Question

You have but one molecule of Hydrogen $\left(H_2\right),$ meandering about in the vast emptiness of a $1c{m}^3$ closed box, at a speed of 500 m/s, bouncing off from one wall to the next. What are the rms speed and the temperature inside the box?

• A. 512 m/s; 20k
• B. 250000 m/s; 20 k
• C. 500 m/s; 20 k
• D. 500 m/s; temperature cannot be determined

Answer 1

The correct option is D

500 m/s; temperature cannot be determined

Well, finding the vrmsfor this one is easy! Since we have only one molecule its speed will be its own "rms” speed -

$v_{rms}=\sqrt{\left(v^2\right)}=\sqrt{\left({500}^2\right)}\frac{m}{s}=500m/s.$

Now, what can we say about the temperature, T ?

Kinetic theory says, for a simple, ideal gas at a fixed volume and pressure, T and $v_{rms}$ are related as -

$v_{rms}=\sqrt{\frac{3RT}{M}}=\frac{1}{3}MR{v}_{rms}^2.$

But can we even use this definition here? If we have just one or a few gas molecules in a fixed volume, the assumption of kinetic theory that - at any given instant the velocity distribution of the molecules is homogenous (uniform density throughout) and isotropic (no special average direction) - will be immediately violated if we have just one molecule in the system, or if it's a small number. We cannot apply our kinetic theory of gases here to determine the temperature from the $v_{rms}$.

In such cases, the definition of temperature comes from a much more generalized statistical physics point of view - as the slope of a heat gained versus entropy function, at any given value of entropy. We will have another look at this in the section for the second law of thermodynamics.