Understanding Energy of Particles with 'x' Degrees of Freedom

[tanaygupta2000]

Homework Statement

Consider a system with a large number of non interacting particles at temperature T. Each particle has energy given by E = ax^6 associated with the degree of freedom x. Evaluate the mean energy per particle associated with this degree of freedom.

Relevant Equations

E = 3/2 kT

I've learned that for a particle having 3 degrees of freedom, its average energy is 3/2 kT.
So for a particle having 'x' degrees of freedom, its energy should be xkT/2.
So what is the use of given E = ax⁶ here?
Please help!

 

[vela]

Look up what exactly the equipartition theorem says. The devil is in the details.

 

[tanaygupta2000]

Look up what exactly the equipartition theorem says. The devil is in the details.

According to Law of Equipartition of Energy, each degree of freedom in a system of particles contributes (1/2)kT to the thermal average energy of the system.
So for x degrees of freedom of each particle in a system of particles, the thermal average energy of the system should be xkT/2.
And it is given that each particle has energy given by E = ax^6.

 

[vela]

That's based on certain assumptions. What are those assumptions?

 

[tanaygupta2000]

That's based on certain assumptions. What are those assumptions?

Okay now I understood.
The above expression holds only when each term is proportional to either a coordinate or momentum squared.
In this case, we similarly have E = ax⁶ ∝ (x³)²

So we have to use for Partition function,
Z = ∫e^-ax6/kTdx = ∫e^{(-a/kT)× (x6)}dx

Let (a/kT) = α
=> Z = ∫e^-αx6 dx

Since

So we can obtain Z and finally E using E = kT² d[ln Z]/dT
Is this correct ?