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Effect Of Nucleus Motion On Energy - Practice Questions & MCQ

Edited By admin | Updated on Sep 25, 2023 25:23 PM | #NEET

Syllabus

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Effect of Nucleus motion on Energy is considered one the most difficult concept.
5 Questions around this concept.

Solve by difficulty

EASY

The wavelengths involved in the spectrum of deuterium $\left ( _{1}^{2} D\right )$ are slightly different from that of hydrogen spectrum, because

size of the two nuclei are different

nuclear forces are different in the two cases

masses of the two nuclei are different

attraction between the electron and the nucleus is different in the two cases.

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Concepts Covered - 1

Effect of Nucleus motion on Energy

Effect of Nucleus motion on Energy-

Till now in Bohr's model we have assumed that the all the mass of the atom is situated at the center of the atom. As the mass of the electron is very much small and negligible as compare to the mass of the nucleus so all thye mass is assumed to be concetrated at the center of the nucleus. But actually, centre of mass of nucleus-electron system is close to nucleus as it is heavy and to keep the centre of mass at rest, both electron and nucleus revolve around their centre of mass like a double star system as shown in figure. If r is the distance of electron from nucleus, the distances of nucleus and electron from the centre of mass, $r_{1}$ and $r_{2}$ , can be given as -

$\begin{aligned} r_{1} &=\frac{m_{e} r}{m_{N}+m_{e}} \\ \\ \text { and } \quad r_{2} &=\frac{m_{N} r}{m_{N}+m_{e}} \end{aligned}$

We can see that in the atom, nucleus and electron revolve around their centre of mass in concentric circles of radii $r_1$ and $r_2$ to keep the centre of mass at rest. In above system, we can analyze the motion of electron with respect to nucleus by assuming nucleus to be at rest and the mass of electron replaced by its reduced mass $\mu_{\mathrm{e}}$ , given as -

$\mu_{\mathrm{e}}=\frac{m_{N} m_{e}}{m_{N}+m_{e}}$

Now we can change our assumption and the system will look like as shown in the figure with reduced mass -

Now we can derive the equation obtained by Bohr with the reduced mass also -

$r_{n}=\frac{n^{2} h^{2}}{4 \pi^{2} k Z e^{2} m_{e}}$

Now after replacing the electron mass by its reduced mass, the equation become -

$\begin{aligned} r_{n}^{\prime} &=\frac{n^{2} h^{2}}{4 \pi^{2} k Z e^{2} \mu_{e}} \Rightarrow r_{n}^{\prime}=\frac{n^{2} h^{2}\left({m}_{N}+m_{e}\right)}{4 \pi^{2} k Z e^{2} {m}_{e} m_{N}} \\ \\ \text { or } \quad r_{n}^{\prime} &=r_{n} \times \frac{m_{e}}{\mu_{e}} \Rightarrow r=(0.529 \mathrm{A}) \frac{m n^{2}}{\mu Z} \end{aligned}$

But there will be no effect on the velocity because the term of mass is not present there -

$v_{n}=\frac{2 \pi k Z e^{2}}{n h}$

Similarly for the energy, we can write that -

$E_{n}=-\frac{2 \pi^{2} k^{2} Z^{2} e^{4} m_{e}}{n^{2} h^{2}}$

After putting the reduced mass in the equation -

$\begin{array}{l}{E_{n}^{\prime}=-\frac{2 \pi^{2} k^{2} Z^{2} e^{4} m_{N} m_{e}}{n^{2} h^{2}\left(m_{N}+m_{e}\right)}} \\ \\ {E_{n}^{\prime}=E_{n} \times \frac{\mu_{e}}{m_{e}} \Rightarrow E_{n}=-(13.6 \mathrm{eV}) \frac{Z^{2}}{n^{2}}\left(\frac{\mu}{m}\right)}\end{array}$

Thus, we can say that the energy of electron will be slightly less compared to what we have derived earlier. But for numerical
calculations this small change can be neglected unless in a given problem it is asked to consider the effect of motion of nucleus.