Question

The ratio of the wavelengths of the light absorbed by a hydrogen atom when it undergoes n = 2 → n = 3 and n = 4 → n = 6 transitions, respectively, is ______.

Options

• `1/36`

• `1/16`

• `1/9`

• `1/4`

Answer 1

The ratio of the wavelengths of the light absorbed by a hydrogen atom when it undergoes n = 2 → n = 3 and n = 4 → n = 6 transitions, respectively, is `bbunderline(1/4)`.

Explanation:

`1/λ = RZ^2(1/(n_1^2)  - 1/(n_2^2))`

For 2 → 3 ⇒ `1/λ_1 = R xx (1)^2 (1/(2^2)  - 1/(3^2))`   ...(1)

For 4 → 6 ⇒ `1/λ_2 = R xx (1)^2 (1/(4^2)  - 1/(6^2))`   ...(2)

Divide (1) & (2)

`(λ_2)/(λ_1) = (1/(2^2)  - 1/(3^2))/(1/(4^2)  - 1/(6^2)`

⇒ `(λ_2)/(λ_1) = (1/4  - 1/9)/(1/16  - 1/36`

⇒ `(λ_2)/(λ_1) = ((9 - 4)/36)/((36 - 16)/576)`

⇒ `(λ_2)/(λ_1) = (5/36)/(20/576)`

⇒ `(λ_2)/(λ_1) = 5/36 xx 576/20`

⇒ `(λ_2)/(λ_1) = 16/4`

⇒ `(λ_2)/(λ_1) = 4/1`

∴ `(λ_1)/(λ_2) = 1/4`