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4.2: Conditions under which Integrals Approximate Partition Functions

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    206328
  • The Boltzmann equation gives the equilibrium fraction of particles in the \(i^{th}\) energy level, \(\epsilon_i\), as

    \[\frac{N^{\textrm{⦁}}_i}{N}=\frac{g_i}{z}\mathrm{exp}\left(\frac{-\epsilon_i}{kT}\right)\]

    so the fraction of particles in energy levels less than \(\epsilon_n\) is

    \[f\left(\epsilon_n\right)=z^{-1}\sum^{n-1}_{i=1}{g_i}\mathrm{exp}\left(\frac{-\epsilon_i}{kT}\right)\]

    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,

    \[\sum^{n-1}_{i=1}g_i\mathrm{exp}\left(\frac{-\epsilon_i}{kT}\right)\approx \int^n_{i=0}{g_i\mathrm{exp}\left(\frac{-\epsilon_i}{kT}\right)}di\]

    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

    \[e^{-\epsilon_{i-1}/kT}-e^{-\epsilon_i/kT}>0\]

    For the integral to be a good approximation, we must have

    \[e^{-\epsilon_{i-1}/kT}\gg e^{-\epsilon_{i-1}/kT}-e^{-\epsilon_i/kT}>0,\]

    which means that

    \[1\gg 1-e^{-\Delta \epsilon /kT}>0\]

    where \(\Delta \epsilon =\epsilon_i-\epsilon_{i-1}\). Now,

    \[e^x\approx 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots\]

    so that the approximation will be good if

    \[1\gg 1-\left(1-\frac{\Delta \epsilon }{kT}+\dots \right)\] or \[1\gg \frac{\Delta \epsilon }{kT}\] or \[kT\gg \Delta \epsilon\]

    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

    \[\Delta \epsilon =\epsilon_i-\epsilon_{i-1}\ll kT.\]

    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.