Notes : Energy Levels of Hydrogen Atom - Class 12 Physics Chapter 12 Atoms

According to Bohr's theory, an electron in a hydrogen atom can occupy only certain discrete energy states. The energy of an electron in the n^th orbit is given by:

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

where,

E_n = Energy of the electron in the n^th orbit
n = Principal quantum number (1, 2, 3, ...)
13.6 eV = Ionization energy of hydrogen atom in the ground state

Energy of Different Orbits

(i) When n = 1 ( K Shell ) :

$$E_1=-\frac{13.6}{(1)^2}=-13.6\;eV$$

The electron revolves in the innermost orbit (K-shell). This is the lowest possible energy of the hydrogen atom. Therefore, the atom is said to be in its ground state.

(ii) When n = 2 ( L Shell ) : 

$$E_2=-\frac{13.6}{(2)^2}=-\frac{13.6}{4}=-3.4\;eV$$

This corresponds to the first excited state of the hydrogen atom.

(iii) When n = 3 ( M Shell ) : 

$$E_3=-\frac{13.6}{(3)^2}=-\frac{13.6}{9}=-1.51\;eV$$

This is the energy of the second excited state of the hydrogen atom.

(iv) When n = 4 ( N Shell ) : 

$$E_4=-\frac{13.6}{(4)^2}=-\frac{13.6}{16}=-0.85\;eV$$

This represents the third excited state of the hydrogen atom.

(v) When n = 5 ( O Shell ) : 

$$E_5=-\frac{13.6}{(5)^2}=-\frac{13.6}{25}=-0.54\;eV$$

This represents the fourth excited state of the hydrogen atom.

(vi) When n = ∞

$$E_{\infty}=-\frac{13.6}{(\infty)^2}=0$$

This is the highest energy state of the hydrogen atom. The electron is no longer bound to the nucleus and becomes a free electron. The atom is said to be ionized.

Energy Level Diagram of Hydrogen Atom

n = 1 - Ground State     → −13.6 eV
n = 2 - First Excited State     → −3.4 eV
n = 3 - Second Excited State     → −1.51 eV
n = 4 - Third Excited State     → −0.85 eV
n = 5 - Fourth Excited State     → −0.54 eV
n = 6 - Fifth Excited State     → −0.38 eV
n = 7 - Sixth Excited State     → −0.28 eV
n = 8 - Seventh Excited State     → −0.21 eV
n = ∞ - Unbound Ionized Atom   → 0 eV

Important Observations

1. The energy values are negative, indicating that the electron is bound to the nucleus.
2. As the principal quantum number (n) increases, the energy becomes less negative and approaches zero.
3. The difference between successive energy levels decreases with increasing value of n. Therefore, the energy levels come closer and closer to each other.
4. The state corresponding to n = 1 is called the ground state, while states with n > 1 are called excited states.
5. The energy required to remove the electron completely from the ground state is:
   
   $$0-(-13.6)=13.6\;eV$$
   This is known as the ionization energy of the hydrogen atom.

Transition Between Energy Levels

When an electron transitions from one orbit to another, energy is either absorbed or emitted.

$$\Delta E=E_f-E_i$$

For hydrogen atom,

$$\Delta E=13.6\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)eV$$

• If n₂ > n₁, energy is absorbed.
• If n₂ < n₁, energy is emitted.

Comparison of Energy Levels of H and He⁺

The energy levels of hydrogen-like atoms are given by:

$$E_n=-\frac{13.6Z^2}{n^2}\;eV$$

For hydrogen atom, Z = 1:

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

For He⁺ ion, Z = 2:

$$E_n=-\frac{54.4}{n^2}\;eV$$

Some energy levels of He⁺ coincide with those of hydrogen atom. Therefore, some wavelengths in the emission spectrum of He⁺ are equal to wavelengths in the hydrogen spectrum.

Key Point

With increase in principal quantum number (n), the energy levels of hydrogen atom come closer and closer to each other because the energy difference between successive levels decreases.