x + 4 x is the ratio of energies of photons produced due to transition of an electron of hydrogen atom from its (i) third permitted energy level to the second level and (ii) the highest permitted en

Question

x + 4 x ​ is the ratio of energies of photons produced due to transition of an electron of hydrogen atom from its (i) third permitted energy level to the second level and (ii) the highest permitted energy level to the second permitted level. The value of x will be

TestHub · Accepted Answer

Energy of a photon released in transition of electron in hydrogen atom is given by,
$E = 13.6 (\frac{1}{( n _{2} ) ^{2}} - \frac{1}{( n _{1} ) ^{2}}) e V$
$\Rightarrow \frac{E _{1}}{E _{2}} = \frac{13.6 ( \frac{1}{4} - \frac{1}{9} )}{13.6 ( \frac{1}{4} - \frac{1}{\infty} )} = \frac{x}{x + 4} \Rightarrow \frac{\frac{1}{4} - \frac{1}{9}}{\frac{1}{4}} = \frac{x}{x + 4}$
$\Rightarrow \frac{5}{9} = \frac{x}{x + 4}$
$\Rightarrow 5 x + 20 = 9 x$
$\Rightarrow x = 5$

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