Question

How to calculate the root mean square speed, vrms, in m·s^-1 at 227 Celsius and 900 torr, of a gas having a molar mass of 16.0 g·mol^-1 ?

Answer 1

There are two important things to look out for when doing root-mean-square speed calculations

1. You must use the value for `R` expressed in Joules per mol K `->` `R = 8.31446 "J"/("mol" * "K")`;
2. You must use molar mass in kilograms;

Using these units will get you to the required units for gas velocity, `m * s^(-1)`.

So, root-mean-square speed allows you to have some measure on the speed of particles in a gas. Mathematically, the equation looks like this

`v_("rms") = sqrt((3RT)/M_m)`, where

`R` - the universal gas constant;
`T` - the temperature of the gas in Kelvin;
`M_m` - the molar mass of the gas;

I'll convert the molar mass from g per mole to kg per mole first, and then plug all the value into the equation for root-mean-square speed

`16.0"g"/"mol" * "1 kg"/"1000 g" = 0.016"kg"/"mol"`

Therefore,

`v_("rms") = sqrt((3 * 8.31446"J"/("mol" * "K") * (273.15 + 227)"K")/(0.016"kg"/"mol"))`

`v_("rms") = 883.01 * sqrt("J"/"kg")`

Use the fact that `"Joule" = ("kg" * "m"^2)/("s"^2)` to get

`v_("rms") = 883.01 * sqrt(("kg" * "m"^2)/("kg" * "s"^2)) = 883.01 "m"/"s" = "111.7 m" * "s"^(-1)`

Rounded to three sig figs, the answer will be

`v_("rms") = "883 m" * "s"^(-1)`