A diatomic molecule is made of two masses  m_{1} and m_{2} which are separated by a distance r . If we calculate its rotational energy by applying Bohr's rule of angular momentum quantization, its energy will be given by (n is an integer)

  • Option 1)

    \frac{(m_{1}+m_{2})^{2}n^{2}h^{2}}{2m_{1}^{2}m_{2}^{2}r^{2}}

  • Option 2)

    \frac{n^{2}h^{2}}{2(m_{1}+m_{2})r^{2}}

  • Option 3)

    \frac{2n^{2}h^{2}}{(m_{1}+m_{2})r^{2}}

  • Option 4)

    \frac{(m_{1}+m_{2})n^{2}h^{2}}{2m_{1}\, m_{2}\, r^{2}}

 

Answers (1)

A diatomic molecule consists of two atoms of  masses m_{1} and m_{2} at a distance r apart. Let r_{1}and r_{2}  be the distances of the atoms from the centre of mass 

The moment of inertia of this molecule about an axis passing through its centre of mass and perpendicular to a line joining the atoms is

The solution is correct. So no need to change it

I= m_{1}r_{1}^{2}+m_{2}r_{2}^{2}

As\: \: \: m_{1}r_{1}= m_{2}r_{2}\: \: \:or\: \: r_{1}= \frac{m_{2}}{m_{1}}r_{2}

\because \: \: r_{1}+r_{2}= r

\therefore \: \: r_{1}= \frac{m_{2}}{m_{1}}\left ( r-r_{1} \right )

On rearranging, we get

r_{1}= \frac{m_{2}r}{m_{1}+m_{2}}

Similarly \: \: r_{2}= \frac{m_{1}r}{m_{1}+m_{2}}

Therefore,\: the \: moment\: of\: inertia\: can \: be\: written\: as

I= m_{1}\left ( \frac{m_{2}r}{m_{1}+m_{2}} \right )^{2}+m_{2}\left ( \frac{m_{1}r}{m_{1}+m_{2}} \right )^{2}

= \frac{m_{1}m_{2}}{m_{1}+m_{2}}r^{2}\cdots \cdots \cdots \cdots (i)

According to Bohr’s quantisation condition

L= \frac{nh}{2\pi }

or\: \: L^{2}= \frac{n^{2}h^{2}}{4\pi ^{2}}

Rotational\: \: energy, E= \frac{L^{2}}{2l}

E= \frac{n^{2}h^{2}}{8\pi ^{2}I}\cdots \cdots \cdots \cdots using(ii)

E= \frac{n^{2}h^{2}\left ( m_{1}+m_{2} \right )}{8\pi ^{2}\left ( m_{1}m_{2} \right )r^{2}}\cdots \cdots \cdots \cdots using(i)

= \frac{n^{2}h^{2}\left ( m_{1}+m_{2} \right )}{2m_{1}m_{2} r^{2}}            

In the question instead of  h, should be given.

 


Option 1)

\frac{(m_{1}+m_{2})^{2}n^{2}h^{2}}{2m_{1}^{2}m_{2}^{2}r^{2}}

Incorrect

Option 2)

\frac{n^{2}h^{2}}{2(m_{1}+m_{2})r^{2}}

Incorrect

Option 3)

\frac{2n^{2}h^{2}}{(m_{1}+m_{2})r^{2}}

Incorrect

Option 4)

\frac{(m_{1}+m_{2})n^{2}h^{2}}{2m_{1}\, m_{2}\, r^{2}}

Correct

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