Partition Theory of Nitrigen to determine ther quantities (eg U, H, F)

[TFM]

Homework Statement

For a mole of nitrogen (N2) gas at room temperature and atmospheric pressure, compute the following: U, H, F, G, S and μ. The internal partition function is purely rotational, and the rotational constant ε for N2 is 0.00025 eV. The electronic ground state is not degenerate.

Homework Equations

[tex] Z_{total} = \frac{1}{N!}_1^N [/tex]

[tex] Z_1 = Z_{int} = \sum{exp(-E_{int}(s)/k_BT)} [/tex]

[tex] U = -\frac{\partial}{\partial \beta} ln z [/tex]

The Attempt at a Solution

Okay, I am having a slight problem calculating the Z value. I have used:


[tex] Z_1 = Z_{int} = \sum{exp(-E_{int}(s)/k_BT)} [/tex]

and inserted 300 K, E = 0.00025 eV abd the Boltzmann Constant (in eV) into this, and have got 9.67 *10^{-3}

I now want to use:

[tex] Z_{total} = \frac{1}{N!}_1^N [/tex]

but N is the number of particles, which is 1 Mole x Avagadros Number. This leaves with a large number which I have to find the factorial of, but it is too large for Excel. Have I gone wrong somewhere?

TFM

 

[anathema]

:

Hello, thank you for your post. Your calculation for Z_1 looks correct. However, for Z_{total}, you do not need to calculate the factorial of Avogadro's number. Instead, you can use the formula:

Z_{total} = \frac{1}{N!} (Z_1)^N

where N is the number of particles (in this case, 1 mole), and Z_1 is the internal partition function that you calculated. This should give you a more manageable number for Z_{total}.

Once you have calculated Z_{total}, you can use the equations you listed to calculate U, H, F, G, S, and μ. I hope this helps! Let me know if you have any further questions.

 

[nlightner]

:

Thank you for your question. It seems like you are on the right track with your calculations. However, there are a few things to consider when using the partition function to determine quantities like U, H, F, G, S, and μ.

First, it is important to note that the partition function for a single molecule (Z1) is not the same as the total partition function (Ztotal). The Ztotal takes into account the number of molecules present, while Z1 only considers the internal states of a single molecule. So, when using Ztotal, you do not need to use the factor of N! in the equation.

Secondly, when calculating U, H, F, G, S, and μ, you will need to use the following equations:

U = -kBT^2\frac{\partial}{\partial T}(lnZtotal)

H = U + kBT^2\frac{\partial}{\partial T}(lnZtotal)

F = -kBTlnZtotal

G = U + PV - TS = F + PV

S = -\frac{\partial F}{\partial T}

μ = \frac{\partial F}{\partial N}

where P is the pressure, V is the volume, and N is the number of molecules.

So, to summarize, when using the partition function to determine quantities, make sure you are using the correct equations and taking into account the number of molecules present. I hope this helps. Good luck with your calculations!