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Suppose an electron is attracted towards the origin by a force \frac{\mathrm{k}}{\mathrm{r}}, where \mathrm{k} is a constant and \mathrm{r} is the distance of the electron from the origin. By applying Bohr model to this system, the radius of the nth orbital of the electron is found to be r_n and the kinetic energy of the electron to be T_n. Then, which of the following is true?

Option: 1

T_n$ independent of $n, r_n \propto n


Option: 2

\mathrm{T}_{\mathrm{n}} \propto \frac{1}{\mathrm{n}}, \mathrm{r}_{\mathrm{n}} \propto \mathrm{n}


Option: 3

\mathrm{T}_{\mathrm{n}} \propto \frac{1}{\mathrm{n}}, \mathrm{r}_{\mathrm{n}} \propto \mathrm{n}^2


Option: 4

\mathrm{T}_{\mathrm{n}} \propto \frac{1}{\mathrm{n}^2}, \mathrm{r}_{\mathrm{n}} \propto \mathrm{n}^2


Answers (1)

best_answer

According to Bohr theory, \mathrm{mvr}=\mathrm{n} \frac{\mathrm{h}}{2 \pi} \Rightarrow \mathrm{v}=\frac{\mathrm{nh}}{2 \pi \mathrm{mr}}

and  \frac{\mathrm{mv}^2}{\mathrm{r}} \propto \frac{\mathrm{k}}{\mathrm{r}} \Rightarrow \frac{\mathrm{m}}{\mathrm{r}}\left(\frac{\mathrm{n}^2 \mathrm{~h}^2}{4 \pi^2 \mathrm{~m}^2 \mathrm{r}^2}\right) \propto \frac{\mathrm{k}}{\mathrm{r}}

\Rightarrow r_n \propto n

Kinetic energy of the electron,

\mathrm{T}=\frac{1}{2} \mathrm{mv}^2=\frac{1}{2} \mathrm{~m}\left(\frac{\mathrm{n}^2 \mathrm{~h}^2}{4 \pi^2 \mathrm{~m}^2 \mathrm{r}^2}\right) \Rightarrow \mathrm{T}_{\mathrm{n}} \propto \frac{\mathrm{n}^2}{\mathrm{r}^2}

\text { But as } \mathrm{r} \propto \mathrm{n} \text {, therefore } \mathrm{T} \propto \mathrm{n}^0

 

 

 

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Nehul

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