Partition Function: Which Energy Relationship?

In summary, there are two forms of the energy equation, E = kT^2 \frac{\partial \ln Z}{\partial T} and E = - \frac{\partial \ln Z}{\partial \beta}, which can be converted to each other using the chain rule since \beta=\frac{1}{k\,T}. The first equation is the correct one and the second equation is incorrect, even though it was mistakenly used twice in a solution on the MIT website.
  • #1
touqra
287
0
Is the energy given by the first or the second? I have seen both relationships in different websites, and I am confused.

[tex] E = kT^2 \frac{\partial Z}{\partial T} [/tex]

or

[tex] E = - \frac{\partial ln Z}{\partial \beta} [/tex]
 
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  • #2
The first one should be

[tex] E = kT^2 \frac{\partial \ln Z}{\partial T} [/tex]

You can go from one to the other by the chain rule since [tex]\beta=\frac{1}{k\,T}[/tex]
 
  • #3
Rainbow Child said:
The first one should be

[tex] E = kT^2 \frac{\partial \ln Z}{\partial T} [/tex]

You can go from one to the other by the chain rule since [tex]\beta=\frac{1}{k\,T}[/tex]

I did realize you can go from one to the other before I post this up. But the MIT website solution happily used the same equation (1) twice. For example, page 6 of http://web.mit.edu/Physics/graduate/gen1sol_S01.pdf
 
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  • #4
It's simply wrong! :smile:
 
  • #5
Yes, Rainbow Child is right.
 

1. What is the partition function in thermodynamics?

The partition function is a mathematical concept used in thermodynamics to describe the distribution of energy among the different states of a system. It is a function of temperature and other thermodynamic variables, and it allows us to calculate the average energy of a system and the probability of finding the system in a particular state.

2. How is energy related to the partition function?

The partition function is directly related to the energy of a system. It represents the sum of the energies of all possible states that a system can occupy. The higher the energy of a state, the lower its probability of being occupied, according to the Boltzmann distribution.

3. What is the significance of the partition function in statistical mechanics?

The partition function is a crucial concept in statistical mechanics because it allows us to calculate thermodynamic properties of a system, such as entropy and free energy. It also provides a link between the microscopic and macroscopic properties of a system, making it an essential tool for understanding the behavior of matter at the atomic and molecular level.

4. How does the partition function change with temperature?

The partition function is a function of temperature; therefore, it changes with temperature. As the temperature increases, the partition function also increases because more states become available to the system. This means that the system has a higher average energy and a higher entropy at higher temperatures.

5. Can the partition function be used for all types of systems?

Yes, the partition function can be used for all types of systems, as long as they are in thermal equilibrium. This includes ideal gases, solids, liquids, and even complex systems such as proteins and polymers. However, for more complicated systems, the calculations may become more challenging, and approximations may need to be made.

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