT\), we can use integration to approximate the sums that appear in the Boltzmann equation. In Section 24.3 , we use this approach and the energy levels for a particle in a box to find the partition function for an ideal gas.