Chapter 12: Q53P (page 472)

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?

Expert verified

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

Step by step solution

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

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,

$V_{r m s} = \sqrt{\frac{3 R T}{M}}$ (i)

The average speed of a gas is given by,

$V_{a v} = \sqrt{\frac{8 R T}{π M}}$ (ii)

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

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

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 $\frac{1}{10}$ .

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