Chapter 28, Problem 51A

To determine

To calculate:The minimum electron energy needed for the hydrogen atoms to emit red light when the atom goes from $n=3$ to $n=2$ state.

Answer to Problem 51A

The hydrogen atoms require $1.89\text{ eV}$ of energy to emit red light.

Explanation of Solution

Given info:

The hydrogen gas emits red light when the atom makes a transition from $n=3$ to $n=2$ state.

Formula used:

The energy of $n$ th energy level of hydrogen is determined by

  $E_n=\frac{-13.6\text{ eV}}{{\text{n}}^2}$ .

Where $E_n$ represents the energy of the $\text{n}$ th energy level.

Calculation:

As per problem,

Calculate the energy of the level $n=3$

  $\begin{array}{l}{E}_n=\frac{-13.6\text{ eV}}{{\text{n}}^2}\\ E_3=\frac{-13.6\text{ eV}}{{(3)}^2}\\ E_3=-1.51\text{ eV}\end{array}$

Calculate the energy of the level $n=2$

  $\begin{array}{l}{E}_n=\frac{-13.6\text{ eV}}{{\text{n}}^2}\\ E_2=\frac{-13.6\text{ eV}}{{(2)}^2}\\ E_2=-3.4\text{ eV}\end{array}$

The energy difference between the two levels is

  $\begin{array}{l}\Delta E=E_3-E_{\begin{array}{l}2\end{array}}\\ \Delta E=(-1.51\text{ eV})-(-3.4\text{ eV)}\\ \Delta E=1.89\text{ eV}\end{array}$

Hence, $1.89\text{ eV}$ of energy is required by the atoms to emit red light.

Conclusion:

Thus, the hydrogen atoms require $1.89\text{ eV}$ of energy to emit red light.