1.19. - An Introduction to Thermal Physics

Question

Suppose you have a gas containing hydrogen molecules and oxygen molecules, in thermal equilibrium. Which molecule are moving faster, on average? By what factor?

Step-by-Step Solution

Answer

In thermal equilibrium, hydrogen molecules moves faster by a factor of 4

Step 1 : Concept of thermal equilibrium which relates with the rms speed of molecules.

Thermal equilibrium refers that both the molecules have same temperature, Hence we can write

$$\begin{gathered} \frac{3}{2}KT=\frac{1}{2}m{v}^2 \\ v=\sqrt{\frac{3KT}{m}} \end{gathered}$$

Step 2 : Relation of speed with molar mass.

From the above equation we can say that $v\alpha \frac{1}{\sqrt{m}}$

Thus the molecule with least mass will have maximum rms speed.

Now O₂ had molecular mass of 32 & H₂ has molecular mass of 2

$\frac{{m_H}_2}{{m_O}_2}=16$

Step 3 : Arriving at the result.

Thus we can say that hydrogen with least mass has the maximum rms speed and

$$\begin{gathered} \frac{v_{H_2}}{v_{O_2}}=\sqrt{\frac{m_{O_2}}{m_{H_2}}=}\sqrt{\frac{1}{16}} \\ {v}_{O_2}=4v_{H_2} \end{gathered}$$