Question

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

Answer 1

It increases by a factor of `sqrt3`

Explanation:

The root-mean-square speed `u_"rms"` of gas particles is given by the equation

`u_"rms" = sqrt((3RT)/(MM))`

where

• `R` is the universal gas constant, for this case `8.314("kg"·"m"^2)/("s"^2·"mol"·"K")`

• `T` is the absolute temperature of the system, in `"K"`

• `MM` is the molar mass of the gas, in `"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 `(3R)/(MM)` a constant, `k`:

`u_"rms-1" = sqrt(kT)`

If the temperature is tripled, then this becomes

`u_"rms-2" = sqrt(3kT)`

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`.