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
Calculations should yield an average speed of approximately 2.72 m/s and a root-mean-square speed of approximately 2.87 m/s. The root-mean-square value is generally larger because squaring emphasizes greater values more than smaller ones.

Step by step solution

01

Calculate the average speed

Add up all the speeds and divide by the total count to find the average speed. In this case the speeds are given as: 2.0 m/s, 2.2 m/s, 2.6 m/s, 2.7 m/s, 3.3 m/s and 3.5 m/s. The average speed can be calculated as follows: \( \frac {(2.0 + 2.2 + 2.6 + 2.7 + 3.3 + 3.5)}{6} \) m/s.
02

Calculate the root-mean-square speed

To calculate the root-mean-square speed, first square all the speeds, then find the average of these squared speeds and finally take the square root. The formula used is: \(\sqrt{\frac{(2.0^2 + 2.2^2 + 2.6^2 + 2.7^2 + 3.3^2 + 3.5^2)}{6}}\) m/s.
03

Compare the two average values

The root-mean-square speed is generally greater than the average speed. If all speed values were the same, the RMS speed would be equal to the average speed. But because there's variability in the speeds, squaring emphasizes larger values more than smaller ones, making the RMS generally higher than a simple average. This is why the root-mean-square speed value is always larger.

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