Einstein's heat capacity model .and law of corresponding

[terp.asessed]

Homework Statement

B/c the textbook mentions that Einstein's heat capacity equation obeys a law of corresponding states...but, I don't really understand how this can be...I've tried to figure out, to no avail...so, any help would be welcome!

Homework Equations

Cv/Nk_B = 3R(hv/(k_BT))²e^-[hv/(k_BT)]/(1-e^-[hv/(k_BT)])²

The Attempt at a Solution

I tried to use compressibility factor, Z = PV/RT, but it doesn't work...
Plus, I understand that the law of corresponding states = properties of ALL gases are the same if compared under same conditions relative to their critical point. So, considering it is ALL gases, the Einstein heat capacity should also work. But, how? Is it a matter of normalization or something?

 

[TSny]

The Einstein model gives the heat capacity due to vibrations of atoms in a solid. So it is not dealing with gases.

Nevertheless, the equation for the heat capacity does appear to obey a "principle of corresponding states" in a general sense. Can you think of a way to "normalize" the temperature in C_V such that the equation would have exactly the same form for any solid?

 

[terp.asessed]

Hello, thank you for correcting me that the Einstein model is for SOLID, not gas!
Should I think in term of temperature b/c volume (hence pressure) is held constant?

 

[TSny]

In the case of gases, you normalized the temperature relative to the critical temperature of the gas. Is there a temperature that you can use in the Einstein solid that could take the place of critical temperature in a gas? Hint: Is there a way to associate a temperature with the vibration frequency $\small \nu$ of the the atoms in the solid? (I need to go eat. Back later)

 

[terp.asessed]

looking at Einstein equation, frequency (v) values within the same bracket as T are [hv/(k_BT)]...so, is this one way to associate v with T?

 

[TSny]

You're heading in the right direction. You want to define an "Einstein temperature", Θ_E, determined by the physical properties of the solid. Then normalize the temperature of the system ,T, by Θ_E. In the Einstein model there is only one physical property of the system that is relevant; namely, the frequency of vibration of the atoms in the solid. So, what would be a natural way to define Θ_E? You should be able to answer this by considering the dimensionless quantity hv/(k_BT) that appears throughout the expression for C_V.