Lesson 45 notes – Boltzmann’s Constant
Objectives
Be able to select and solve problems using the ideal gas equation expressed
as pV = NkT and pV = nRT, where N is the number of atoms and n is the
number of moles;
Be able to explain that the mean translational kinetic energy of an atom of an
ideal gas is directly proportional to the temperature of the gas in kelvin;
Be able to select and apply the equation E = 3/2kT for the mean translational
kinetic energy of atoms.
Outcomes
Understand the difference between R and k and when to apply them.
Be able to state that the Boltzmann constant is the gas constant for individual
molecules whilst the molar gas constant is the constant used when dealing
with quantities in moles.
Be able to explain that the mean translational kinetic energy of an atom of an
ideal gas is directly proportional to the temperature of the gas in Kelvin.
Be able to select and apply the equation E = 3/2kT for the mean translational
kinetic energy of atoms correctly in different situations.
Be able to derive equation for the translational KE of an atom in an ideal gas
E = 3/2kT.
The Boltzmann Constant
The Boltzmann constant is used when we are looking at amounts of
molecules rather than amounts of moles.
From lesson 44 we know that
pV = nRT
We can now write that
pV = NkT
where N is the number of atoms and n is the number of moles
So we can see that
nR=Nk
Also
N=nNA
Therefore
k=R/NA
Check that you understand this from the definitions of these symbols.
Translational kinetic energy of an atom of an ideal gas
From lesson 33 we saw that from the kinetic theory, gas pressure is given by:
P = ⅓ρ<c>2
Letting <c> = the mean square speed
which is the same as saying:
P  Nmv2 / 3V .
From lesson 44 we saw that the Ideal Gas equation is:
PV = nRT
Therefore:
nRT = 1/3 Nm<c>2
Now,
KE = 1/2m v2
Rearranging
3nRT / N = m<c>2
So,
1/2 m<c>2= (3/2) nRT/N
We know that
NA = N/n so n/N = 1/NA
(N is the number of molecules in a volume of gas, n is the amount of gas in
moles, NA is Avagadro’s number).
Therefore:
E = 1/2 m<c>2= (3/2) RT / NA = (3/2)kT
This tells us that the mean kinetic energy of a molecule of an ideal gas is
proportional to the thermodynamic temperature. (lesson 35).