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Question: The ground state energy of a hydrogen atom is \( - 13.6\,{\text{eV}}\). Determine the energy of the ...

The ground state energy of a hydrogen atom is −13.6 eV - 13.6\,{\text{eV}}. Determine the energy of the level corresponding to the quantum number n = 5n\, = \,5 is:
A. −0.54 eV - 0.54\,{\text{eV}}
B. −5.40 eV - 5.40\,{\text{eV}}
C. −0.85 eV - 0.85\,{\text{eV}}
D. −2.72 eV - 2.72\,{\text{eV}}

Explanation

Solution

The hydrogen atom has quantized energy levels. The energy of an electron in each level depends upon the principal quantum number and atomic number.

Formula used:
E= E∘ Z2n2eVE = \,{E_ \circ }\,\dfrac{{{Z^2}}}{{{n^2}}}{\text{eV}}

Complete step by step answer:
The energy formula to determine the energy of the electron of the hydrogen atom is as follows.
E= E∘ Z2n2eVE = \,{E_ \circ }\,\dfrac{{{Z^2}}}{{{n^2}}}{\text{eV}}

Where,
EE is the energy of the electron in the nth level.
ZZ is the atomic number.
E∘{E_ \circ} is the energy of the group state.
nn is the principal quantum number.

Energy is inversely proportional to the quantum number. As the quantum number increases the energy of the level decreases.
For the hydrogen atom, the value of ZZ is11, the given value of nn is 55 and the given value of ground state energy is −13.6eV - 13.6{\text{eV}}. On substituting values we get,
⇒E= −13.6 1252eV\Rightarrow E = \, - 13.6\,\dfrac{{{1^2}}}{{{5^2}}}{\text{eV}}
⇒E= −13.6 1225eV\Rightarrow E = \, - 13.6\,\dfrac{{{1^2}}}{{25}}{\text{eV}}
⇒E= −0.54 eV\Rightarrow E = \, - 0.54\,{\text{eV}}
So, the energy of the principal quantum number of hydrogen atom is −0.54 eV - 0.54\,{\text{eV}}.

Therefore the option (A) is correct.

Additional information : The unit of the formula should be noticed. The above formula is in eV. The ground state energy varies in different units.
The formula of the energy of an electron of the hydrogen atom in the atomic unit is as follows:
E= −0.5 Z2n2a.uE = \, - 0.5\,\dfrac{{{Z^2}}}{{{n^2}}}{\text{a}}{\text{.u}}
The formula of the energy of an electron of the hydrogen atom in term of wavelength or cm unit is as follows:
E= −RHZ2n2cm−1E = \, - {R_H}\dfrac{{{Z^2}}}{{{n^2}}}{\text{c}}{{\text{m}}^{ - 1}}
Where,
RH{R_H}is the Rydberg constant.

Note:
The energy value of any level of hydrogen atom always comes negative. The negative sign indicates the stabilization of that energy level. As the value of the quantum number increases the energy decreases.