Question

When an electron has a transition from the state ( n + 1) to state n where n is quite large, then the frequency of the emitted radiation (v) will vary with n as

• A. $v\propto \frac{1}{n^3}$
• B. $v\propto \frac{1}{n^2}$
• C. $v\propto \frac{1}{n}$
• D. $v\propto n$

Answer 1

Answer: (A)

$v\propto \frac{1}{n^3}$

$\begin{array}{c}\text{Given,}\\ \text{electron emitted from}(n+1)\text{to}n\text{state}\\ \text{and}n\text{is very large}\end{array}$

$\text{Now,}$

$hv=13.6(\frac{1}{n_1^2}-\frac{1}{n_2^2})$

$\begin{array}{c}hv=13.6(\frac{1}{(n+1{)}^2}-\frac{1}{n^2})\\ hv=13\cdot 6\left(\frac{n^2-(n+1{)}^2}{(n+1{)}^2\left(n^2\right)}\right)\end{array}$

$\begin{array}{c}=13\cdot 6\left(\frac{n^2-(n^2+1+2n)}{(n+1{)}^2\left(n^2\right)}\right)\\ =13\cdot 6\left(\frac{-2h}{n+1}\right)n+1\approx n\end{array}$

$v\propto \frac{1}{n^3}$