Question

The ground state energy of a hydrogen atom is \[ - 13.6eV\]. What is the kinetic energy of an electron in the ${2^{nd}}$ excited state?

Answer 1

Hint: In order to solve this you have to know the concept of the total energy of ${n^{th}}$ excited state of the hydrogen atom. Also, remember that the total energy of any excited state of the hydrogen atom is equal to the negative of its kinetic energy in that excited state.

Formula used:
The formula for the total energy of ${n^{th}}$ excited state of hydrogen atom is given by
${E_n} = \dfrac{{{E_0}}}{{{n^2}}}$
Where, ${E_0}$ is the energy at the ground state of hydrogen atom
$n$ is the excited state level

Complete step by step solution:
It is given in the question, the ground state energy of the hydrogen atom is
${E_0} = - 13.6eV$
And we have to find the kinetic energy of an electron in the ${2^{nd}}$ excited state
So, the ${2^{nd}}$ excited state means \[{3^{rd}}\] normal state or we can say that
$n = 3$
By using the formula for the total energy of ${n^{th}}$ excited state of hydrogen atom is given by
${E_n} = \dfrac{{{E_0}}}{{{n^2}}}$
Where, ${E_0}$ is the energy at the ground state of hydrogen atom
$n$ is the excited state level
Now, on putting the values in the above formula, we get
$ \Rightarrow {E_n} = \dfrac{{ - 13.6}}{{{{\left( 3 \right)}^2}}}$
$ \Rightarrow {E_n} = \dfrac{{ - 13.6}}{9}$
On further solving, we get
$ \Rightarrow {E_n} = - 1.51eV$
As we know that the total energy of any excited state of hydrogen atom is equal to the negative of its kinetic energy in that excited state
Total energy $ = $ $ - $ kinetic energy
So, kinetic energy $ = $ $ - $ total energy
Thus, the kinetic energy of an electron in the ${2^{nd}}$ excited state is given by,
\[K.E = -1.51eV\]
$ \therefore K.E = 1.51eV$

Therefore, the kinetic energy of an hydrogen electron in the ${2^{nd}}$ excited state is $1.51eV$.

Note: Always remember that if the electron wants to jump from $n = 1$, the first energy level to the, $n = 2$ , the second energy level then the second energy level has higher energy than the first and the electron needs to gain energy as it moves from lower energy level to the high energy level. The electron can gain the energy it needs by absorbing light. If the electron jumps from a higher energy level down to a lower energy level, it must lose some energy by emitting light.