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(C) The energy of a transition in a hydrogen-like species is given by $\Delta E = 13.6 \ Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \ eV$.Fo

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$He^{+}$

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(C) The energy of a transition in a hydrogen-like species is given by $\Delta E = 13.6 \ Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} ight) \ eV$.For the first Lyman transition in $H$ $(Z=1)$: $\Delta E = 13.6 	imes 1^2 \left( \frac{1}{1^2} - \frac{1}{2^2} ight) = 13.6 	imes \frac{3}{4} = 10.2 \ eV$.For the second Balmer transition,$n_1 = 2$ and $n_2 = 4$ (since the first is $2 	o 3$ and the second is $2 	o 4$).Setting the energy equal: $10.2 = 13.6 	imes Z^2 \left( \frac{1}{2^2} - \frac{1}{4^2} ight)$.$10.2 = 13.6 	imes Z^2 \left( \frac{1}{4} - \frac{1}{16} ight) = 13.6 	imes Z^2 \left( \frac{3}{16} ight)$.$10.2 = 13.6 	imes Z^2 	imes 0.1875 = 2.55 	imes Z^2$.$Z^2 = \frac{10.2}{2.55} = 4$,so $Z = 2$.The species with atomic number $Z = 2$ is $He^+$,which is a hydrogen-like ion.

In the hydrogen spectrum,the energy during the first Lyman transition is $E = 10.2 \ eV$. The same energy is observed during the second Balmer series transition in which of the following species?

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