Atomic Structure: Particles, Orbitals, and Quantum Models

Fundamental Particle Properties

• Mass of Electron ($m_e$): $9.109534 \times 10^{-31}$ kg
• Charge of Electron ($q_e$): $-1.602189 \times 10^{-19}$ C
• Mass of Proton ($m_p$): $1.672649 \times 10^{-27}$ kg
• Charge of Proton ($q_p$): $1.602189 \times 10^{-19}$ C
• Mass of Neutron ($m_n$): $1.674954 \times 10^{-27}$ kg

Rays/channels are formed by positively charged particles. The relationship between charge and mass varies according to the gas used in the tube.

Quantum Numbers and Orbitals

Quantum numbers describe the state of an electron:

• Principal Quantum Number ($n$): Energy level
• Orbital Angular Momentum ($l$): Orbital shape (ranges from 0 to $n-1$)
• Magnetic Quantum Number ($m_l$): Orbital orientation
• Spin Quantum Number ($m_s$): Electron spin ($\pm 1/2$)

The quantum-mechanical model uses equations describing the behavior of electrons inside the atom, acknowledging both their wave character and the impossibility of predicting their exact trajectories. Orbitals are typically represented by imaginary surfaces where the probability of finding the electron with a given energy is very large.

Rutherford Model Concepts

In the Rutherford core:

• The positive charge and almost all the mass reside in the nucleus.
• The electron 'crust' is formed by electrons orbiting the nucleus, similar to a solar system.
• The core radius is approximately $100,000$ times smaller than the radius of the atom.

Limitations of the Rutherford Model

The model fails because:

1. Electrons moving in circular orbits undergo acceleration, and classical physics dictates that accelerating electric charges must emit electromagnetic energy (radiation).
2. Consequently, the electron should spiral into the nucleus, causing the atom to lose energy continuously. The emitted radiation should be continuous.

Bohr Model Postulates

Niels Bohr introduced key concepts to address Rutherford's limitations:

Quantization of Energy

The energy of electrons inside the atom is quantized; the electron occupies specific, stationary states around the nucleus with discrete energy values.

Stationary Orbits

The electron moves in circular orbits, each corresponding to a stationary or allowed energy level associated with a natural number ($n$).

Angular Momentum Condition

Allowed energy levels occur when the electron's angular momentum is an integer multiple of $h/2\pi$, where $h$ is Planck's constant.

Energy Transitions

Energy is only absorbed or emitted when an electron moves from one energy level to another. If $E_i$ is the starting energy level and $E_f$ is the arrival energy level, the corresponding energy change ($\Delta E$) and frequency ($\nu$) are related by:

$$\Delta E = E_f - E_i = h\nu$$