Question

Figure 12.59 shows the distribution of speeds of atoms in a particular gas at a particular temperature. Approximately what is the average speed? Is the RMS (root-mean-square) speed bigger or smaller than this? Approximately what fraction of the molecules have speeds greater than 1000 m/s?

Short answer

• The average speed of an atom is$\stackrel{-}{\nu }=600m/s$.
• The RMS speed is bigger than the average speed.

• The fraction of the molecules has a speed greater than 1000 m/s is roughly $\frac{1}{10}$.

Step-by-step solution

Identification of given data

• The graph between v(m/s) and f(v) .

Concept of root mean square speed 

Molecules in gas travel at an average velocity-squared, which is defined as the square root of the velocity-squared squared for each molecule.

Determination of root mean square speed of gas

The average speed of gas from the graph is .

The RMS speed is always larger than the average speed.

The root mean square speed is given by,

$V_{rms}=\sqrt{\frac{3RT}{M}}$ (i)

The average speed of a gas is given by,

$V_{av}=\sqrt{\frac{8RT}{\pi M}}$ (ii)

Comparing equation (ii) with equation (i), we get,

$\begin{array}{l}\sqrt{2}=\sqrt{\frac{8}{\pi }}\\ 144:1.595\end{array}$

Thus, we can conclude that the RMS value is always bigger than the average speed.

From looking at the graph, we can conclude that the fraction of the molecules has a speed greater than 1000m/s is approximate $\frac{1}{10}$.