Question

What will be the ratio of the wavelength of the first line to that of the second line of paschen series of H atom? A)256:175 B)175:256 C)15:16 D)24:27

Answer 1

The answer is (A) `256:175`

Explanation:

Your tool of choice here will be the Rydberg equation, which tells you the wavelength, `lamda`, of the photon emitted by an electron that makes a `n_i -> n_f` transition in a hydrogen atom.

`1/(lamda) = R * (1/n_f^2 - 1/n_i^2)`

Here

• `R` is the Rydberg constant, equal to `1.097 * 10^(7)` `"m"^(-1)`
• `n_i` is the initial energy level of the electron
• `n_f` is the final energy level of the electron

Now, the Paschen series is characterized by `n_f = 3`.

The first transition in the Paschen series corresponds to

`n_i = 4" " -> " " n_f = 3`

In this transition, the electron drops from the fourth energy level to the third energy level.

You will have

`1/(lamda_1) = R * (1/3^2 - 1/4^2)`

The second transition in the Paschen series corresponds to

`n_i = 5 " " -> " " n_f = 3`

This time, you have

`1/(lamda_2) = R * (1/3^2 - 1/5^2)`

Now, to get the ratio of the first line to that of the second line, you need to divide the second equation by the first one.

`(1/(lamda_2))/(1/(lamda_1)) = (color(red)(cancel(color(black)(R))) * (1/3^2 - 1/4^2))/(color(red)(cancel(color(black)(R))) * (1/3^2 - 1/5^2))`

This will be equivalent to

`(lamda_1)/(lamda_2) =(1/9 - 1/25)/ (1/9 - 1/16)`

`(lamda_1)/(lamda_2) = ((25 - 9)/(25 * 9))/((16 - 9)/(16 * 9)) = 16/(25 * color(red)(cancel(color(black)(9)))) * (16 * color(red)(cancel(color(black)(9))))/7 = 16^2/(25 * 7)`

Therefore, you can say that you have

`(lamda_1)/(lamda_2) = 256/175`