The amount of energy needed to excite a hydrogen atom from its ground state to the second excited state is 

Energy expression which is used to determine the energy of the hydrogen atom present in n^th shell is:

${\mathrm{E}}_{\mathrm{n}}=-13.6 \frac{1}{{\mathrm{n}}^2} \mathrm{eV}$

Electron moving from ground to second excited state $\left(1\to 3\right)$

${\mathrm{E}}_1=\frac{-13.6}{1^2}\mathrm{eV}; {\mathrm{E}}_3=\frac{-13.6}{3^2}\mathrm{eV}$

Energy required is calculated as:

 $∆\mathrm{E}={\mathrm{E}}_3-{\mathrm{E}}_1$
        $=13.6\left[1-\frac{1}{9}\right]=13.6\times \frac{8}{9}\cong 12.1 \mathrm{eV}$