Question

Which of the following electronic transitions requires that the greatest quantity energy be absorbed by the hydrogen atom?
A. \[n = 1\] to $n = 2$
B. $n = 2$ to $n = 4$
C. $n = 3$ to $n = 1$
D. $n = 1$ to $n = \infty $

Answer 1

Hint: To solve this question one must have a concept of postulates of Bohr atomic model and energy levels and then you can easily solve the following question. Firstly, in this question we used the formula to eliminate the incorrect option and then simply calculated the energies of the rest of the transitions and got our required solution.

Formula used:
$E = - 13.6\dfrac{{{Z^2}}}{{{n^2}}}$
Where, $E$ is the energy, $Z$ is the atomic number and $n$ is the principal quantum number.

Complete step by step answer:
As we know that the energy is given by,
$E = - 13.6\dfrac{{{Z^2}}}{{{n^2}}}$
And here we can see that the energy is inversely proportional to $E \propto \dfrac{1}{{{n^2}}}$. Because energy is inversely proportional to $n$. So, we can easily eliminate option C and option D.

Now, let us check which transitions require the greatest quantity of energy.
${E_1} = {E_2} - {E_1} \\
\Rightarrow {E_1} = \left( {\dfrac{{ - 1}}{{{2^2}}}} \right) - \left( {\dfrac{{ - 1}}{{{1^2}}}} \right) \\
\Rightarrow {E_1} = \dfrac{3}{4} \\ $
Similarly, for ${E_2}$
${E_2} = {E_4} - {E_2} \\
\Rightarrow {E_2} = \left( {\dfrac{{ - 1}}{{{4^2}}}} \right) - \left( {\dfrac{{ - 1}}{{{2^2}}}} \right) \\
\therefore {E_2} = \dfrac{3}{{16}} $
Here we can clearly see that, ${E_1} > {E_2}$. Hence the first transition would require the most quantity of energy to be absorbed by the hydrogen atom.

Thus, the correct option is A.

Note: We can easily get confused in eliminating the options and one can easily solve by finding all the energies of the transitions and after comparing we will get the required solutions. And note that $13.6\,eV$ is the smallest possible energy constant value of an electron.