That’s a good question, and thanks for asking it! I know we briefly talked about this a little bit in class yesterday, but I wanted to put it here again. Who knows, maybe it will help you.

The kinetic molecular theory (KMT) of gases is based on the statistical distribution of molecular speeds at different temperatures. The statistical distribution for KMT is the Maxwell distribution of molecular speeds:

F(\upsilon)d\upsilon = 4\pi \left(\frac{m}{2\pi kT}\right)^{3/2} \upsilon^{2} e^{-m\upsilon^{2}/2kT}

Just like any distribution, there are certain metrics that can be used to describe it. Some of these are quite familiar to you:

  • Mean – the first moment of the distribution
  • Mode – the most probable value of the distribution
  • Median – the value for which exactly half the distribution is greater and half is less
  • Root mean square – the second moment of the distribution (aka the standard deviation).

For a symmetrical distribution, like the Gaussian distribution, or the one dimensional velocity distribution of molecular velocities

f(\upsilon_{j}) = \left(\frac{m}{2\pi kT}\right)^{1/2}e^{-m\upsilon_{j}^{2}/2kT}

(where j = x, y, or z), the mean, mode and median are the same. The standard deviation describes the width of the distribution.

The Maxwell distribution of speeds is an asymmetrical distribution (actually it is a \chi^{2} distribution), so in this case the mode (most probable) and mean are different from one another. The root mean square velocity still describes the standard deviation of the distribution from the mean though.