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Protons are positively charged particles found in the nucleus, with a charge of \( +1e \) and a mass of \( 1.672 \times 10^{-27} \) kg. Ernest Rutherford discovered the proton during his gold foil experiment in 1917. Protons determine the atomic number (Z) of an element and, along with neutrons, contribute to the atomic mass.

Numerical Example:

- Determine the number of protons, neutrons, and electrons in \(_{17}^{35}Cl^-\). 

  Solution: 

  Protons = 17, Neutrons = \( 35 - 17 = 18 \), Electrons = \( 17 + 1 = 18 \) (due to the negative charge).

The Pauli Exclusion Principle, introduced in 1926, asserts that no two electrons in an atom can share the same set of quantum numbers. This limits each orbital to two electrons with opposite spins. The principle is vital for understanding electronic configurations, the periodic table, and the structure of atoms.

Numerical Example:

- Assign quantum numbers for the two electrons in the 2s orbital of a helium atom. 

  Solution: 

  First electron: \( n = 2, l = 0, m_l = 0, m_s = +\frac{1}{2} \). 

  Second electron: \( n = 2, l = 0, m_l = 0, m_s = -\frac{1}{2} \).

Atoms consist of three primary subatomic particles: protons, neutrons, and electrons. Protons and neutrons are located in the nucleus, while electrons orbit the nucleus in designated energy levels. Protons have a positive charge, neutrons are neutral, and electrons carry a negative charge. The relative masses and charges of these particles govern atomic structure and behavior.

Numerical Example:

- For \(_{15}^{31}P\), calculate the number of protons, neutrons, and electrons. 

  Solution: 

  Protons = 15, Neutrons = \( 31 - 15 = 16 \), Electrons = 15.

In 1911, Ernest Rutherford proposed the nuclear model of the atom based on his gold foil experiment. He demonstrated that atoms consist of a small, dense, positively charged nucleus surrounded by electrons in orbits. Most of the atom's mass is concentrated in the nucleus, with electrons occupying most of the atom's volume.

Numerical Example:

- If alpha particles are scattered by a gold foil, calculate the percentage of particles that pass through undeflected, given most of the atom is empty space. 

  Solution: 

  Approximately 99% pass through undeflected due to the atom’s empty space.

Louis de Broglie’s hypothesis in 1924 proposed that particles exhibit wave-like behavior. The de Broglie wavelength \( \lambda \) is given by \( \lambda = \frac{h}{p} \), where \( h \) is Planck’s constant and \( p \) is the momentum of the particle. This relation is fundamental in quantum mechanics and explains phenomena such as electron diffraction.

Numerical Example:

- Calculate the de Broglie wavelength of an electron traveling at \( 2 \times 10^6 \) m/s. 

  Solution: 

  \( \lambda = \frac{6.626 \times 10^{-34}}{9.1 \times 10^{-31} \times 2 \times 10^6} = 3.64 \times 10^{-10} \) m.

Relevant Image: Use wave-particle duality equations from NCERT.

In 1898, J.J. Thomson proposed that atoms consist of a positively charged “pudding” with negatively charged electrons embedded like “plums.” This model explained atomic neutrality but was later disproved by Rutherford’s gold foil experiment, which showed that positive charge is concentrated in a nucleus.

Numerical Example:

- Why did Thomson's model fail? 

  Solution: 

  Thomson's model couldn’t explain the deflection of alpha particles in Rutherford’s experiment.

J.J. Thomson discovered the electron in 1897 through experiments with cathode rays. He showed that cathode rays were made of negatively charged particles (electrons). His discovery laid the foundation for modern atomic theory, revealing that atoms are divisible and consist of smaller charged particles.

Numerical Example:

- In a cathode ray tube, calculate the charge-to-mass ratio of an electron if the applied electric field is \( 1.0 \times 10^4 \) V/m and the electron is deflected by \( 5 \times 10^{-4} \) m. 

  Solution: 

  Use \( e/m = 1.76 \times 10^{11} \, \text{C/kg} \).

Niels Bohr’s model introduced the idea of quantized electron orbits in 1913. Electrons occupy specific energy levels, and they emit or absorb energy when transitioning between levels. Bohr’s model explained the spectral lines of hydrogen but had limitations when applied to multi-electron atoms.

Numerical Example:

- Calculate the energy difference when an electron in a hydrogen atom transitions from \( n = 3 \) to \( n = 2 \). 

  Solution: 

  \( \Delta E = 13.6 \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = 1.89 \, \text{eV} \).

Electromagnetic radiation consists of waves with electric and magnetic fields oscillating perpendicular to each other. The energy of electromagnetic radiation is given by \( E = h\nu \), where \( \nu \) is the frequency. This radiation spans a broad spectrum, from radio waves to gamma rays, and is involved in phenomena like the photoelectric effect and atomic spectra.

Numerical Example:

- Calculate the energy of a photon with a frequency of \( 5 \times 10^{14} \) Hz. 

  Solution: 

  \( E = 6.626 \times 10^{-34} \times 5 \times 10^{14} = 3.313 \times 10^{-19} \) J.

The uncertainty principle, formulated by Werner Heisenberg in 1927, states that it is impossible to know both the exact position and momentum of a particle simultaneously. The principle is expressed as \( \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \). It explains the limitations of measuring quantum systems and is foundational in quantum mechanics.

Numerical Example:

- Calculate the uncertainty in momentum if the uncertainty in position is \( 1 \times 10^{-10} \) m. 

  Solution: 

  \( \Delta p \geq \frac{6.626 \times 10^{-34}}{4\pi \times 1 \times 10^{-10}} = 5.27 \times 10^{-25} \) kg·m/s.

Developed by Schrödinger, this model describes electrons as wavefunctions rather than fixed particles. Electrons occupy orbitals, which are probability distributions. The model uses quantum numbers to describe the electron’s position and energy. The four quantum numbers—principal \( n \), angular momentum \( l \), magnetic \( m_l \), and spin \( m_s \)—define each electron’s state and behavior

within an atom. The quantum mechanical model successfully explains the electronic structure of multi-electron atoms and the shapes of orbitals.

Numerical Example:

- Assign the quantum numbers for an electron in the 3d orbital. 

  Solution: 

  \( n = 3, l = 2, m_l = -2, -1, 0, +1, +2, m_s = +\frac{1}{2} \) or \( -\frac{1}{2} \).

When electrons in an atom transition between energy levels, they emit or absorb light at specific wavelengths, producing atomic spectra. These spectra consist of discrete lines, corresponding to particular transitions. In the hydrogen atom, spectral lines are grouped into series (Lyman, Balmer, Paschen, etc.). The energy difference between levels is given by the formula:

\[

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

\]

where \( n_1 \) and \( n_2 \) are the initial and final principal quantum numbers.

Numerical Example:

- Calculate the wavelength of light emitted when an electron in a hydrogen atom transitions from \( n = 4 \) to \( n = 2 \) (Balmer series). 

  Solution: 

  \( \Delta E = 13.6 \left( \frac{1}{2^2} - \frac{1}{4^2} \right) = 2.55 \, \text{eV} \). 

  Using \( E = \frac{hc}{\lambda} \), 

  \( \lambda = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{2.55 \times 1.6 \times 10^{-19}} = 486 \, \text{nm} \).