Revolutions per second for hydrogen molecule

[Effect]

Homework Statement

The distance between the both nucleons in the H₂ molecule is 0.0741 nm
The energy for the 3 lowest radiation levels of the molecule are:
$E_{0}=0 \\ E_{1}=0.0151 eV \\ E_{2}=0.0453 eV$
How many revolutions per second does the molecule rotate with in the different states?

Homework Equations

See attachment

The Attempt at a Solution

How many revolutions per second does the molecule rotate with? I thought that if f in the last equation above is frequency then the rotational speed would be omega=f*2pi. So if we rewrite the last equation we get
$f=\frac{E_{v}}{(\nu+1/2)h} \Leftrightarrow \omega=\frac{2\pi* E_{v}}{(\nu+1/2)h}$

Of course the first level gives zero rev/s but for the others I get the wrong answer. When putting in the above values, I get

$\omega=\frac{2\pi* 0.0151* 1.606*10^{-19}}{(1+1/2)*6.626*10^{-34}}=1.53*10^{14} rev/s$ which is incorrect. Also the other value gives the wrong answer in this equation.

The correct answers should be:
$\omega_0 = 0 \\ \omega_1 = 5.2*10^{12} \\ \omega_2 = 9.0*10^{12} \\ (rev/s)$

I don't understand how to get to these answers. Shouldn't the above equation be used? Or is it something else that needs to be taken into consideration? Please help how to get to these answers.

 

[vela]

Homework Statement

The distance between the both nucleons in the H₂ molecule is 0.0741 nm
The energy for the 3 lowest radiation levels of the molecule are:
$E_{0}=0 \\ E_{1}=0.0151 eV \\ E_{2}=0.0453 eV$
How many revolutions per second does the molecule rotate with in the different states?

Homework Equations

See attachment

The Attempt at a Solution

How many revolutions per second does the molecule rotate with? I thought that if f in the last equation above is frequency then the rotational speed would be omega=f*2pi. So if we rewrite the last equation we get
$f=\frac{E_{v}}{(\nu+1/2)h} \Leftrightarrow \omega=\frac{2\pi* E_{v}}{(\nu+1/2)h}$

Of course the first level gives zero rev/s but for the others I get the wrong answer. When putting in the above values, I get

$\omega=\frac{2\pi* 0.0151* 1.606*10^{-19}}{(1+1/2)*6.626*10^{-34}}=1.53*10^{14} rev/s$ which is incorrect. Also the other value gives the wrong answer in this equation.

The correct answers should be:
$\omega_0 = 0 \\ \omega_1 = 5.2*10^{12} \\ \omega_2 = 9.0*10^{12} \\ (rev/s)$

I don't understand how to get to these answers. Shouldn't the above equation be used? Or is it something else that needs to be taken into consideration? Please help how to get to these answers.

The equation you started with applies to vibrations of the molecule, not rotations. It doesn't apply here.

You need to go back and review rotational kinematics to find out how the energy is related to the frequency of rotation.

 

[Effect]

Do you mean rotational dynamics $$E_{rotational}=1/2 I\omega^2$$
This?

I still don't get the right answer.

 

[vela]

Yes, that's right. It worked for me. Show your work.

 

[Effect]

$E_{rotational}=1/2 I\omega^2 \Leftrightarrow \omega=\sqrt{\frac{2E}{I}} \\ I=1/2mr^2$
$$\Rightarrow \omega=\sqrt{\frac{4E}{mr^2}}$$
$\Rightarrow \omega=\sqrt{\frac{4*0.0151*1.606*10^{-19}}{1.66*10^{-27}*(7.41*10^{-11})^2}}=1.03*10^{13}$

 

[vela]

Divide by $2\pi$ since you want the result in rev/s.

 

[Effect]

$\frac{1.03*10^{13}}{2\pi}=1.64*10^{12}$
Still not correct, correct answer should be $5.2*10^{12}$

 

[vela]

I don't get $1.03 \times 10^{13}$ in the numerator from the expression you gave above. I get $3.26 \times 10^{13}$.

 

[Effect]

You are correct, I must have made a careless mistake somewhere. Thank you for your assistance!

 

[ehild]

$\Rightarrow \omega=\sqrt{\frac{4*0.0151*1.606*10^{-19}}{1.66*10^{-27}*(7.41*10^{-11})^2}}=1.03*10^{13}$

Under the square root, the result is 1.0642x10²⁷. You took the square root of 1.0642, and added the magnitude 10¹³ instead of the square root of 10.642 x 10 ²⁶.