2.4 Distribution of molecular speeds  (Page 2/9)

[link] shows that the curve is shifted to higher speeds at higher temperatures, with a broader range of speeds.

We can use a probability distribution to calculate average values by multiplying the distribution function by the quantity to be averaged and integrating the product over all possible speeds. (This is analogous to calculating averages of discrete distributions, where you multiply each value by the number of times it occurs, add the results, and divide by the number of values. The integral is analogous to the first two steps, and the normalization is analogous to dividing by the number of values.) Thus the average velocity is

Similarly,

as in Pressure, Temperature, and RMS Speed . The most probable speed    , also called the peak speed     $v_p,$ is the speed at the peak of the velocity distribution. (In statistics it would be called the mode.) It is less than the rms speed $v_{\text{rms}}.$ The most probable speed can be calculated by the more familiar method of setting the derivative of the distribution function, with respect to v , equal to 0. The result is

which is less than $v_{\text{rms}}.$ In fact, the rms speed is greater than both the most probable speed and the average speed.

The peak speed provides a sometimes more convenient way to write the Maxwell-Boltzmann distribution function:

In the factor $e^{\text{−}m{v}^2\text{/}2k_{\text{B}}T}$ , it is easy to recognize the translational kinetic energy. Thus, that expression is equal to $e^{\text{−}K\text{/}{k}_{\text{B}}T}.$ The distribution f ( v ) can be transformed into a kinetic energy distribution by requiring that $f(K)dK=f(v)dv.$ Boltzmann showed that the resulting formula is much more generally applicable if we replace the kinetic energy of translation with the total mechanical energy E . Boltzmann’s result is

The first part of this equation, with the negative exponential, is the usual way to write it. We give the second part only to remark that $e^{E\text{/}{k}_{\text{B}}T}$ in the denominator is ubiquitous in quantum as well as classical statistical mechanics.