Bohr's Model — Explained

Detailed Explanation

The journey to understanding atomic structure has been a fascinating one, marked by successive models refining our comprehension. Ernest Rutherford's nuclear model, while revolutionary for establishing the existence of a dense, positively charged nucleus, faced two critical challenges that classical physics could not resolve: atomic stability and the nature of atomic spectra.

1. The Problem of Atomic Stability: According to classical electromagnetic theory (Maxwell's equations), an electron, being a charged particle accelerating in a circular orbit around the nucleus, should continuously radiate energy. As it loses energy, its orbit would shrink, and it would spiral into the nucleus in a fraction of a second ($10^{-8}$ seconds). This would imply that atoms are inherently unstable, which contradicts the observed stability of matter.

2. The Problem of Atomic Spectra: When atoms are excited (e.g., by heating or electric discharge), they emit light. However, this emitted light is not a continuous spectrum (like a rainbow from white light) but rather consists of discrete lines, each corresponding to a specific wavelength. This 'line spectrum' is unique for each element and was inexplicable by classical physics, which would predict a continuous range of frequencies from a spiraling electron.

Niels Bohr, a student of Rutherford, addressed these issues in 1913 by proposing a new model for the hydrogen atom, incorporating Planck's quantum theory. His model was based on three fundamental postulates:

Bohr's Postulates:

Postulate 1: Stationary Orbits (Non-radiating Orbits): — Electrons revolve around the nucleus in certain definite circular paths called 'stationary orbits' or 'stationary states.' While in these orbits, electrons do not radiate energy, defying classical electromagnetism. Each stationary orbit is associated with a definite amount of energy, meaning the energy of the electron in an atom is quantized.
Postulate 2: Quantization of Angular Momentum: — An electron can revolve only in those orbits for which its angular momentum is an integral multiple of $rac{h}{2pi}$, where $h$ is Planck's constant ($6.626 \times 10^{-34} \text{ J s}$). Mathematically, this is expressed as:

$$L = mvr = n\frac{h}{2pi}$$

where $m$ is the mass of the electron, $v$ is its velocity, $r$ is the radius of the orbit, and $n$ is a positive integer (1, 2, 3, ...), known as the principal quantum number. Each value of $n$ corresponds to a specific stationary orbit (e.g., $n=1$ for the first orbit, $n=2$ for the second, and so on).

Postulate 3: Energy Transitions (Frequency Condition): — An electron can jump from one stationary orbit to another only by absorbing or emitting a photon of energy. When an electron jumps from a lower energy orbit ($E_i$) to a higher energy orbit ($E_f$), it absorbs a photon of energy $h

u = E_f - E_i$. Conversely, when it jumps from a higher energy orbit ($E_f$) to a lower energy orbit ($E_i$), it emits a photon of energy$h u = E_f - E_i$. This relationship is known as Bohr's frequency condition.

Derivations from Bohr's Model:

For a hydrogen-like species (one electron, Z protons in the nucleus), the electrostatic force of attraction between the nucleus (charge $+Ze$) and the electron (charge $-e$) provides the necessary centripetal force for the electron to orbit.

1
Force Balance:

$$rac{k (Ze)(e)}{r^2} = \frac{mv^2}{r}$$

where $k = \frac{1}{4piepsilon_0}$ is Coulomb's constant. So, $rac{Ze^2}{4piepsilon_0 r^2} = \frac{mv^2}{r} quad (1)$

1
Quantization of Angular Momentum:

$$mvr = n\frac{h}{2pi} implies v = \frac{nh}{2pi mr} quad (2)$$

Substitute (2) into (1) and solve for $r$:

rac{Ze^2}{4piepsilon_0 r^2} = m left(\frac{nh}{2pi mr}\right)^2 \frac{1}{r}

$$rac{Ze^2}{4piepsilon_0 r^2} = \frac{m n^2 h^2}{4pi^2 m^2 r^2} \frac{1}{r}$$

$$r = \frac{n^2 h^2 epsilon_0}{pi m Z e^2}$$

This gives the **radius of the $n^{th}$ Bohr orbit**:

$$r_n = 0.529 \frac{n^2}{Z} \text{ Å}$$

For hydrogen ($Z=1$), the radius of the first orbit ($n=1$) is $r_1 = 0.529 \text{ Å}$, known as the Bohr radius ($a_0$).

