Question

Assume that the potential energy of an hydrogen atom in its ground state is zero. Then its energy in the first excited state will be:
A.$23.8$ $eV$
B.$27.5$ $eV$
C.$30.4$ $eV$
D.$34.8$ $eV$

Answer 1

Hint:Potential energy of atoms result from the movement of electrons in an orbit. In an excited state atoms move to their higher energy orbital. The potential energy increases as the electrons move far away from the nucleus.

Complete step by step answer:
Here it is given that the potential energy of the hydrogen atom in its ground state is zero.
Ground state is defined as the lowest energy of all the possible states whereas the excited state is defined as the energy having the highest value than the ground state.
$P.E=-2\times \dfrac{13.6}{{{n}^{2}}}$
Where, $P.E$ is the potential energy
$K.E$ is the kinetic energy
Kinetic energy of hydrogen atom in ground state is $\dfrac{13.6}{{{n}^{2}}}$ $eV$
$n=1$ for hydrogen atom
Where, $n$ is principal quantum number
On substituting the value in above formula we get,
$K.E=\dfrac{13.6}{{{\left( 1 \right)}^{2}}}$
$K.E=13.6$ $eV$
Kinetic energy $=13.6$ $eV$
On substituting the value in potential energy we get,
In ground state, the potential energy of a hydrogen atom is $-2\times \dfrac{13.6}{{{1}^{2}}}$ \[\]
Potential energy of a hydrogen atom in ground state is $-27.2$ $eV$
To make this potential energy equal to zero we need to add extra $27.2$ $eV$ energy
The total energy of a electron in first excited state of hydrogen atom is $\dfrac{\left( -13.6 \right)}{{{n}^{2}}}$
$T.E=\dfrac{\left( -13.6 \right)}{{{2}^{2}}}$
$T.E=-3.4$ $eV$
Where, $T.E$ is the total energy of the hydrogen atom and $n$ is the principal quantum number.
The total energy of an electron in the first excited state of a hydrogen atom is $-3.4$ $eV$
The energy level of a hydrogen atom is increased by $27.2$$eV$ therefore, the total energy in the first excited state will be $-3.4+27.2=23.8$$eV$
Hence, the correct option is $\left( A \right)$ that is $23.8$ $eV$ .

Additional information
Hydrogen atom is the simplest atom that has one proton and one electron. When an electron moves from energy level to another energy level then the energy of an electron also changes. Energy is required by the electron to move from state to another.

Note:
-Principal quantum number is a quantum number that is assigned to each electron in an atom that describes the electron state. It is denoted with the symbol $n$.
-Potential energy is an energy associated with an object because of its position.