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

Jun 20, 2017

It increases by a factor of $\sqrt{3}$

#### Explanation:

The root-mean-square speed ${u}_{\text{rms}}$ of gas particles is given by the equation

${u}_{\text{rms}} = \sqrt{\frac{3 R T}{M M}}$

where

• $R$ is the universal gas constant, for this case $8.314 \left(\text{kg"·"m"^2)/("s"^2·"mol"·"K}\right)$

• $T$ is the absolute temperature of the system, in $\text{K}$

• $M M$ is the molar mass of the gas, in $\text{kg"/"mol}$

The question is nonspecific for which gas, but we're just asked to find what generally happens to the r.m.s. speed if only the temperature changes, so we'll call the quantity $\frac{3 R}{M M}$ a constant, $k$:

${u}_{\text{rms-1}} = \sqrt{k T}$

If the temperature is tripled, then this becomes

${u}_{\text{rms-2}} = \sqrt{3 k T}$

To find what happens, let's divide this value by the original equation:

(u_"rms-2")/(u_"rms-1") = (sqrt(3kt))/(sqrt(kt)) = color(red)(sqrt3

Thus, if the temperature is tripled, the root-mean-square speed of the gas particles increases by a factor of color(red)(sqrt3.