The difference between the radii of $3^{\text{rd}}$ and $4^{\text{th}}$ orbits of $\ce{Li^{2+}}$ in $\Delta R_{1}$. The difference between the radii of $3^{\text{rd}}$ and $4^{\text{th}}$ orbits of $\ce{He^{+}}$ is $\Delta R_{2}$. Ratio $\Delta R_{1}: \Delta R_{2}$ is:

JEE Main · 2020 · 05 Sep 2020 · Shift 1 · Atomic Structure

The difference between the radii of

3^{rd}

and

4^{th}

orbits of

Li X^{2 +}

Δ R_{1}

. The difference between the radii of

3^{rd}

and

4^{th}

orbits of

He X^{+}

Δ R_{2}

. Ratio

Δ R_{1} : Δ R_{2}

is:

Answer: (B)

For a hydrogen-like ion the orbit radius is

r_{n} = \frac{0.529 n ^{2}}{Z}

A, so the gap between the third and fourth orbits is

Δ r = \frac{0.529}{Z} (16 - 9) = \frac{0.529 \times 7}{Z}

, which scales simply as

\frac{1}{Z}

. For

Li X^{2 +}

Z = 3

, and for

He X^{+}

Z = 2

, so

Δ R_{1} : Δ R_{2} = \frac{1}{3} : \frac{1}{2} = 2 : 3

JEE Main · 2020 · 05 Sep 2020 · Shift 1 · Single correct option