Question

Let $v_n$ and $E_n$ be the respective speed and energy of an electron in the nth orbit of radius $r_n$, in a hydrogen atom, as predicted by Bohr's model. Then

• A. Plot of $E_n{r}_n/E_1{r}_1$ as a function of $n$ is a straight line of slope $0$.
• B. Plot of $r_n{v}_n/r_1{v}_1$ as a function of $n$ is a straight line of slope $1$.
• C. Plot of $\ln \left(\frac{r_n}{r_1}\right)$ as a function of $\ln \left(n\right)$ is a straight line of slope $2$.
• D. Plot of $\ln \left(\frac{r_n{E}_1}{{E_{n}r}_1}\right)$ as a function of $\ln \left(n\right)$ is a straight line of slope $4$.

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A Plot of $E_n{r}_n/E_1{r}_1$ as a function of $n$ is a straight line of slope $0$.
B Plot of $r_n{v}_n/r_1{v}_1$ as a function of $n$ is a straight line of slope $1$.
C Plot of $\ln \left(\frac{r_n}{r_1}\right)$ as a function of $\ln \left(n\right)$ is a straight line of slope $2$.
D Plot of $\ln \left(\frac{r_n{E}_1}{{E_{n}r}_1}\right)$ as a function of $\ln \left(n\right)$ is a straight line of slope $4$.

$v_n\propto \frac{1}{n}{E}_n\propto \frac{1}{n^2}{r}_n\propto n^2$
$\therefore E_n{r}_n\propto n^0\therefore E_n{r}_n\propto E_1{r}_1\frac{E_n{r}_n}{E_1{r}_1}=$ constant ) (slope $=0$)
$r_n{v}_n\propto n^2\times \frac{1}{n}\propto n\therefore \frac{r_n{v}_n}{r_1{v}_1}=n$ (slope $=1$)
$r_n\propto n^2\therefore \frac{r_n}{r_1}=n^2\ln \frac{r_n}{r_1}=2\ln \left(n\right)$ (slope $=2$)
$\frac{r_n}{E_n}\times \frac{E_1}{r_1}=n^4$
$\ln \left(\frac{r_n{E}_1}{E_n{r}_1}\right)=4\ln \left(n\right)$ (slope $=4$)