# 4.2: Conditions under which Integrals Approximate Partition Functions

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.