What Is the Temperature of Each Block Before They Are Brought into Contact?

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SUMMARY

The discussion centers on calculating the temperature of two blocks modeled as Einstein oscillators before and after they reach thermal equilibrium. The first block contains N1 oscillators with frequency omega and total energy E1, while the second block has N2 oscillators with frequency 2omega and total energy E2. The relationship between temperature, entropy, and energy is established through the equation 1/T = dS/dE. The challenge lies in determining the changes in entropy (dS) and energy (dE) to find the temperatures of each block prior to contact and the common temperature post-equilibrium.

PREREQUISITES
  • Understanding of Einstein solid model and its implications on thermal properties
  • Familiarity with thermodynamic concepts, particularly entropy and internal energy
  • Knowledge of statistical mechanics, specifically the relationship between temperature, energy, and entropy
  • Proficiency in calculus for calculating derivatives of entropy and energy
NEXT STEPS
  • Study the derivation of the entropy formula for Einstein solids
  • Learn how to calculate changes in entropy (dS) and energy (dE) in thermodynamic systems
  • Explore the concept of thermal equilibrium and its mathematical implications
  • Investigate the implications of large N limits in statistical mechanics
USEFUL FOR

This discussion is beneficial for physicists, particularly those specializing in statistical mechanics, thermodynamics, and anyone interested in the behavior of systems at thermal equilibrium.

tysonk
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I'm kind of stuck on this problem, if someone could help me out that would be appreciated.

Consider 2 blocks treated as collections of Einstein oscillators. The first block has N1 oscillators of frequency omega. The second block has N2 oscillators of frequency 2omega. Initially the first block has a total energy E1 and the second has a total energy E2. Both N1 and N2 are very large, of order Avagadro's number. E1/(ℏ omega ) and E2/(ℏ omega ) are also very large. The blocks are brought into contact and reach thermal equilibrium without any energy escaping to the environment.
  • What is the temperature of each block before they are brought into contact?
  • What is the common temperature after they reach thermal equilibrium?

Thank you.
 
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For einstein solid,

1/T = dS/dE

Where E is internal energy and S is entropy. I can find relevant equations for E and S. But how do I calculate dS and dE?
Still not sure how to find their temp when in contact at equilibrium.
 

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