A free hydrogen atom after absorbing a photon of wavelength λ a gets excited from the state n = 1 to the state n = 4 . Immediately after that the electron jumps to n = m state by emitting a photon of wavelength λ e . Let the change in momentum of atom due to the absorption and the emission are Δ p a and Δ p e , respectively. If λ a λ e = 1 5 . Which of the option(s) is/are correct? [Use

Step-by-step solution

Energy for transition of electron from one orbit to other is given by- E 2 - E 1 = 13.6 Z 2 1 n 1 2 - 1 n 2 2 = h c λ ⇒ 1 λ = 13.6 h c . Z 2 1 n 1 2 - 1 n 2 2 Now as per question 1 λ a = 13.6 h c . Z 2 1 1 2 - 1 4 2 and 1 λ e = 13.6 h c . Z 2 1 m 2 - 1 4 2 ⇒ λ e λ a = 1 - 1 16 1 m 2 - 1 16 = 15 m 2 16 - m 2 But λ a λ e = 1 5 g i v e n ⇒ 16 - m 2 15 m 2 = 1 5 ⇒ 16 - m 2 = 3 m 2 ⇒ m 2 = 4 ⇒ m = 2 ⇒ C is correct Now, 1 λ e = 13.6 h c . Z 2 1 m 2 - 1 16 = 13.6 1242 × 1 1 4 - 1 16 a s h c = 1242 a n d z = 1 ⇒ λ e = 1242