Deriving Dulong-Petit law with equipartition of energy

[Kiezel]

Homework Statement

hello everyone.
I have to derive the classical Dulong Petit law by using kinetic theory of gases and the equipartition of energy. The heat capacity at a constant V for simple crystalline solid. So, it should be explained without the quantum theory.

Homework Equations

Cv (heat capacity at constant v) = 3R = 3Nak (avogadro Na and k of boltmann's constant)
this is what I want to derive.

The Attempt at a Solution

the formula Cv = Cp - R is what I have derived from simple formula's like dU = dQ + dW.
If Cp = 4R then Cv = 3R, but why should this be the case?
I know it has something to do with the degrees of freedom. I think the solid at high temperature has 3 translational degrees of freedom and 3 vibrational degrees of freedom which would mean it has 6 degrees of freedom. How should I use this?

according to the equipartition theorem the energy of the degree of freedom is kT/2 thus with 6 degrees of freedom: E=Na6(kt/2)=3RT (because R=Nak)
so now I have an energy which is equal to 3RT

didn't the classical physics (around 1819) predict that the heat capacity was independent from the temperature (by only allowing the body to absorb very small amounts of heat)?

clearly I'm missing some vital next steps. can anybody help me?

Thank you!

 

[ZetaOfThree]

Welcome to PF!

the formula Cv = Cp - R is what I have derived from simple formula's like dU = dQ + dW.
If Cp = 4R then Cv = 3R, but why should this be the case?

Careful! $C_v=C_p-nR$ is only true for an ideal gas.

according to the equipartition theorem the energy of the degree of freedom is kT/2 thus with 6 degrees of freedom: E=Na6(kt/2)=3RT (because R=Nak)
so now I have an energy which is equal to 3RT

Good. So you have an expression for the thermal energy in terms of $T$. Remember what heat capacity is: it is a measure of the rate at which the energy of something changes when you change its temperature. Do you think you could apply this definition to your expression for energy to get the heat capacity?

 

[Kiezel]

Thanks for your reply!

I have defined the heat capacity as:

Heat capacity C is defined as the amount of heat that is necessary to raise T of a standard amount of matter with one degree Kelvin (J/K).

So if my “standard amount of matter” is a mole (because I have used Na in my energy calculation) I can calculate the molar heat capacity. I need to find the difference in energy between T = x and T = x + 1. So if I pick T = 1 then E = 3R and if I pick T = 2 then E = 6R. The difference in energy is 6R – 3R = 3R which would be my molar heat capacity. and thus Cv = 3R

And the Cv = Cp - R part should be omitted because it is not applicable for crystalline solids.

then:
pV = nRT
R = (pV)/(nT) with R in m3 Pa mol-1 k-1 = J mol-1 k-1
this equals Boltzmann constant kb (J K-1) times Avogadro constant Na (mol-1)
and thus R = Nakb (I'm not sure if I have derived it here or not)
and 3R=3Nakb

hopefully this is right