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Chapter 5: Problem 84

At a certain temperature the speeds of six gaseous molecules in a container are \(2.0 \mathrm{~m} / \mathrm{s}, 2.2 \mathrm{~m} / \mathrm{s}, 2.6 \mathrm{~m} / \mathrm{s}\) \(2.7 \mathrm{~m} / \mathrm{s}, 3.3 \mathrm{~m} / \mathrm{s},\) and \(3.5 \mathrm{~m} / \mathrm{s} .\) Calculate the root- mean-square speed and the average speed of the molecules. These two average values are close to each other, but the root-mean-square value is always the larger of the two. Why?

Short Answer

Expert verified
The average speed of the gas molecules is 2.72 m/s and the root mean square speed is 2.23 m/s. The root mean square speed is larger than the average speed because when calculating the root mean square, the speeds are squared before averaging, which gives more weight to higher speeds.

Step by step solution

01

Calculate Average Speed

Find the sum of all the speeds: \(2.0m/s + 2.2m/s + 2.6m/s + 2.7m/s + 3.3m/s + 3.5m/s = 16.3m/s\). Divide the sum by the total number of molecules, which is 6, to find the average speed: \(16.3m/s ÷ 6 = 2.72m/s\).
02

Calculate Root Mean Square Speed

Find the square of each of the speeds, then find their sum: \(2.0m/s^2 + 2.2m/s^2 + 2.6m/s^2 + 2.7m/s^2 + 3.3m/s^2 + 3.5m/s^2 = 29.8(m/s)^2. Divide this sum by the total number of molecules, which is 6, to find the mean square speed: \(29.8(m/s)^2 ÷ 6 = 4.97(m/s)^2. Finally, find the root of this number to get the rms speed: √4.97(m/s)^2 = 2.23m/s.
03

Reason for Difference in Values

RMS value is larger than the average value because when calculating rms, the speeds are squared before averaging. This gives more weight to higher speeds compared to lower speeds. The weights are then equalized by taking the square root, but the result is still higher compared to the simple average.

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