$\mathbf{X}^{\mathbf{a}+}$ and $\mathbf{Y}^{\mathbf{b}+}$ are hydrogen-like species. The wavelength of light absorbed during the transition between the states with principal quantum numbers $n=1$ and $n=2$ of $\mathbf{X}^{\mathbf{a}+}$ is $\lambda$. The wavelength of light absorbed during the transition between the states with principal quantum numbers $n=2$ and $n=4$ of $\mathbf{Y}^{\mathbf{b}+}$ is $9\lambda$. The lowest possible value of $(\mathbf{a}+\mathbf{b})$ is ____.

JEE Advanced · 2026 · 17 May 2026 · Paper 2 · Atomic Structure

X^{a +}

and

Y^{b +}

are hydrogen-like species. The wavelength of light absorbed during the transition between the states with principal quantum numbers

n = 1

and

n = 2

X^{a +}

λ

. The wavelength of light absorbed during the transition between the states with principal quantum numbers

n = 2

and

n = 4

Y^{b +}

9 λ

. The lowest possible value of

(a + b)

is ____.

Answer: 3

For hydrogen-like species, the wavelength is given by:

\frac{1}{λ} = R_{H} Z^{2} [\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}}]

For

X^{a +}

(transition

n = 1

n = 2

\frac{1}{λ} = R_{H} Z_{x}^{2} [\frac{1}{1 ^{2}} - \frac{1}{2 ^{2}}] = R_{H} Z_{x}^{2} \cdot \frac{3}{4}

For

Y^{b +}

(transition

n = 2

n = 4

\frac{1}{9 λ} = R_{H} Z_{y}^{2} [\frac{1}{2 ^{2}} - \frac{1}{4 ^{2}}] = R_{H} Z_{y}^{2} \cdot \frac{3}{16}

Dividing the first equation by the second:

9 = \frac{Z _{x}^{2}}{Z _{y}^{2}} \times \frac{3/4}{3/16} = \frac{Z _{x}^{2}}{Z _{y}^{2}} \times 4

\frac{Z _{x}^{2}}{Z _{y}^{2}} = \frac{9}{4} \Rightarrow \frac{Z _{x}}{Z _{y}} = \frac{3}{2}

For minimum values:

Z_{x} = 3

and

Z_{y} = 2

Since these are hydrogen-like species (one electron),

a = Z_{x} - 1 = 2

and

b = Z_{y} - 1 = 1

Therefore,

a + b = 2 + 1 = 3

JEE Advanced · 2026 · 17 May 2026 · Paper 2 · Numerical answer