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 ?

1 Answer
Feb 28, 2015

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)#