Question

The force acting on the electron in a hydrogen atom depends on the principal quantum number as :

• A. $\mathrm{F}\propto {\mathrm{n}}^2$
• B. $\mathrm{F}\propto {\mathrm{n}}^4$
• C. $\mathrm{F}\propto \frac{1}{{\mathrm{n}}^2}$
• D. $\mathrm{F}\propto \frac{1}{{\mathrm{n}}^4}$

Answer 1

The correct option is D

$\mathrm{F}\propto \frac{1}{{\mathrm{n}}^4}$

Step 1 : Concept :

1. The principal quantum number represents the principal energy level in which an electron revolves around the nucleus.
2. It is denoted by the letter $\mathrm{n}$ and can have any integral value except $0$ i.e $\mathrm{n}=1,2,3.......$.etc.

Step 2 : Explanation :

The force acting on the electron in a hydrogen atom is :

$\mathrm{F}=\frac{{\mathrm{mv}}^2}{\mathrm{r}}$

Where, $\mathrm{F}$ = force, $\mathrm{m}$ = mass, $\mathrm{v}$ = velocity, $\mathrm{r}$ = radius.

As $\mathrm{v}\propto \frac{1}{\mathrm{n}}$ and $\mathrm{r}\propto {\mathrm{n}}^2$ (from the idea of Bohr's orbit)

$\therefore \mathrm{F}\propto \frac{1}{{\mathrm{n}}^4}$

Hence option $\left(\mathrm{D}\right)$is correct option.