Mass and Distribution of Speeds

On the previous page, we saw that temperature is directly related to the kinetic energy of a gas sample, and since kinetic energy is related to speed ($K = \frac{1}{2}mv^2$) we can see how temperature is related to the speed of particles.

Particles at the same temperature will have the same (average) kinetic energy – but speed isn’t the only factor. If we have particles of different molecular/atomic mass ($m$ in the equation above), they will have a different average speed – and a different speed distribution – at the same temperature, in order to keep the kinetic energy equal.

The figure below shows the speed distributions of noble gas samples, all at the same temperature:

A graph is shown with four positively or right-skewed curves of varying heights. The horizontal axis is labeled, “Speed u ( m divided by s ).” This axis is marked by increments of 500 beginning at 0 and extending up to 3000. The vertical axis is labeled, “Fraction of molecules.” The tallest and narrowest of these curves is labeled, “X e.” Its right end appears to touch the horizontal axis around 600 m per s. It is followed by a slightly wider curve which is labeled, “A r,” that is about half the height of the initial curve. Its right end appears to touch the horizontal axis around 900 m per s. The third curve is significantly wider and just over a third of the height of the initial curve. It is labeled, “N e.” Its right end appears to touch the horizontal axis around 1200 m per s. The final curve is only about one fourth the height of the initial curve. It is much wider than the others, so much so that its right reaches the horizontal axis around 2500 m per s. This curve is labeled, “H e.”
Particle velocity is directly related to molecular/atomic mass. At a given temperature, lighter particles move faster on average than heavier particles.

Each of these gases is at the same temperature, and has the same average kinetic energy. However, since He is the lightest of the noble gases (4.003 g/mol), it needs a relatively fast speed to achieve the same kinetic energy as say Xe (131.3 g/mol), which carries the same kinetic energy at a much lower speed.

Notice also that – similarly to the trend with temperature – as the average speed increases, the distribution of speeds tends to also get broader. Some slow-moving particles remain even when the average speed is high.