Atomic Structure: Bohr Model, Wave-Particle Duality, and Quantum Numbers

The Bohr Model of the Atom

Limitations of the Rutherford Model

The Rutherford (nuclear) model of the atom described a dense, positively charged nucleus surrounded by electrons, but it could not explain how atoms change when they gain or lose energy.

• Key Limitation: Did not account for the mechanism of energy absorption or emission by atoms.

Bohr's Model and Quantization of Energy

Niels Bohr developed a new atomic model to explain the discrete energy changes observed in atoms. His major idea was that the energy of the atom is quantized, meaning atoms can only have specific, fixed amounts of energy.

• Quantized Energy: Electrons can only occupy certain allowed orbits (energy levels) at fixed distances from the nucleus.

• Stationary States: Electrons in these orbits do not radiate energy.

• Energy Transitions: Electrons emit or absorb energy as photons when they move between orbits.

• Photon Energy: The energy of the emitted or absorbed photon corresponds to the difference in energy between the two orbits.

Example: The emission spectrum of hydrogen shows discrete lines, each corresponding to an electron transition between specific energy levels.

Wave Behavior of Electrons

de Broglie Hypothesis

Louis de Broglie proposed that particles, such as electrons, exhibit wave-like properties. He predicted that the wavelength of a particle is inversely proportional to its momentum.

• de Broglie Relation:

$\lambda = \frac{h}{mv}$

• h: Planck's constant ($6.626 \times 10^{-34}$ J·s)

• m: mass of the particle

• v: velocity of the particle

Because electrons have very small mass, their wave character is significant and observable.

Electron Diffraction

Electron diffraction experiments demonstrate the wave nature of electrons. When a beam of electrons passes through two closely spaced slits, it produces an interference pattern, similar to light waves.

• Expected for Particles: Only two bright spots should appear if electrons behaved only as particles.

• Observed: An interference pattern is observed, confirming the wave-like behavior of electrons.

Application: Calculating de Broglie Wavelength

To calculate the wavelength of an electron with a given speed, use the de Broglie equation:

$\lambda = \frac{h}{mv}$

• Example: For an electron traveling at $2.65 \times 10^6$ m/s, substitute the values for $h$ and $m$ (electron mass $= 9.11 \times 10^{-31}$ kg) to find $\lambda$.

Wave-Particle Duality and the Uncertainty Principle

Complementary Properties

Electrons exhibit both particle-like and wave-like properties, but these are complementary: the more precisely one property is known, the less precisely the other can be known.

• Particle Nature: Position

• Wave Nature: Interference pattern (wavelength, frequency)

Heisenberg's Uncertainty Principle

The Heisenberg Uncertainty Principle states that it is impossible to simultaneously know both the exact position and exact velocity (momentum) of a particle such as an electron.

$\Delta x \times m \Delta v \geq \frac{h}{4\pi}$

• $\Delta x$: Uncertainty in position

• $\Delta v$: Uncertainty in velocity

• m: Mass of the particle

This principle implies that the more accurately we know the position of an electron, the less accurately we can know its velocity, and vice versa.

Determinacy vs. Indeterminacy

• Classical Physics: Particles have definite, predictable paths (determinacy).

• Quantum Mechanics: Because of the uncertainty principle, the path of an electron is indeterminate; we can only predict the probability of finding it in a certain region.

Example: The probability distribution of an electron in an atom is described by a statistical function, not a definite path.

Quantum Mechanical Model and Quantum Numbers

Schrödinger's Equation and Orbitals

Erwin Schrödinger developed a mathematical equation to describe the behavior of electrons in atoms. The solutions to this equation are called wave functions ($\psi$), and the square of the wave function ($|\psi|^2$) gives the probability density of finding an electron at a particular location.

$\hat{H}\psi = E\psi$

• Orbitals: Regions in space where the probability of finding an electron is high; visualized as probability distribution maps.

Quantum Numbers

Quantum numbers arise from the solutions to Schrödinger's equation and describe the properties of atomic orbitals and electrons:

• Principal Quantum Number ($n$): Indicates the energy level and size of the orbital. $n = 1, 2, 3, ...$

• Angular Momentum Quantum Number ($l$): Determines the shape of the orbital. $l = 0, 1, ..., n-1$

• Magnetic Quantum Number ($m_l$): Specifies the orientation of the orbital. $m_l = -l, ..., 0, ..., +l$

• Spin Quantum Number ($m_s$): Describes the spin of the electron. $m_s = +\frac{1}{2}$ or $-\frac{1}{2}$

Summary Table: Quantum Numbers and Orbital Types

Quantum Number	Symbol	Allowed Values	Physical Meaning
Principal	n	1, 2, 3, ...	Energy level, size of orbital
Angular Momentum	l	0 to n-1	Shape of orbital (s, p, d, f)
Magnetic	m_l	-l to +l	Orientation of orbital
Spin	m_s	+1/2, -1/2	Spin direction of electron

Orbital Types and Sublevels

• s-orbitals ($l=0$): Spherical shape

• p-orbitals ($l=1$): Dumbbell shape (two lobes)

• d-orbitals ($l=2$): Cloverleaf shape (four lobes)

• f-orbitals ($l=3$): Complex, multi-lobed shapes

For a given $n$, the number of possible $l$ values is $n$, and for each $l$, the number of $m_l$ values is $2l+1$.

Example: n = 4 Principal Energy Level

l	Orbital Name	Possible ml Values	Number of Orbitals
0	4s	0	1
1	4p	-1, 0, +1	3
2	4d	-2, -1, 0, +1, +2	5
3	4f	-3, -2, -1, 0, +1, +2, +3	7

Total number of orbitals for n = 4: 1 + 3 + 5 + 7 = 16

Summary: Describing an Orbital

• Each orbital is described by a unique set of quantum numbers ($n$, $l$, $m_l$).

• Orbitals with the same $n$ are in the same principal energy level (shell).

• Orbitals with the same $n$ and $l$ are in the same sublevel (subshell).

Practice and Conceptual Connections

• For $n = 3$, possible $l$ values are 0, 1, and 2 (s, p, and d orbitals).

• For $l = 2$, possible $m_l$ values are -2, -1, 0, +1, +2 (five d orbitals).

• Each electron in an atom is uniquely described by its set of four quantum numbers.

Additional info: The notes also discuss why macroscopic objects (like baseballs) do not exhibit observable wave properties: their mass is so large that their de Broglie wavelength is negligible compared to their size.