At a certain temperature the speeds of six gaseous molecules in a container are $2.0, 2.2, 2.6, 2.7, 3.3$ , and $3.5 m s^{- 1}$ . Calculate the roots mean square speed and the average speed of the molecules.

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Solution

Solution:
$X_{r m s} = \sqrt{\frac{1}{n (x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + . . . . . .)}}$

In case we have 6 molecules
Therefore,
$V_{r m s} = \sqrt{\frac{1}{6 ({2.0}^{2} + {2.2}^{2} + {2.6}^{2} + {2.7}^{2} + {3.3}^{2} + {3.5}^{2}}}$

$V_{r m s} = \sqrt{0.16667 \times (4.00 + 4.84 + 6.76 + 7.29 + 10.89 + 12.25)}$
$V_{r m s} = \sqrt{0.16667 \times 46.03}$

$V_{r m s} = \sqrt{7.672} = 2.77 m / s$

The average speed $V_{a v} = \frac{1}{6} \times (2 + 2.2 + 2.6 + 2.7 + 3.3 + 3.5)$

$= 2.44 m / s$
The reason the RMS speed is greater than the average speed because the RMS takes account of the variance of the spread of data. Therefore the increase is related to the fact the data varies from sample to sample.

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At a certain temperature the speeds of six gaseous molecules in a container are 2.0,2.2,2.6,2.7,3.3, and 3.5 ms−1. Calculate the roots mean square speed and the average speed of the molecules.

At a certain temperature the speeds of six gaseous molecules in a container are $2.0, 2.2, 2.6, 2.7, 3.3$ , and $3.5 m s^{- 1}$ . Calculate the roots mean square speed and the average speed of the molecules.