Bohr's Model for Hydrogen Atom

Bohr’s Model for Hydrogen Atom

In 1913, Niels Bohr became the first person to give a quantitative explanation of the hydrogen atom’s structure and its spectrum. He built his model on Planck’s idea of quantised energy. While Bohr’s theory has since been superseded by modern quantum mechanics, it remains valuable for understanding many features of atomic structure and spectra.

Bohr’s model rests on four fundamental postulates.

Postulate I: Stationary Orbits

The electron in a hydrogen atom moves around the nucleus only in certain circular paths of fixed radius and fixed energy. These paths are called orbits, stationary states, or allowed energy states. They are arranged concentrically around the nucleus.

Postulate II: No Energy Change in a Stationary State

The energy of an electron in a given orbit does not change with time. Energy is absorbed only when the electron jumps from a lower stationary state to a higher one, and energy is emitted only when it falls from a higher stationary state to a lower one. The energy change is not continuous — it happens in discrete packets.

Postulate III: Bohr’s Frequency Rule

When an electron transitions between two stationary states whose energies differ by $\Delta E$, the frequency $\nu$ of the radiation absorbed or emitted is given by:

where $E_1$ and $E_2$ are the energies of the lower and higher allowed states respectively, and $h$ is Planck’s constant. This is known as Bohr’s frequency rule.

Postulate IV: Quantisation of Angular Momentum

The angular momentum of the electron is quantised. In any given stationary state, it can only take values that are integral multiples of $h/2\pi$:

Here $m_e$ is the mass of the electron, $v$ is its velocity, $r$ is the radius of the orbit, and $n$ is a positive integer called the principal quantum number.

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Consequences of Bohr’s Model for the Hydrogen Atom

From these postulates, Bohr derived a set of quantitative results for the hydrogen atom. The full derivation is mathematically involved and is typically covered in higher classes, but the results themselves are essential.

(a) Principal Quantum Number

The stationary states are numbered $n = 1, 2, 3, \dots$. These integers are called principal quantum numbers. The state with $n=1$ is the lowest energy state (closest to the nucleus), and as $n$ increases, the electron is found further from the nucleus.

(b) Radii of Stationary States

The radius of the $n$th stationary state is given by:

where $a_0 = 52.9 \text{ pm}$ is the Bohr radius — the radius of the first orbit ($n=1$). So the radius of the first orbit is $52.9$ pm. For $n=2$, the radius is $2^2 \times 52.9 = 211.6$ pm, and so on.

(c) Energies of Stationary States

The energy of the electron in the $n$th stationary state of a hydrogen atom is:

where $R_H$ is the Rydberg constant for hydrogen, with a value of $2.18 \times 10^{-18} \text{ J}$.

What Does the Negative Sign Mean?

The negative sign in the energy expression is not arbitrary — it carries physical meaning. A free electron at rest, infinitely far from the nucleus, is taken to have zero energy. Mathematically, this corresponds to setting $n = \infty$ in the formula, giving $E_\infty = 0$.

When the electron is attracted by the nucleus and occupies an orbit with a finite $n$, its energy is lower than that of the free electron. Since the free electron’s energy is zero, a lower energy must be negative. As the electron gets closer to the nucleus (as $n$ decreases), the absolute value of $E_n$ becomes larger, and the energy becomes more negative. The most negative value occurs at $n=1$, which corresponds to the most stable orbit — the ground state.

Numerical Values

For the ground state ($n=1$):

For the first excited state ($n=2$):

The energies of different stationary states are often shown in an energy level diagram, where each horizontal line represents an allowed energy level. The diagram for hydrogen shows the Lyman, Balmer, and Paschen series of spectral lines, which correspond to transitions ending at $n=1$, $n=2$, and $n=3$ respectively.

(d) Extension to Hydrogen-like Species …