Partition Function for system with 3 energy levels

In summary, the conversation discusses determining the partition function for particles A, B, and C. It also includes a diagram of possible arrangements and the creation of two partition functions for distinguishable and indistinguishable particles. The question raised is why the method used does not work, and there is a disagreement on the interpretation of the restriction for indistinguishable particles.
  • #1
MigMRF
15
0
Homework Statement
A system contains 3 particles A, B and C. A can have the energies (0, Delta) while B and C can have the energies (0,Delta,6 Delta). Determine the partition function if the particles and distinguisable. Then determine the partition function if the particles are indistinguishable
Relevant Equations
Z=sum(e^(-beta*E))
I determined the partition function of the particle A, B and C.
1652369926116.png

1652369953371.png

C should be the same as B.
I then considered the situation, where all particles are in the system at the same time, and drew a diagram of all possible arrangements:
1652370142390.png

The grey boxes are the different partitions, given that we can't tell the difference on the particles. The number at the bottom of the table is the sum of all the energies.
From this table i created the two partition functions as shown below:
Distinguishable:
1652370259304.png

Indistinguishable:
1652370313306.png

The correct answer is the following partition functions:
1652370488128.png

1652370510644.png

So my question is. Why does my method not work?

Kind Regards
 
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  • #2
I agree with your answer for part (a). It looks like the given solution does not include any states where particle A is in the state with energy ##\Delta##.

For part (b), I'm not sure. When the particles are indistinguishable, we can no longer pick out "particle A" and restrict it to the first two energy levels, while allowing the other two particles to be in any of the three levels. This would be treating one of the particles differently than the other two, which doesn't make sense if the particles are indistinguishable. But, maybe they intend for us to interpret the restriction for part (b) as saying that the level with energy ##6 \Delta## is only allowed to hold at most two particles. So, we disallow the state where all three particles occupy this energy level. This appears to be how you interpreted it. If this is what they want, then I agree with your answer for part (b). It looks like their answer leaves out states where there are no particles in the lowest energy level.
 

FAQ: Partition Function for system with 3 energy levels

1. What is the partition function for a system with 3 energy levels?

The partition function for a system with 3 energy levels is the sum of the Boltzmann factors for each energy level. It is represented by the symbol Z and is given by the equation Z = e-E1/kBT + e-E2/kBT + e-E3/kBT, where E1, E2, and E3 are the energies of the three levels, kB is the Boltzmann constant, and T is the temperature.

2. How is the partition function related to the thermodynamic properties of a system with 3 energy levels?

The partition function is related to the thermodynamic properties of a system with 3 energy levels through the equation Z = e-E1/kBT + e-E2/kBT + e-E3/kBT. This equation allows us to calculate the average energy, entropy, and other thermodynamic properties of the system.

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

The partition function is a fundamental concept in statistical mechanics. It is used to calculate the thermodynamic properties of a system and provides a link between the microscopic properties of individual particles and the macroscopic properties of the system. It also plays a crucial role in the calculation of the equilibrium distribution of energy among the different energy levels of a system.

4. How does the partition function change with temperature?

The partition function is directly proportional to temperature. As the temperature increases, the value of e-Ei/kBT decreases, resulting in a decrease in the overall value of the partition function. This means that as the temperature increases, the system is more likely to occupy higher energy states.

5. Can the partition function be used for systems with more than 3 energy levels?

Yes, the partition function can be used for systems with any number of energy levels. The general equation for the partition function is Z = Σi e-Ei/kBT, where i ranges from 1 to the total number of energy levels in the system. This equation can be applied to systems with 3, 10, or even an infinite number of energy levels.

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