Question

For a hydrogen atom, the energies that an electron can have are given by the expression, $E = - 13.58/{n^2}eV$ , where n is an integer, The smallest amount of energy that a hydrogen atom in the ground state can absorb is:
(A) $1.00eV$
(B) $3.39eV$
(C) $6.79eV$
(D) $10.19eV$

Answer 1

The energy of the ground state of the atom can be determined by the difference between the energies at second energy level and first energy level. Hydrogen atoms have only one electron in its outermost orbital or outermost shell. This electron can jump from the first energy shell to the second energy level called excitation.
$E = - 13.58/{n^2}eV$
Where E is the energy of a hydrogen atom in its ground state
n is the energy level that takes the value of an integer.

Complete answer:
The atomic number of Hydrogen is $1$ . It has only one electron in its outermost shell.
Hydrogen is the element belonging to period $1$ and group $1$ .
Thus, electrons are excited from one energy shell to another energy shell when energy is supplied to the atom.
After a certain time, the electron returns to ground state from the second energy level to first energy level.
The second energy level was represented by ${E_2}$ , the first energy level was represented by ${E_1}$ .
${E_1} = - 13.58/{1^2} = - 13.58/1 = - 13.58eV$
${E_2} = - 13.58/{2^2} = - 13.58/4 = - 3.39eV$
The difference between the two energies of ${E_1}$ and ${E_2}$ gives the smallest amount of energy that a hydrogen atom in the ground state can absorb.
$E = {E_2} - {E_1} = - 3.39 - \left( { - 13.58} \right) = 10.19eV$
Thus, option D is the correct one.

Note:
The energies of the shell must be taken by substituting the value of $n$ in the given formula.
Ground state energy is obtained by the difference between the energies of only second and first energy levels. Energy must be in electron Volts or eV.