The energy required by electrons, present in the first Bohr orbit...

The energy of an electron in the $n^{th}$ Bohr orbit of a hydrogen atom is given by $E_n = -R_H \left( \frac{1}{n^2} \right)$.
The energy required for excitation from $n=1$ to $n=2$ for a single atom is $\Delta E = E_2 - E_1 = R_H \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R_H \left( 1 - \frac{1}{4} \right) = \frac{3}{4} R_H$.
Given $R_H = 2.18 \times 10^{-11}$ ergs.
Converting ergs to Joules: $1 \text{ erg} = 10^{-7} \text{ J}$.
So, $R_H = 2.18 \times 10^{-11} \times 10^{-7} = 2.18 \times 10^{-18} \text{ J}$.
The energy required per atom is $\Delta E = \frac{3}{4} \times 2.18 \times 10^{-18} = 1.635 \times 10^{-18} \text{ J}$.
To find the energy required per mole of electrons, multiply by Avogadro's number $N_A \approx 6.022 \times 10^{23} \text{ mol}^{-1}$.
Energy per mole $= 1.635 \times 10^{-18} \times 6.022 \times 10^{23} \text{ J mol}^{-1}$.
Energy per mole $\approx 9.846 \times 10^5 \text{ J mol}^{-1}$.
Comparing with the given options, $9.835 \times 10^5$ is the closest value.