The root mean square speed and average kinetic energy of an ideal gas as derived
from the Kinetic Molecular Theory
The quantitative kinetic molecular model gives:
P = pressure of gas
N = number of gas molecules
m = mass of a single gas molecule
u = speed of a gas molecule
V = volume of gas
N m u2
P=
3V
This equation expresses pressure in terms of its molecular parameters based on the
kinetic – molecular theory.
In contrast, the ideal gas equation (PV = nRT) expresses pressure from experimental
measurements (V, n and T)
Molecular speeds of ideal gases (The root mean square speed of gas molecules)
N m ū2
From kinetic – molecular theory,
PV=
3
From the ideal gas law, P V = nRT, therefore
N m ū2
= nRT
3
For one mole of the gas,
n= 1
N = Avogadros’s number (NA)
NA m = Molar mass (M)
M ū2
= RT
3
½
3RT
ū = ūrms =
M
The average kinetic energy of one mole of gas
Now that the kinetic energy (K.E.) of one gas molecule is given by:
K.E. = ½ m ū2
For one mole of gas:
NA = Avogadro’s number
K.E. = ½ NAm ū2
NAm = M (molar mass)
K.E. = ½ M ū2
From the kinetic – molecular theory we derive that,
3RT
ū2 =
M
Therefore kinetic energy for one mole of gas is:
1
K.E. =
3 RT
M
2
3 RT
K.E. =
2
3 RT
=
M
2