Einstein's heat capacity model .and law of corresponding

In summary, the conversation is discussing the Einstein heat capacity equation and how it obeys the law of corresponding states. The equation is used for solids, not gases, but it appears to have a similar principle. The speaker suggests normalizing the temperature in the equation by a quantity called the "Einstein temperature", which could be determined by the frequency of vibration of atoms in the solid.
  • #1
terp.asessed
127
3

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/NkB = 3R(hv/(kBT))2e-[hv/(kBT)]/(1-e-[hv/(kBT)])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?
 
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  • #2
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 CV such that the equation would have exactly the same form for any solid?
 
  • #3
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?
 
  • #4
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)
 
  • #5
looking at Einstein equation, frequency (v) values within the same bracket as T are [hv/(kBT)]...so, is this one way to associate v with T?
 
  • #6
You're heading in the right direction. You want to define an "Einstein temperature", ΘE, determined by the physical properties of the solid. Then normalize the temperature of the system ,T, by ΘE. 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 ΘE? You should be able to answer this by considering the dimensionless quantity hv/(kBT) that appears throughout the expression for CV.
 

1. What is Einstein's heat capacity model?

Einstein's heat capacity model is a theoretical model developed by Albert Einstein in 1907 to explain the heat capacity of solids. It states that the heat capacity of a solid at a constant volume is directly proportional to its absolute temperature and is independent of the type of material.

2. How does the law of corresponding states relate to Einstein's heat capacity model?

The law of corresponding states is a generalization of Einstein's heat capacity model, stating that all substances will exhibit similar behavior at the same reduced temperature and pressure. This means that the heat capacity of a solid will follow the same trend regardless of the material, as long as it is at the same reduced temperature.

3. What is the significance of Einstein's heat capacity model in thermodynamics?

Einstein's heat capacity model was a breakthrough in thermodynamics, as it provided a theoretical explanation for the behavior of solids at high temperatures. It also helped to pave the way for further developments in the field of statistical mechanics, which is the study of the behavior of a large number of particles.

4. How accurate is Einstein's heat capacity model?

Einstein's heat capacity model is a simplified model and is not accurate for all materials at all temperatures. It is most accurate for higher temperatures and for materials with simple crystal structures. At lower temperatures, more complex models are needed to accurately describe the behavior of solids.

5. Can Einstein's heat capacity model be applied to liquids and gases?

No, Einstein's heat capacity model is specifically designed for solids and does not apply to liquids or gases. The behavior of liquids and gases at high temperatures is better described by other models, such as the ideal gas law or the van der Waals equation.

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