Question

Of the following transitions in the hydrogen atom, which gives an emission line of the highest frequency
a) n=1 and n=3
b) n=2 and n=1
c) n=3 and n=10
d) n=10 and n=3

Answer 1

Hint: The electrons when moved from lower to higher energy state absorb energy from the incident radiation. But when there is a transition from higher to lower energy state it emits energy radiation of a particular frequency. Greater the difference in transition from higher energy state to lower energy state greater is the frequency of the radiation emitted.

Complete step-by-step answer:
In a hydrogen atom, the energy in the nth energy state is given by,
${{E}_{n}}=\dfrac{-13.6}{{{n}^{2}}}$ . From this equation we can clearly see that as we move towards the higher permitted orbits the energy becomes less and less negative. From this we can imply that the energy basically increases. Now let us say an electron makes a transition from higher energy state m, to lower energy state n. Then the energy radiation emitted from this transition is given by,
$\begin{align}
  & \text{Energy emitted=}{{E}_{m}}-{{E}_{n}}=\dfrac{-13.6}{{{m}^{2}}}-\dfrac{-13.6}{{{n}^{2}}}=-13.6\left( \dfrac{1}{{{m}^{2}}}-\dfrac{1}{{{n}^{2}}} \right) \\
 & {{E}_{m}}-{{E}_{n}}=-13.6\left( \dfrac{{{n}^{2}}-{{m}^{2}}}{{{m}^{2}}\times {{n}^{2}}} \right) \\
\end{align}$
It is clear from the above equation that if the difference between the permitted orbits or the energy states is greater, then the energy emitted will also be greater. The energy emitted can also be written as,
$E=h\gamma $ where h is the Planck’s constant and $\gamma $ is the frequency of the emitted radiation. From this we can conclude that if the difference between the transition from higher to lower energy states is large, then the frequency of emitted radiation will also be more. If we see the above given options, then the transition from the higher energy state i.e. 10 to lower energy state i.e. 3 corresponds to maximum energy radiation.

So, the correct answer is “Option d”.

Note: When an electron undergoes transition from lower to higher energy state, it absorbs energy from the incident radiation. The energy of the incident radiation is given by $E=h\gamma $. When the electron makes the transition from the same higher to the same lower energy state, whatever energy it absorbs it emits back and hence we conclude that the frequency of emission is $\gamma $.