Question

Another Hydrogen atom Due on Friday, Dec 7 at 11:59 pm (EST) Five possible transitions for a hydrogen atom are listed below: Select whether the atom gains or loses energy for each transition. Submit Answer Tries 0/10 Find the transition where the atom gains the most energy. How much energy does the atom gain? Subemit Answer Tries 0/10 Find the transition which will emit the shortest wavelength photon. What is the energy of the photon of this transition? Submit Answer Tries 0/10 Post DisCUussion

Answer 1

The energy levels of the hydrogen atom is
   
And the transition energy from the level n to level m is
    $\frac{hc}{\lambda_{n_i \rightarrow n_f}}=E_{n_i}-E_{n_f}=\frac{13.6}{n_f^2}-\frac{13.6}{n_i^2}$ ..................(1)
    $\Rightarrow \frac{1}{\lambda_{n_i \rightarrow n_f}}=\frac{13.6}{hc}\left (\frac{1}{n_f^2}-\frac{1}{n_i^2} \right )$
Let use give a label for the overall constant
    $A=\frac{hc}{13.6}$
And so,
    $\Rightarrow \frac{1}{\lambda_{n_i \rightarrow n_f}}=\frac{1}{A}\left (\frac{1}{n_f^2}-\frac{1}{n_i^2} \right )$
   $\Rightarrow {\lambda_{n_i \rightarrow n_f}}=\frac{A}{\frac{1}{n_f^2}-\frac{1}{n_i^2}}$ ..........(2)

And if $n_f<n_i$ , then the atom emits photon and so loses energy.

If $n_f>n_i$ , then, the atom absorbs photon and so gains energy.

In the tabular form
  

options	$n_i$	$n_f$	Check $n_f>n_i$ or $n_f<n_i$	Energy gain / Energy loss	${\lambda_{{n_i}\rightarrow {n_f}}}$ (eqn 2)	Comments	Energy (eqn 1)
a	2	6	$n_f>n_i$ Absorption	Energy gain	- 4.2 A
b	6	4	$n_f<n_i$ emission	Energy loss	28.8 A
c	4	7	$n_f>n_i$ Absorption	Energy gain	-23.76 A	Maximum energy gain	0.572 eV
d	7	3	$n_f<n_i$ emission	Energy loss	11.025 A	Shortest wavelength of photon	1.234 eV
e	5	3	$n_f<n_i$ emission	Energy loss	14.0625 A

So, the transition from n_i = 4 to n_f = 7 has the maximum energy gain. And the corresponding energy is 0.572 eV.

The transition from n_i = 7 to n_f = 3 has the shortest wavelength of photon. And the corresponding energy is 1.234 eV.