Partition Function of N particles in an assymetrical box

In summary, the conversation discusses the calculation of the probability and energy of molecules in a gas, as well as the average energy using a specific expression. The partition function is introduced and used to demonstrate the equivalence of the average energy expression, with the partition function then being found and used to calculate the average energy. The use of integrals instead of sums in the partition function is also explained.
  • #1
FranciscoSili
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Homework Statement


Consider a gas sufficiently diluted containing N identical molecules of mass m in a box of dimensions Lx, Ly, Lz.

  • Calculate the probability of finding the molecules in any of their quantum states.
  • Calculate the energy of each quantum state εr, as a function of the quantum numbers ##n_x##, ##n_y## and ##n_z##.
  • Calculate the average energy using the expression
    $$\bar \varepsilon = \sum_{r} P_r \varepsilon_r$$
    Demonstrate that the last expression is equivalent to $$\bar \varepsilon = -\frac {\partial ln(Z)} {\partial \beta}$$
    where ## Z= \sum_{r} exp\{-\beta\varepsilon_r\} ##, it's called the partition function.
  • Found the partition function and calculate the average energy.

    Help: the sum over ##r## can be replaced by an integral. Why?

    2. The attempt at a solution
    I have tried solving this problem this way, but I am not sure this is the correct solution and/or way.

    First, I calculated the energy for a single particle in this box as $$\varepsilon = {\frac {h^2} {8m}} \left[ \left( \frac {n_x} {L_x} \right)^2 + \left( \frac {n_y} {L_y} \right)^2 + \left( \frac {n_z} {L_z} \right)^2 \right]$$
    Because these particles are indistinguishable I can use the property of the partition function that reads $$Z\left(T,V,N\right) = \frac 1 {N!} Z\left(T,V,1\right)^N$$ that will let me calculate only the partition function of a single particle and then obtain it to N of them.
    Now I calculate the partition function for one molecule as indicated in the problem statement, $$Z\left(T,V,1\right)=\sum_{n_x,n_y,n_z} exp\left\{\ {\frac {-h^2\beta} {8m}} \left[ \left( \frac {n_x} {L_x} \right)^2 + \left( \frac {n_y} {L_y} \right)^2 + \left( \frac {n_z} {L_z} \right)^2 \right]\right\}$$ which can be split into three different sums, multiplying one another: $$Z\left(T,V,1\right)=\sum_{n_x=1}^\infty exp\left\{\ {\frac {-\beta h^2} {8m}}\left( \frac {n_x} {L_x} \right)^2\right\} \sum_{n_y=1}^\infty exp\left\{\ {\frac {-\beta h^2} {8m}}\left( \frac {n_y} {L_y} \right)^2\right\} \sum_{n_z=1}^\infty exp\left\{\ {\frac {-\beta h^2} {8m}}\left( \frac {n_z} {L_z} \right)^2\right\}.$$ If I can change the sums for integrals which, approximately, run from 0 to infinity, the partition function becomes
    \begin{equation}
    Z\left(T,V,1\right)=\int_0^\infty exp\left\{\ {\frac {-\beta h^2} {8m}}\left( \frac {n_x} {L_x} \right)^2\right\} \, dn_x \int_0^\infty exp\left\{\ {\frac {-\beta h^2} {8m}}\left( \frac {n_y} {L_y} \right)^2\right\} \, dn_y \int_0^\infty exp\left\{\ {\frac {-\beta h^2} {8m}}\left( \frac {n_z} {L_z} \right)^2\right\} \, dn_z,
    \end{equation}
    and solving them we get
    \begin{equation}
    Z\left(T,V,1\right)= \left(\frac {2m\pi} {h^2\beta} \right)^{3/2} L_x L_y L_z.
    \end{equation}
    Then I can calculate the partition function for the hole system of N particles, then the probability ##P_r## and finally the average energy,##\bar \varepsilon##, as in the statement

    Any help or comments that you might have is welcomed, since this is my second week of my Statistical Mechanics course. Thank you very much :D
 
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  • #2
I think your work looks good.

The order of the questions indicates to me that you are to answer the first 3 dotted questions before working out the partition function. I'm guessing that in the first question, you are to write out a general expression for ##P_r## in terms of the symbols ##\epsilon_r## and ##\beta## but you are not meant to evaluate it explicitly. Likewise in the third question, express the average energy in terms of ##\epsilon_r## and ##\beta## using your expression for ##P_r## from the first question. The partition function ##Z## is defined here and you should show the identity involving the derivative of ##Z## with respect to ##\beta##. Finally, in the fourth (last) question, evaluate ##Z## as you have done and then evaluate explicitly the average energy. Hope I'm not misleading you here.
 
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  • #3
TSny said:
I think your work looks good.
Ohh ok, that's good to hear. Thank you.

TSny said:
The order of the questions indicates to me that you are to answer the first 3 dotted questions before working out the partition function. I'm guessing that in the first question, you are to write out a general expression for ##P_r## in terms of the symbols ##\epsilon_r## and ##\beta## but you are not meant to evaluate it explicitly. Likewise in the third question, express the average energy in terms of ##\epsilon_r## and ##\beta## using your expression for ##P_r## from the first question. The partition function ##Z## is defined here and you should show the identity involving the derivative of ##Z## with respect to ##\beta##. Finally, in the fourth (last) question, evaluate ##Z## as you have done and then evaluate explicitly the average energy. Hope I'm not misleading you here.

Yes I know that I did everything in reverse, but the most complicated calculations were the ones for the partition function, so that's what I wanted to ask, if it was correct. I should have said that in the beginning, sorry for that.

Thank you.
 
  • #4
FranciscoSili said:
Yes I know that I did everything in reverse, but the most complicated calculations were the ones for the partition function, so that's what I wanted to ask, if it was correct. I should have said that in the beginning, sorry for that.

Thank you.
That's fine. Good work.
 
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Related to Partition Function of N particles in an assymetrical box

1. What is the partition function of N particles in an assymetrical box?

The partition function of N particles in an assymetrical box is a mathematical concept used in statistical mechanics to describe the distribution of particles in a given system. It represents the total number of possible ways that the particles can be arranged within the box, taking into account their energy levels and interactions with each other.

2. How is the partition function calculated?

The partition function is calculated by summing over all possible energy states of the particles in the system, using the Boltzmann distribution formula. This takes into account the energy levels of each particle and their corresponding probabilities of occupying each state.

3. Why is the partition function important in statistical mechanics?

The partition function is important in statistical mechanics because it allows us to calculate important thermodynamic quantities, such as the entropy, free energy, and average energy of a system. These quantities provide valuable insights into the behavior of the system and can be used to make predictions about its macroscopic properties.

4. How does the shape of the box affect the partition function?

The shape of the box can affect the partition function by altering the possible energy states that the particles can occupy. In an asymmetrical box, the particles may have different energy levels depending on their position, leading to a different partition function compared to a symmetrical box.

5. Can the partition function be used for systems with a large number of particles?

Yes, the partition function can be used for systems with a large number of particles. In fact, it is often used to describe macroscopic systems with millions or even billions of particles. However, as the number of particles increases, the calculation of the partition function becomes more complex and may require advanced mathematical techniques or computer simulations.

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