# Kinetic Energy of atoms

1. Jul 27, 2013

### Knightycloud

1. The problem statement, all variables and given/known data
Write an expression E for the average kinetic energy of Helium atoms using the Boltzmann constant.

2. Relevant equations
PV = nRT

PV = $\frac{1}{3}$mN$\overline{c^2}$

k = $\frac{R}{N_A}$

[P - Pressure ; V - Volume ; m - Mass of an atom ; N - Number of atoms ; others have their general meanings.]

3. The attempt at a solution
My problem is they're asking for an expression for the average kinetic energy of Helium atoms. It's plural. I can build an expression for the kinetic energy of a single atom.

nRT = $\frac{1}{3}$mN$\bar{c^2}$

$\frac{3}{2}$nRT = $\frac{1}{2}$mnNA$\overline{c^2}$

$\frac{3}{2}$RT = $\frac{1}{2}$mNA$\overline{c^2}$

$\frac{3}{2}$kT = $\frac{1}{2}$m$\overline{c^2}$ = E (Kinetic Energy of an atom)

Rest of the question is easy if this is the expression they're asking.
Do I have to take mN as the total mass of the gas and calculate it from there taking R as kNA? That way it's making the sum a bit difficult because they haven't defined the constant NA in the question, giving out a hint that it's not involving in this, may be.

2. Jul 27, 2013

### ehild

The average kinetic energy of an atom is the sum of the KE of the individual atoms divided by the number of the atoms KEav=ƩKEi/N

You certainly know the Equipartition Principle. The average energy of a particle in an ideal gas is (f/2)kT where f is the degrees of freedom. http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/eqpar.html. A He atom can only translate, its kinetic energy is translational energy with three degrees of freedom. Your final formula is the answer to the problem: The average kinetic energy of the He atoms (of a He atom in the He gas) is 3/2 kT, expressed with the Boltzmann constant k.

ehild

3. Jul 27, 2013

### Knightycloud

Thanks for the help and the link! :D