Does Quantum Molar Internal Energy Converge to Classical at High Temperatures?

[plonker]

Homework Statement

I'm so confused please help :\

Show that the contribution to the total energy from molar internal energy Um reverts to the classical expression at high T.

Homework Equations

Classical: Um = 3NakT Quantum Um = 3NAhv/e^(hv/kT)-1

The Attempt at a Solution

Manipulating variables- E=hv
Quantum rearranging: hv/kT= ln(3Nahv)-ln(Um)
Very confused on what is meant by total energy though. Is that supposed to be E+Um?

 

[Bryson]

Pardon my ignorance (I have never heard of this in all my years in physics), but what is Um?

 

[plonker]

My teacher said it was "internal energy, U" but while in the context of the failures of classical physics in terms of heat capacities. Apparently Einstein calculated the contribution of the oscillation of the atoms to the total molar energy of metal and obtained the quantum equation above in place of the classical one? I'm so nervous for this course now :\

 

[Bryson]

I am assuming that for high T you can manipulate $e^{\frac{h \nu}{k T}}$. Try this.

 

[paranormal]

Hi there,

I understand that this may be a confusing topic, but I will try my best to explain it to you. The content is discussing the difference between classical and quantum expressions for molar internal energy (Um). Molar internal energy is the average energy per mole of a substance. In classical mechanics, Um is given by Um = 3NakT, where N is the number of particles, a is the average energy per particle, k is the Boltzmann constant, and T is the temperature.

In quantum mechanics, the expression for Um is more complex and takes into account the quantum nature of particles. It is given by Um = 3NAhv/e^(hv/kT)-1, where h is the Planck constant and v is the frequency of the particles. This expression becomes important at low temperatures when the quantum effects become more significant.

Now, the homework statement is asking you to show that at high temperatures, the quantum expression for Um reverts back to the classical expression. This is because at high temperatures, the classical expression becomes dominant and the quantum effects become negligible. To show this mathematically, you can take the limit of the quantum expression as T→∞. This will result in the quantum expression simplifying to Um = 3NakT, which is the classical expression.

I hope this helps to clarify the concept. If you are still confused, I suggest reviewing your notes or textbook for a more detailed explanation. Good luck with your homework!