Bohr Atom

The Bohr model of the atom, introduced by Niels Bohr in 1913, is a quantum physics-based model that describes the structure of hydrogen atoms. This model was a significant step in the development of quantum mechanics and is still used as a simplified model for teaching purposes. Below, we will explore the key concepts, formulas, and differences that are central to the Bohr model.

Key Concepts

• Energy Levels: Electrons orbit the nucleus in specific energy levels or shells without radiating energy. These levels are quantized, meaning electrons can only occupy certain allowed energy states.
• Quantum of Action: The angular momentum of an electron in orbit is quantized and is an integral multiple of the reduced Planck constant divided by 2π.
• Photon Emission/Absorption: When an electron transitions between energy levels, it emits or absorbs a photon whose energy is equal to the difference in energy levels.

Important Formulas

Energy of an Electron in an Orbit

The energy of an electron in the nth orbit is given by:

$$ E_n = -\frac{Z^2 \cdot R_H \cdot h \cdot c}{n^2} $$

where:

• ( E_n ) is the energy of the nth orbit.
• ( Z ) is the atomic number (for hydrogen, ( Z = 1 )).
• ( R_H ) is the Rydberg constant for hydrogen (( R_H \approx 1.097 \times 10^7 \, \text{m}^{-1} )).
• ( h ) is the Planck constant (( h \approx 6.626 \times 10^{-34} \, \text{J} \cdot \text{s} )).
• ( c ) is the speed of light in a vacuum (( c \approx 3.00 \times 10^8 \, \text{m/s} )).
• ( n ) is the principal quantum number (integer).

Radius of an Electron Orbit

The radius of the nth orbit is given by:

$$ r_n = \frac{n^2 \cdot h^2}{4 \pi^2 \cdot k \cdot m_e \cdot Z \cdot e^2} $$

where:

• ( r_n ) is the radius of the nth orbit.
• ( k ) is the Coulomb's constant (( k \approx 8.988 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2 )).
• ( m_e ) is the mass of an electron (( m_e \approx 9.109 \times 10^{-31} \, \text{kg} )).
• ( e ) is the elementary charge (( e \approx 1.602 \times 10^{-19} \, \text{C} )).

Angular Momentum Quantization

The angular momentum of an electron in the nth orbit is quantized and given by:

$$ L_n = n \cdot \frac{h}{2 \pi} $$

where:

• ( L_n ) is the angular momentum of the nth orbit.
• ( n ) is the principal quantum number.

Frequency of Emitted/Absorbed Photon

When an electron transitions from a higher energy level ( n_i ) to a lower energy level ( n_f ), the frequency of the emitted or absorbed photon is:

$$ f = \frac{E_{n_i} - E_{n_f}}{h} $$

Wavelength of Emitted/Absorbed Photon

Using the Rydberg formula, the wavelength ( \lambda ) of the photon associated with the transition from ( n_i ) to ( n_f ) is:

$$ \frac{1}{\lambda} = R_H \cdot Z^2 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) $$

Differences and Important Points

Aspect	Bohr Model	Modern Quantum Mechanics
Energy Levels	Quantized orbits with fixed radii and energies	Probability clouds with no defined orbits
Electron Position	Electrons have definite orbits	Electrons have no definite path (Heisenberg uncertainty)
Angular Momentum	Quantized angular momentum	Angular momentum is still quantized
Photon Emission	Occurs when an electron jumps between orbits	Emission/Absorption described by wave functions
Applicability	Works well for hydrogen-like atoms	Applicable to all atoms and molecules
Mathematical Complexity	Relatively simple formulas	Involves complex wave equations

Examples

Example 1: Energy of an Electron

Calculate the energy of an electron in the second orbit of a hydrogen atom.

Using the formula for ( E_n ):

$$ E_2 = -\frac{1^2 \cdot R_H \cdot h \cdot c}{2^2} $$

Plugging in the constants:

$$ E_2 = -\frac{1 \cdot (1.097 \times 10^7) \cdot (6.626 \times 10^{-34}) \cdot (3.00 \times 10^8)}{4} \approx -5.45 \times 10^{-19} \, \text{J} $$

Example 2: Wavelength of Photon

Determine the wavelength of the photon emitted when an electron in a hydrogen atom transitions from the third orbit to the second orbit.

Using the Rydberg formula:

$$ \frac{1}{\lambda} = R_H \left( \frac{1}{2^2} - \frac{1}{3^2} \right) $$

Solving for ( \lambda ):

$$ \frac{1}{\lambda} = (1.097 \times 10^7) \left( \frac{1}{4} - \frac{1}{9} \right) $$ $$ \frac{1}{\lambda} = (1.097 \times 10^7) \left( \frac{5}{36} \right) $$ $$ \lambda \approx 6.56 \times 10^{-7} \, \text{m} $$

The Bohr model, despite its limitations, provides a foundation for understanding atomic structure and the quantization of energy levels. It is a stepping stone to more advanced quantum mechanical models that describe the behavior of electrons in atoms with greater accuracy.