Why is it not possible to derive Cv from equation of state?

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SUMMARY

The discussion centers on the inability to derive the heat capacity at constant volume (Cv) from the equation of state for a pure material, specifically when only the equation of state and the number of particles are known. It is established that while relationships like Cp - Cv = R can be derived, determining Cv or Cp requires additional information about the molecular degrees of freedom (monatomic, diatomic, polyatomic) and experimental data on how these heat capacities vary with temperature. This highlights a fundamental difference in the nature of thermodynamic quantities derived from the equation of state.

PREREQUISITES
  • Understanding of thermodynamic concepts such as heat capacity (Cv and Cp)
  • Familiarity with the ideal gas law (PV = nRT)
  • Knowledge of molecular structure and degrees of freedom
  • Basic principles of statistical mechanics
NEXT STEPS
  • Research the relationship between heat capacities for different molecular types (monatomic, diatomic, polyatomic)
  • Study the derivation of Cv and Cp from statistical mechanics principles
  • Explore experimental methods for measuring heat capacities at varying temperatures
  • Learn about the implications of the equipartition theorem in thermodynamics
USEFUL FOR

This discussion is beneficial for students and professionals in thermodynamics, physicists studying molecular behavior, and engineers involved in material science and heat transfer applications.

Frank0
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Assume you have no knowledge about the microscopic information, all you know about the system of pure material is it's equation of state and number of particles. Why can't we get Cv=(∂U/∂T)V from here(if this is wrong how do we solve it?)? Why can we obtained some other quantities just from equation of state(Cp-Cv, (∂U/∂V)T for example)? Is there any fundamental difference between these different types of thermodynamic quantities?
 
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Frank0 said:
Assume you have no knowledge about the microscopic information, all you know about the system of pure material is it's equation of state and number of particles. Why can't we get Cv=(∂U/∂T)V from here(if this is wrong how do we solve it?)? Why can we obtained some other quantities just from equation of state(Cp-Cv, (∂U/∂V)T for example)? Is there any fundamental difference between these different types of thermodynamic quantities?
A simple equation of state would be: PV=nRT

You would be able to determine that Cp(T)-Cv(T) = R. But it is not enough to allow you to determine Cv or Cp. To determine Cp or Cv you first would need information about the degrees of freedom of the molecules (monatomic, diatomic, polyatomic). If it is not monatomic, one would also need to have experimental data as to how Cv and Cp varied with temperature.

AM
 

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