Q. 1.19 - An Introduction to Thermal Physics

Question

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

Step-by-Step Solution

Answer

Hydrogen molecule will be moving faster

Step1: Given information

At room temp (298 K ) Hydrogen and Oxygen molecule is given

Step2: Explanation

Root mean square velocity of molecule is given by 

$v=\sqrt{\frac{3kT}{m}} .....................\left(1\right)$

Use this to find the rms of Oxygen and and Hydrogen, And then get their ratio

As both molecules hydrogen and oxygen are in equilibrium so they both have the same temperature 298 K. and Boltzmann constant is same .

$$\begin{gathered} \frac{v_H}{v_o}=\sqrt{\frac{\frac{3k{T}_H}{m_H}}{\frac{3k{T}_O}{m_O}}} \\ \frac{v_H}{v_o}=\sqrt{\frac{m_O}{m_H}}.......................................\left(2\right) \end{gathered}$$

Substitute m_O=32 and m_H =2 in equation (2) and calculate ratio

$$\begin{gathered} \frac{v_H}{v_O}=\sqrt{\frac{32}{2}} \\ \frac{v_H}{v_o}=4 \\ {v}_H=4v_O \end{gathered}$$

So velocity of Hydrogen is four times more than that of oxygen molecules.