Rotational spectra - thermal population of rotational levels

In summary, rotational spectra is the measurement and analysis of electromagnetic radiation emitted or absorbed by molecules as they rotate. This allows for the determination of rotational energy levels and transitions. These levels are populated thermally based on the Boltzmann distribution, which is affected by temperature, moment of inertia, and degeneracy. Rotational spectra is used in molecular identification by comparing observed levels to known values, and is significant in astrophysics for studying the composition and physical conditions of celestial objects.
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Homework Statement



If the thermal population of the rotational levels is given by:

Nj/No = (2J+1)*exp(-hcBJ(J+1)/kT)

Calculate which state has the highest thermal population at a given temperature T. Calculation needs to be shown, not just a result.

Homework Equations



Nj/No = (2J+1)*exp(-hcBJ(J+1)/kT)

where:

Nj = population in excited j state
No = population in ground state
J = rotational quantum #
B = rotational constant
k = Boltzmann constant
T = temperature

The Attempt at a Solution



I honestly don't know where to even begin... The given equation appears to be in the form of a Boltzmann distribution. I'd assume a reasonable answer would be a function of J and T; where the function is greater than 1, obviously the Nj state would have the highest thermal population and vice versa. Other than that, I'm pretty much at a loss. Took a stab in the dark, differentiating the right side of the equation with respect to J then setting equal to zero to maximize the function and get it down to a single variable, temperature, but I seem to be getting nowhere fast.

Any help would be greatly appreciated, thanks.
 
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  • #2

Thank you for your question. I am happy to assist you in finding the solution to this problem.

Firstly, let's rewrite the given equation as follows:

Nj/No = (2J+1)*exp(-Ej/kT)

where Ej = hcBJ(J+1)

Now, we can see that the thermal population of each state is dependent on two factors: the rotational quantum number J and the temperature T. To determine which state has the highest thermal population at a given temperature T, we need to find the value of J that maximizes the function (2J+1)*exp(-Ej/kT).

To do this, we can differentiate the function with respect to J and set it equal to zero:

d/dJ [(2J+1)*exp(-Ej/kT)] = 0

Simplifying this equation, we get:

(2J+1)*(-Ej/kT)*exp(-Ej/kT) = 0

From this, we can see that the only way for the function to equal zero is if (-Ej/kT) = 0. This means that the value of J that maximizes the function is the one that makes (-Ej/kT) equal to zero. In other words, we need to find the value of J that makes the energy term (-Ej) equal to zero.

To find this value, we can use the equation for Ej:

Ej = hcBJ(J+1)

Setting this equal to zero, we get:

hcBJ(J+1) = 0

Solving for J, we get:

J = 0

Therefore, at a given temperature T, the state with the highest thermal population is the one with the rotational quantum number J = 0. This makes sense, as the ground state (J = 0) is always the most populated state at any temperature.

I hope this explanation helps. Let me know if you have any further questions.
 

1. What is rotational spectra?

Rotational spectra is the measurement and analysis of the electromagnetic radiation emitted or absorbed by molecules as they rotate around their own axis. This allows for the determination of the rotational energy levels and transitions between them.

2. How are rotational levels populated thermally?

Rotational levels are populated thermally based on the Boltzmann distribution, which states that at a given temperature, the higher energy levels will have a lower population compared to the lower energy levels. This is due to the fact that molecules at higher energy levels have a higher probability of transitioning to lower energy levels.

3. What factors affect the thermal population of rotational levels?

The thermal population of rotational levels is affected by the temperature of the system, the moment of inertia of the molecule, and the degeneracy of rotational states. Higher temperatures, larger moment of inertia, and higher degeneracy will result in a higher population of rotational levels.

4. How is rotational spectra used in molecular identification?

Rotational spectra is used in molecular identification by comparing the observed rotational energy levels and transitions to known values for different molecules. Each molecule has a unique rotational spectrum, allowing for its identification based on the observed peaks and transitions.

5. What is the significance of rotational spectra in astrophysics?

Rotational spectra is significant in astrophysics as it allows for the identification and study of molecules in space. The rotational spectra of molecules can be observed in the emission or absorption lines of light from celestial objects, providing valuable information about the composition and physical conditions of these objects.

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