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(C) For a hydrogen atom, the radius of the $n^{th}$ orbit is given by $r_n \propto n^2$ and the energy is given by $E_n \propto \frac{1}{n^2}$.For the

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$\frac{9}{4} r_2, \frac{4}{9} E_2$

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(C) For a hydrogen atom, the radius of the $n^{th}$ orbit is given by $r_n \propto n^2$ and the energy is given by $E_n \propto \frac{1}{n^2}$.For the second orbit $(n=2)$: $r_2 \propto (2)^2 = 4$ and $E_2 \propto \frac{1}{(2)^2} = \frac{1}{4}$.For the third orbit $(n=3)$: $r_3 \propto (3)^2 = 9$ and $E_3 \propto \frac{1}{(3)^2} = \frac{1}{9}$.Calculating the ratios:$\frac{r_3}{r_2} = \frac{9}{4} \implies r_3 = \frac{9}{4} r_2$$\frac{E_3}{E_2} = \frac{1/9}{1/4} = \frac{4}{9} \implies E_3 = \frac{4}{9} E_2$Thus, the radius and energy of the third Bohr orbit are $\frac{9}{4} r_2$ and $\frac{4}{9} E_2$, respectively.

If the radius and energy of the second Bohr orbit of a hydrogen atom are $r_2$ and $E_2$, respectively, then the radius and energy of the third Bohr orbit will be, respectively:

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