How Do You Calculate the Average Kinetic Energy of Multiple Helium Atoms?

[Knightycloud]

Homework Statement

Write an expression E for the average kinetic energy of Helium atoms using the Boltzmann constant.

Homework Equations

PV = nRT

PV = [itex]\frac{1}{3}[/itex]mN[itex]\overline{c^2}[/itex]

k = [itex]\frac{R}{N_A}[/itex]

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

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 = [itex]\frac{1}{3}[/itex]mN[itex]\bar{c^2}[/itex]

[itex]\frac{3}{2}[/itex]nRT = [itex]\frac{1}{2}[/itex]mnN_A[itex]\overline{c^2}[/itex]

[itex]\frac{3}{2}[/itex]RT = [itex]\frac{1}{2}[/itex]mN_A[itex]\overline{c^2}[/itex]

[itex]\frac{3}{2}[/itex]kT = [itex]\frac{1}{2}[/itex]m[itex]\overline{c^2}[/itex] = 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 kN_A? That way it's making the sum a bit difficult because they haven't defined the constant N_A in the question, giving out a hint that it's not involving in this, may be.

 

[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 KE_av=ƩKE_i/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

 

[Knightycloud]

Thanks for the help and the link! :D