# Speed distribution of lighter vs heavier gases

I cannot understand why the range of molecular speed is not always wider for a lighter gas as compared to a heavier gas .

If the same energy is supplied to both gases then wouldn't the molecular speed of the lighter gas be more?

• Do you have an example of such an exception? I don't doubt that there are exceptions since the Maxwell Boltzmann distribution requires a system to be at thermal equilibrium and assumes ideal gases. However do you have a specific example of a situation where a lower molecular weight gas has a thinner distribution? – Tyberius Apr 16 '17 at 3:09
• @Tyberius If i have the example of exception then why would I have asked the question – search Apr 16 '17 at 6:31
• Let me rephrase my question. If you don't have a specific example, how do you know it happens? What source told you that this could be the case? – Tyberius Apr 16 '17 at 13:30
• Knowing what led to this question might help to provide an explanation that addresses your specific concern. But as I mentioned in the comments to my answer, probably the main cause of deviations from Maxwell Boltzmann would be real gas behavior (attraction and repulsion) and multiatomic molecules, which have internal degrees of freedom where they can store the thermal energy. – Tyberius Apr 16 '17 at 15:47

## 1 Answer

To my understanding, the speed distribution is wider for lighter molecules. The Wikipedia page for the Maxwell Boltzmann distribution has two images at the top that convey this very well.  In the first plot, $a=\sqrt{k_bT/M}$ where $M$ is the molar mass and $k_b$ Boltzmann's constant. For a Maxwell-Boltzmann distribution, the variance is given by $$\sigma^2= \frac{a^2\cdot(3\pi-8)}{\pi}=\frac{k_b\cdot T\cdot(3\pi-8)}{M\cdot\pi}$$ So, this shows that the speed distribution will be wider for higher temperatures and smaller molar mass.

• There might be some exception – search Apr 16 '17 at 1:39
• @Tyberius With infinite temperature (equivalent to an infinitely small mass) every state (velocity) has same probability density. – user43021 Apr 16 '17 at 3:32
• @santimirandarp right, but that still wouldn't lead to smaller speed distribution for lighter molecules. Equal within that limit, but not smaller. I would guess certain effects of real gases, namely attractive forces, could lead to deviations from the Maxwell Boltzmann distribution. – Tyberius Apr 16 '17 at 3:44
• @santimirandarp but would it be smaller than the distribution of a large molecular weight gas at the same very high temperature. Even a large molecule would have a broad speed distribution at large temperature. – Tyberius Apr 16 '17 at 4:01
• I think range is same for both gases . As the graph is extending till infinite . – search Apr 16 '17 at 6:33