The average kinetic energy of the molecules in a gas sample depends only on the temperature, 𝑇.

Question

However, given the same kinetic energies, a lighter molecule will move faster than a heavier molecule, as shown in the equation for rms = square root 3RT/ℳ

where 𝑅=8.314 J/(mol⋅K) and ℳ is molar mass in kilograms per mole. Note that a joule is the same as a kilogram‑meter squared per second squared (kg·m²/s²). 

What is the rms speed of O2 molecules at 505 K?

What is the rms speed of He atoms at 505 K?

Answer 1

For O2:

M = 32 g/mol x 1 kg/1000 g = 0.032 kg/mol

R = 8.314 J/mol-K = 8.314 kgm²/s²-mol-K

rms = sqrt [(3)(8.314)(505) / 0.032

rms = sqrt 393,616

rms = 627 m/s

For He:

M = 4 g/mol x 1 kg/1000 g = 0.004 kg/mol

8.314 J/mol-K = 8.314 kgm²/s²-mol-K

rms = sqrt [(3)8.314)(505) /0.004

rms = sqrt 3,148,928

rms = 1775 m/s

Answer 2

Hello, Bri,

It seems that all we need to do is put the values into the equation and solve. I see that the units all cancel to leave meters/sec. Thanks for including the definition of Joule - it made the units a lot easier to understand.

I had questioned why kilograms per mole, instead of the traditional grams per mole. The equation needed kilograms since Joules are defined in kilograms.

For O2, I get 0.032 kg/mole. Putting the values into the equation, we get rms = ((3*8.314*505)/0.032)^1/2. TI left of the units here since I did them separately. Please leave them in when you do the calculation to ensure everything cancels to meters/second. The value I got for that calculation is 627 m/s.

For He, the process is the same. My result is 1,775 m/s.

Therefore, the lighter atom moves more quickly that the heavier one at the same temperature.

Bob