Question

The difference between n^th and (n+1)^th Bohr's radius of the hydrogen atom is equal to (n−1)^th Bohr's radius. Find the value of n.

Answer 1

Bohr radius is a physical constant value that expresses the most probable distance between the electron and the nucleus in a hydrogen atom in the ground state. The radius of ${\mathrm{n}}^{\mathrm{th}}$ Bohr's orbit is proportional to the ${\mathrm{n}}^2$, where $\mathrm{n}$ is the principal quantum number value.

As per the data given above:

$$\begin{gathered} \Rightarrow {(\mathrm{n}+1)}^2-{\mathrm{n}}^2={(\mathrm{n}-1)}^2 \\ \Rightarrow ({\mathrm{n}}^2+1+2\mathrm{n})-{\mathrm{n}}^2=({\mathrm{n}}^2+1-2\mathrm{n}) \\ \Rightarrow 1+2\mathrm{n}={\mathrm{n}}^2+1-2\mathrm{n} \\ \Rightarrow {\mathrm{n}}^2=4\mathrm{n} \\ \Rightarrow \mathrm{n}=4 \end{gathered}$$

Therefore, the value of $\mathrm{n}$ is $4$.