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Found in: Page 13

### An Introduction to Thermal Physics

Book edition 1st
Author(s) Daniel V. Schroeder
Pages 356 pages
ISBN 9780201380279

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

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

See the step by step solution

## 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

$\frac{3}{2}KT=\frac{1}{2}m{v}^{2}\phantom{\rule{0ex}{0ex}}v=\sqrt{\frac{3KT}{m}}$

## 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

$\frac{{v}_{{H}_{2}}}{{v}_{{O}_{2}}}=\sqrt{\frac{{m}_{{O}_{2}}}{{m}_{{H}_{2}}}=}\sqrt{\frac{1}{16}}\phantom{\rule{0ex}{0ex}}{v}_{{O}_{2}}=4{v}_{{H}_{2}}$