Question

At a certain temperature the speeds of six gaseous molecules in a container are 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. Calculate the rootmean-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?

Answer 1

Answer:

root-mean-sqaure = 2.77 m/s

average = 2.72 m/s

The root-mean-square is always the largest because it takes account of the variance of the spread of the data. The increase is related to the fact that the data varies to sample.

Explanation:

The rootmean-square (R) is the square root of the squares of the valeus divided by the number of the datas.

$R = \sqrt{\frac{x1^2 + x2^2 +...+xn^2}{n} }$

$R = \sqrt{\frac{(2.0)^2 + (2.2)^2 + (2.6)^2 + (2.7)^2 ^(3.3)^2 + (3.5)^2}{6} }$

R = √(46.03)/6

R = 2.77 m/s

The average speed is the sum of the speeds divided by the number of datas:

$A = \frac{2.0 + 2.2 + 2.6 + 2.7 + 3.3 + 3.5}{6}$

A = 16.3/6

A = 2.72 m/s

Answer 2

Answer:

rms speed = 2.77 m/s

avg speed = 2.717 m/s

The squared values have a higher weight in the calculation of the rms than the non-squared values of the avg.

Explanation:

The root mean square (rms) speed of various gas molecules is calculated as follows:

rms speed = √([v1^2 + v2^2+ v3^2 + v4^2 + v5^2 + v6^2]/n)

where v1 to v6 are the individual speed and n is the number of molecules.

rms speed = √([2^2 + 2.2^2+ 2.6^2 + 2.7^2 + 3.3^2 + 3.5^2]/6) = 2.77 m/s

The average speed is calculated as follows:

avg speed = (v1 + v2 + v3 + v4 + v5 + v6)/n

avg speed = (2 + 2.2 + 2.6 + 2.7 + 3.3 + 3.5)/6 = 2.717 m/s

The rms speed is always greater that the avg speed unless all speeds are equal. The reason is that higher values in the list have a higher weight (because you average the squares) in the calculation of a rms speed compared to the calculation of the avg speed.