Question

Which of the following transitions in $H{e}^{+}$ion will give rise to a spectral line which has the same wavelength as some spectral line in the hydrogen atom?
(1) $n=4$ to $n=2$
(2) $n=8$ to $n=4$
(3) $n=6$ to $n=3$
(4) $n=6$ to $n=2$

• A. 1, 2 and 3 are correct
• B. 1 and 2 are correct
• C. 2 and 4 are correct
• D. 1 and 3 are correct

Answer 1

Answer: (B)

Let the spectral line in the transition from $n=p$ to $n=q$ in hydrogen atom is same as the transition from $n=a$ to $n=b$ in helium atom.
Then $\left(\frac{1}{q^2}-\frac{1}{p^2}\right)=4\left(\frac{1}{b^2}-\frac{1}{a^2}\right)$
or $\frac{1}{q^2}-\frac{1}{p^2}={\left(\frac{2}{b}\right)}^2-{\left(\frac{2}{a}\right)}^2$
This will be satisfied if $b=2q$ and $a=2p$ ie,
$a$ and $b$ should be even integers and greater than $1.$