A monochromatic light is incident on a hydrogen sample in ground state. Hydrogen atoms absorb a fraction of light and subsequently emit radiation of six different wavelengths. The frequency of incident light is $$x \times 10^{15}$$ Hz. The value of x is _______.
(Given $$h = 4.25 \times 10^{-15}$$ eVs)

When hydrogen atoms absorb light and subsequently emit radiation of 6 different wavelengths, we use the formula for number of spectral lines:

$$\text{Number of lines} = \frac{n(n-1)}{2}$$

Setting this equal to 6:

$$\frac{n(n-1)}{2} = 6 \implies n(n-1) = 12 \implies n = 4$$

So the hydrogen atoms are excited from the ground state ($$n = 1$$) to $$n = 4$$.

Energy required:

$$E = 13.6\left(\frac{1}{1^2} - \frac{1}{4^2}\right) = 13.6\left(1 - \frac{1}{16}\right) = 13.6 \times \frac{15}{16}$$

$$E = \frac{204}{16} = 12.75 \text{ eV}$$

Frequency of incident light:

$$f = \frac{E}{h} = \frac{12.75}{4.25 \times 10^{-15}} = 3 \times 10^{15} \text{ Hz}$$

Therefore, $$x = 3$$.