A particular hydrogen-like ion emits the radiation of frequency $$3 \times 10^{15}$$ Hz when it makes transition from $$n = 2$$ to $$n = 1$$. The frequency of radiation emitted in transition from $$n = 3$$ to $$n = 1$$ is $$\frac{x}{9} \times 10^{15}$$ Hz, when $$x$$ = ______.

A hydrogen-like ion emits frequency $$3 \times 10^{15}$$ Hz for $$n=2 \to n=1$$. Find the frequency for $$n=3 \to n=1$$.

$$ \nu = RZ^2\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right) $$

where $$R$$ is the Rydberg constant (in frequency units), $$Z$$ is the atomic number.

For $$n=2 \to n=1$$:

$$ 3 \times 10^{15} = RZ^2\left(\frac{1}{1^2} - \frac{1}{2^2}\right) = RZ^2\left(1 - \frac{1}{4}\right) = RZ^2 \times \frac{3}{4} $$

$$ RZ^2 = \frac{3 \times 10^{15} \times 4}{3} = 4 \times 10^{15} $$

$$ \nu' = RZ^2\left(\frac{1}{1^2} - \frac{1}{3^2}\right) = 4 \times 10^{15} \times \left(1 - \frac{1}{9}\right) = 4 \times 10^{15} \times \frac{8}{9} = \frac{32}{9} \times 10^{15} $$

Comparing with the form $$\frac{x}{9} \times 10^{15}$$: $$x = 32$$.

The answer is 32.