The shortest wavelength of hydrogen atom in Lyman series is $\lambda$. The longest wavelength in Balmer series of $\mathrm{He}^{+}$ is

JEE Main · 2023 · 29 Jan 2023 · Shift 1 · Atomic Structure

The shortest wavelength of hydrogen atom in Lyman series is

λ

. The longest wavelength in Balmer series of

He^{+}

Answer: (B)

The shortest wavelength of the Lyman series of hydrogen is its series limit,

n = \infty \to 1

, for which

\frac{1}{λ} = R_{H}

, so

λ = \frac{1}{R _{H}}

. The longest wavelength of the Balmer series of

H e^{+}

comes from the smallest Balmer jump,

n = 3 \to 2

, with

Z = 2

\frac{1}{λ ^{'}} = R_{H} Z^{2} (\frac{1}{4} - \frac{1}{9}) = 4 R_{H} \cdot \frac{5}{36} = \frac{5 R _{H}}{9}

. Dividing one by the other gives

\frac{λ ^{'}}{λ} = \frac{9}{5}

, so the required wavelength is

\frac{9 λ}{5}

JEE Main · 2023 · 29 Jan 2023 · Shift 1 · Single correct option