Question

Statement (A): The potential energy of an electron = $2\times TE$

Statement (B): The kinetic energy of an electron = $-KE$

Statement (C): $r_n=\frac{n^2{h}^2}{{4\pi }^2{me}^{2}Z}$

Which of the following option(s) is/are correct?

• A. All the statements are correct
• B. All the statements are incorrect
• C. A, B are correct and C is incorrect
• D. A, B are incorrect and C is correct

Answer 1

The correct option is A All the statements are correct

Centripetal force is provided by electrostatic force of attraction.

$\therefore \frac{m{v}^2}{r}=+\frac{Ke\times e}{r^2}$

$\Rightarrow m{v}^2=+\frac{K{e}^2}{r}$

$\Rightarrow v^2=\frac{K{e}^2}{rm}$ .... $\left(I\right)$

$\therefore KE=\frac{1}{2}m{v}^2=+\frac{K{e}^2}{2r}$

Also,

$P.E=\frac{K\times q_1\times q_2}{r}$

$=\frac{k\times e\times (-e)}{r}$

$=-\frac{K{e}^2}{r}$

$Totalenergy=KE+PE$

$\therefore TE=-\frac{K{e}^2}{2r}$ ....$\left(II\right)$

$\therefore TE=-KE$

$\therefore PE=2\times KE$

Also,

$mvr=\frac{nh}{2\pi }$ (Bohr's postulate)

$\Rightarrow m^2{v}^2{r}^2=\frac{n^2{h}^2}{4{\pi }^2}$

$\Rightarrow m^2{r}^2\times \frac{K{e}^2}{rm}=\frac{n^2{h}^2}{4{\pi }^2}$ ....[from $\left(i\right)$]

$\Rightarrow r=\frac{n^2{h}^2}{4{\pi }^{2}mk{e}^2}$.

Therefore, statement $1$, $2$ and $3$ are correct.

Hence, option $\left(A\right)$ is correct.