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Orbital Diagrams

The ratio of ...

Question

The ratio of the radii of first orbit of $H, H e^{+} a n d L i^{+ 2}$ is?

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Solution

The radii are in the ratio 6 : 3: 2

Explanation:
The radii of the first orbits of a hydrogen-like ion are given by the equation
$r = \frac{h^{2}}{Z m_{e} k_{e} e^{2}}$
Since everything except Z is constant, we can rewrite the formula as
$r = \frac{k}{Z}$
Then the ratios of the three radii for Z = 1, 2 and 3 are
$r_{1} : r_{2} : r_{3} = \frac{1}{1} : \frac{1}{2} : \frac{1}{3} = 6 : 3 : 2$

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