1
Velocity of Electron: — Substitute the expression for $r$ back into equation (2):

v_n = \frac{nh}{2pi m} left( \frac{pi m Z e^2}{n^2 h^2 epsilon_0} \right) = \frac{Z e^2}{2 epsilon_0 n h}

This gives the **velocity of the electron in the $n^{th}$ Bohr orbit**:

$$v_n = 2.18 \times 10^6 \frac{Z}{n} \text{ m/s}$$

Notice that velocity decreases as $n$ increases, and increases with $Z$.

1
Energy of Electron: — The total energy ($E$) of an electron in an orbit is the sum of its kinetic energy (KE) and potential energy (PE).

$$KE = \frac{1}{2}mv^2$$

From equation (1), $mv^2 = \frac{Ze^2}{4piepsilon_0 r}$. So, $KE = \frac{Ze^2}{8piepsilon_0 r}$.

$$PE = -\frac{Ze^2}{4piepsilon_0 r}$$

$$E_n = KE + PE = \frac{Ze^2}{8piepsilon_0 r} - \frac{Ze^2}{4piepsilon_0 r} = -\frac{Ze^2}{8piepsilon_0 r}$$

Substitute the expression for $r_n$:

E_n = -\frac{Ze^2}{8piepsilon_0} left( \frac{pi m Z e^2}{n^2 h^2 epsilon_0} \right) = -\frac{m Z^2 e^4}{8 epsilon_0^2 n^2 h^2}

This gives the **total energy of the electron in the $n^{th}$ Bohr orbit**: $$E_n = -13.

6 rac{Z^2}{n^2} ext{ eV}$$The negative sign indicates that the electron is bound to the nucleus. As$n$increases,$E_n$becomes less negative (i.e., higher energy), approaching zero as$n o infty$ (ionization).

Atomic Spectra and Rydberg Formula:

When an electron transitions from a higher energy level ($n_2$) to a lower energy level ($n_1$), it emits a photon. The energy of this photon is:

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

Since $Delta E = h u = \frac{hc}{lambda}$, we can write: $$rac{1}{lambda} = rac{13.

6 Z^2}{hc} left( rac{1}{n_1^2} - rac{1}{n_2^2} ight)$$The term$rac{13.6}{hc}$is the Rydberg constant ($R_H$). So, the **Rydberg formula** for hydrogen-like species is:$rac{1}{lambda} = R_H Z^2 left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$where$R_H = 1.

09677 imes 10^7 ext{ m}^{-1}$.

Spectral Series of Hydrogen:

Different series are observed depending on the final energy level ($n_1$) to which the electron transitions:

Lyman Series: — $n_1 = 1$, $n_2 = 2, 3, 4, ...$ (Ultraviolet region)
Balmer Series: — $n_1 = 2$, $n_2 = 3, 4, 5, ...$ (Visible region)
Paschen Series: — $n_1 = 3$, $n_2 = 4, 5, 6, ...$ (Infrared region)
Brackett Series: — $n_1 = 4$, $n_2 = 5, 6, 7, ...$ (Infrared region)
Pfund Series: — $n_1 = 5$, $n_2 = 6, 7, 8, ...$ (Infrared region)

Real-World Applications & NEET Relevance:

Bohr's model successfully explained:

The stability of the hydrogen atom.
The line spectrum of hydrogen and hydrogen-like ions ($He^+, Li^{2+}$).
The calculation of ionization energy for hydrogen and hydrogen-like species (energy required to remove an electron from $n=1$ to $n=infty$).
The concept of quantized energy levels, which is fundamental to all of quantum chemistry.

For NEET, understanding the derivations is less critical than knowing the final formulas and their dependencies on $n$ and $Z$. Questions frequently involve calculating radii, energies, velocities, or wavelengths of spectral lines for hydrogen and hydrogen-like species. Ratios of these quantities for different $n$ or $Z$ values are also common.

Common Misconceptions:

Electrons orbit like planets: — While a useful analogy, it's misleading. Electrons in Bohr's model are in 'stationary states' with quantized energy, not continuously orbiting like planets. They don't 'travel' between orbits; they 'jump' instantaneously.
Bohr's model applies to all atoms: — It only works perfectly for single-electron systems (hydrogen and hydrogen-like ions). It fails for multi-electron atoms due to electron-electron repulsion and screening effects, which it doesn't account for.
Bohr's model is completely wrong: — It was a crucial stepping stone. While superseded by more advanced quantum mechanics, its fundamental concepts of quantized energy and angular momentum remain valid and are integral to modern atomic theory.
Energy levels are equally spaced: — The energy levels become closer together as $n$ increases ($E_n propto 1/n^2$). This is a common trap in conceptual questions.

Despite its limitations, Bohr's model was a monumental achievement, bridging classical physics with the nascent quantum theory and providing the first successful explanation of atomic structure and spectra.