# Find Moment of Inertia with Rotational Spectrum Wavelengths

• Pengwuino
In summary: It has to be known that only such transitions are allowed where J changes by +1 or -1. If you have absorption spectrum the spectrum lines correspond to the transitions from J to J+1. In an emission spectrum, it is the opposite, the molecule emits a photon while it gets back from the J+1-th rotational level to the J-th one. The energy of a the photon emitted is hf=((J+2)(J+1)-J(J+1))\frac{\hbar^2}{2I}=(J+1)\frac{\hbar^2}{I} The emission spectrum of a two-atomic molecule consists of
Pengwuino
Gold Member
Ok so i have 5 wavelengths of the rotational spectrum of a certain molecule. I need to find the moment of inertia.

I have the equation down to I=(Hbar * wavelength)/(hc)

Do i just use the shortest wavelength to figure out the moment of intertia? No radius was given.

Pengwuino said:
Ok so i have 5 wavelengths of the rotational spectrum of a certain molecule. I need to find the moment of inertia.

I have the equation down to I=(Hbar * wavelength)/(hc)

Do i just use the shortest wavelength to figure out the moment of intertia? No radius was given.

How did you get this formula? Check it. The unit of I should be mass times length squared and you have "second".

ehild

The book gave me the equation and i figured what 'I' would be. Its an HCl module molecule.

Pengwuino said:
I have the equation down to I=(Hbar * wavelength)/(hc)
I don't know where that equation comes from--the units don't make sense.

Treating the HCL molecule as a rigid rotor, the allowable rotational energy levels are:
$$E_J = J (J +1) \frac {\hbar^2}{2I}$$

You should be able to relate the wavelengths to transitions between levels.

Doc Al said:
I don't know where that equation comes from--the units don't make sense.

Treating the HCL molecule as a rigid rotor, the allowable rotational energy levels are:
$$E_J = J (J +1) \frac {\hbar^2}{2I}$$

You should be able to relate the wavelengths to transitions between levels.

To Pendwuino:

It has to be known that only such transitions are allowed where J changes by +1 or -1. If you have absorption spectrum the spectrum lines correspond to the transitions from J to J+1. In an emission spectrum, it is the opposite, the molecule emits a photon while it gets back from the J+1-th rotational level to the J-th one.

The energy of a the photon emitted is

$$hf=((J+2)(J+1)-J(J+1))\frac{\hbar^2}{2I}=(J+1)\frac{\hbar^2}{I}$$
The emission spectrum of a two-atomic molecule consists of equidistant spectral lines, which correspond to transitions on to the levels J=0, J=1...and so on. The difference between the frequencies of two closest lines is
$$\Delta f = \frac{h}{4\pi^2I}$$
You know the wavelength of the spectral lines. Calculate the frequencies from the wavelengths
$$f=c/\lambda$$. Sort the frequencies and calculate the difference between the subsequent ones. Take the average: and calculate I from it.

ehild

## 1. What is the concept of moment of inertia in rotational motion?

Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It is similar to mass in linear motion and depends on the mass, shape, and distribution of mass of the object.

## 2. How can rotational spectrum wavelengths be used to find the moment of inertia?

Rotational spectrum wavelengths are the wavelengths of light absorbed or emitted by a rotating molecule. By analyzing these wavelengths, scientists can determine the rotational energy levels of the molecule and use this information to calculate its moment of inertia.

## 3. What is the relationship between moment of inertia and rotational spectrum wavelengths?

The moment of inertia is directly proportional to the rotational spectrum wavelengths. This means that as the moment of inertia increases, so does the rotational spectrum wavelength.

## 4. Can moment of inertia be calculated using only one rotational spectrum wavelength?

No, moment of inertia cannot be calculated using only one rotational spectrum wavelength. It requires multiple wavelengths to determine the rotational energy levels of the molecule and ultimately the moment of inertia.

## 5. What is the significance of calculating the moment of inertia using rotational spectrum wavelengths?

Calculating the moment of inertia using rotational spectrum wavelengths allows scientists to study the physical properties and behavior of molecules in a non-invasive way. It also provides valuable information for understanding molecular structures and chemical reactions.

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