# Electronic partition function for molecule with degeneracies

• wiveykid
In summary: Also I am not too confident in the equation I found.If anyone understands degeneracy, electronic partition functions or excited electronic state calculations I would greatly appreciate any help. I would assume that you just use E=hf.
wiveykid

## Homework Statement

A atom had a threefold degenerate ground level, a non degenerate electronically excited level at 3500 cm^-1(setting the energy orgin as the ground electronic state energy of the atom ) and a threefold degenerate level at 4700 cm^-1 . Calculate the electronic partition function of this atom at 2000K

## Homework Equations

qel= sumnation{i=0inf}[gel*exp[-(Ei-Ei-1)/(kbT)]]

## The Attempt at a Solution

I see three levels in the problem
I believe, given the problem statement, that g0=3, g1=1 @3500cm-1, g2=3@4700cm-1

I came up with the equation
qel= 3+ 1*exp[-(E1-E0)/(kbT)] + 3*exp[-(E2-E1)/(kbT)]

T= 2000K, kb= Boltzmann constant ~1.38e-23

I am having trouble finding the energy values for each excited state. I am not sure if E=hv applies to this problem.

Also I am not too confident in the equation I found.

If anyone understands degeneracy, electronic partition functions or excited electronic stateI would greatly appreciate any help

I would assume that you just use
E = h f
= h c / lambda
= hbar c k
, no? (I'm assuming that the inverse lengths that the problem specifies are wavenumbers of photons that would be emitted from the corresponding energy differences.)

yes I guess I can just multiply those numbers by c to get the excited state energies. Doing this it seems that all the exponentials go to zero because kbT is so small and c is so large. I guess the electronic partition function ends up being equal to 3

wiveykid said:
yes I guess I can just multiply those numbers by c to get the excited state energies. Doing this it seems that all the exponentials go to zero because kbT is so small and c is so large. I guess the electronic partition function ends up being equal to 3
There is more to the calculation than just kBT and c. Try calculating the value of $\frac{\Delta E}{k_B T}$ for the two excited states.

wiveykid said:
I came up with the equation
qel= 3+ 1*exp[-(E1-E0)/(kbT)] + 3*exp[-(E2-E1)/(kbT)]
That's pretty much correct, except that it is E2-E0 in the final term.

I am having trouble finding the energy values for each excited state. I am not sure if E=hv applies to this problem.
Yes, it does.

Last edited:
!

Hello,

Thank you for your question. I can provide some guidance on how to approach this problem.

Firstly, degeneracy refers to the number of energy levels that have the same energy value. In this case, the threefold degenerate ground level means that there are three energy levels with the same energy value. Similarly, the threefold degenerate level at 4700 cm^-1 means that there are three energy levels with the same energy value of 4700 cm^-1.

Now, for the electronic partition function, we need to consider the energy levels and their degeneracies. The formula you have provided is correct, but we need to know the energy values for each level in order to calculate the partition function.

To find the energy values, we can use the equation E = hv, where E is the energy, h is the Planck's constant, and v is the frequency. We can also use the relation between frequency and wavenumber, which is v = c/λ, where c is the speed of light and λ is the wavelength.

Using these equations, we can find the energy values for the excited levels. For example, the non-degenerate excited level at 3500 cm^-1 would have an energy value of E = (3500 cm^-1)(c/λ), where c = 3.00 x 10^8 m/s and λ = 3.00 x 10^10 cm. Similarly, for the threefold degenerate level at 4700 cm^-1, we would have E = (4700 cm^-1)(c/λ).

Once we have the energy values for each level, we can plug them into the partition function equation and solve for qel. Your equation is correct, but we need to use the correct energy values for each level.

I hope this helps. If you have any further questions, please let me know. Good luck with your homework!

## 1. What is the electronic partition function for a molecule with degeneracies?

The electronic partition function for a molecule with degeneracies is a thermodynamic quantity that represents the sum of all possible energy states of electrons in a molecule, taking into account the degeneracies or multiple ways in which these states can be occupied.

## 2. How is the electronic partition function calculated?

The electronic partition function is calculated using the Boltzmann distribution, which takes into account the energy of each electronic state and the temperature of the system. The formula for calculating the partition function is Z = ∑gie-Ei/kT, where gi is the degeneracy of the ith state, Ei is the energy of the ith state, k is the Boltzmann constant, and T is the temperature in Kelvin.

## 3. What is the significance of the electronic partition function in thermodynamics?

The electronic partition function is a fundamental quantity in thermodynamics as it is used to calculate other thermodynamic properties such as the internal energy, entropy, and free energy of a system. It also provides information about the distribution of energy states within a molecule at a given temperature.

## 4. How does the degeneracy of energy states affect the electronic partition function?

The degeneracy of energy states has a direct effect on the electronic partition function as it determines the number of ways in which the system can distribute its energy. Higher degeneracy leads to a higher partition function, indicating that there are more possible energy states available for the system to occupy.

## 5. Can the electronic partition function be used to predict the behavior of a molecule?

Yes, the electronic partition function can be used to predict the thermodynamic behavior of a molecule, such as its heat capacity and equilibrium constants. It provides valuable information about the distribution of energy states within a molecule, which is crucial in understanding its behavior in different conditions.

Replies
1
Views
2K
Replies
3
Views
2K
Replies
1
Views
1K
Replies
2
Views
2K
Replies
1
Views
1K
Replies
3
Views
2K
Replies
6
Views
1K
Replies
1
Views
1K
Replies
2
Views
3K
Replies
1
Views
2K