If the absolute temperature of a gas is quadrupled, what happens to the root-mean-square speed of the molecules?

The root-mean-square speed measures the average speed of the gas molecules. The relation of root-mean-square speed and the absolute temperature is based on the kinetic molecular theory of gases.

The root-mean-square speed define as ν_rms

where:

ν_rms = √ (3RT/M)

R = universal gas constant; 0.8206 L-atm/mol-K

T = absolute temperature

M = molecular weight of gas particles

So, if the temperature of the gas is quadrupled the root-mean-square speed will be doubled.

Proof:

Since T is quadrupled, then T=4T

Substitute to the formula of root-mean-square speed,

ν_rms = √[3R (4T) / M]

= 2√ (3RT/M) since the square root of 4 is 2

Therefore, root-mean-square speed is doubled when the absolute temperature is quadrupled.