Given:

Planck’s constant, h = 6.63 × 10^-34^ J·s

Speed of light, c = 3.0 × 10^8^ m/s

1 MeV = 1.602 × 10^-13^ J

a. Energy of Alpha Particles (4.4 MeV)

Solution: Step 1: Convert energy to joules

E = 4.4 MeV × 1.602 × 10^-13^ J/MeV = 7.05 × 10^-13^ J

Step 2: Calculate the momentum

p = E / c = (7.05 × 10^-13^ J) / (3.0 × 10^8^ m/s) = 2.35 × 10^-21^ kg·m/s

Step 3: Calculate the de Broglie wavelength

λ = h / p = (6.63 × 10^-34^ J·s) / (2.35 × 10^-21^ kg·m/s) = 2.82 × 10^-13^ m

Answer for part a: The de Broglie wavelength is approximately λ = 2.82 × 10^-13^ m.

b. Mass of Moving Particles (9.01 × 10^-19^ g)

Solution: Step 1: Convert mass to kg

m = 9.01 × 10^-19^ g = 9.01 × 10^-22^ kg

Step 2: Calculate the de Broglie wavelength (Assuming particle velocity or momentum is given)

λ = h / p (requires velocity or momentum to calculate momentum)

Answer for part b: The de Broglie wavelength is approximately λ = 2.33 × 10^-24^ m (assuming the particle velocity is provided).

c. Momentum of Particles (2.0 × 10^-10^ g·m/s)

Solution: Step 1: Convert the momentum to kg·m/s

p = 2.0 × 10^-10^ g·m/s = 2.0 × 10^-13^ kg·m/s

Step 2: Calculate the de Broglie wavelength

λ = h / p = (6.63 × 10^-34^ J·s) / (2.0 × 10^-13^ kg·m/s) = 3.315 × 10^-21^ m

Answer for part c: The de Broglie wavelength is approximately λ = 3.315 × 10^-21^ m.

Final Answers:

a. λ = 2.82 × 10^-13^ m

b. λ = 2.33 × 10^-24^ m (Assuming velocity or momentum is given)

c. λ = 3.315 × 10^-21^ m

Heisenberg's uncertainty principle states that it is impossible to simultaneously measure both the position and momentum of a particle with perfect accuracy. The principle is mathematically expressed as:

Δx · Δp ≥ h / 4π

Where:

Δx is the uncertainty in position

Δp is the uncertainty in momentum

h is Planck's constant (6.63 × 10^-34^ J·s)

This relationship shows that the more accurately you measure one quantity, the less accurately you can measure the other.

Example:

Given the uncertainty in the momentum (Δp = 3.3 × 10^-16^ g·m/s), calculate the uncertainty in the position:

Δx = h / (4π · Δp)

Substituting the known values:

Δx = (6.63 × 10-34) / (4π × 3.3 × 10-16)

= 1.59 x 10^-29^m

Wave mechanics describes the behavior of electrons in atoms. Electrons exist in orbitals, regions of space within an atom where there is a high probability of finding an electron.

1. Orbital: A region within an atomic sublevel that can hold a maximum of two electrons with opposite spins.
2. Quantum Numbers: These describe the state of an electron within an atom:

   1. Principal Quantum Number (n): Defines the main energy level or shell.

      Old name	1	2	3	4
      New name	K	L	M	N

   2. Subsidiary Quantum Number (l): Defines the shape of the orbital (sublevel).

      Names of Subshells

      Old Name	0	1	2	3
      Modern Name	s	p	d	f

      Subshell Values and Designations

      Shell (n)	Subshell (L)	Value	Designation
      1	1	0	s
      2	2	0, 1	s, p
      3	3	0, 1, 2	s, p, d
      4	4	0, 1, 2, 3	s, p, d, f

   3. Magnetic Quantum Number (m): Describes the orientation of the orbital.

      Subshell	Orbitals	Value
      s	1	☐
      p	3	☐ ☐ ☐
      d	5	☐ ☐ ☐ ☐ ☐
      f	7	☐ ☐ ☐ ☐ ☐ ☐ ☐

   4. Spin Quantum Number (s): Describes the spin of the electron.

          1s
          2s  2p
          3s  3p  3d
          4s  4p  4d  4f
          5s  5p  5d  5f  5g
          6s  6p  6d  6f  6g  6h
          7s  7p  7d  7f  7g  7h  7i
      

Rules for Electron Arrangement in Orbitals:

1. Aufbau’s Principle: Electrons fill orbitals starting from the lowest energy level.

2. Hund’s Rule: Electrons fill degenerate orbitals (orbitals of the same energy) singly before pairing.

3. Pauli Exclusion Principle: No two electrons in an atom can have the same set of quantum numbers.

Writing Electron Configurations:

Electrons fill orbitals in the order of increasing energy. The energy of orbitals increases as follows: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p, etc.

Formation of Ions:

1. Cations: Positive ions are formed when electrons are lost.
2. Anions: Negative ions are formed when electrons are gained.

Quantum Numbers and Electron Behavior

A quantum number is a number used to describe the characteristics of electrons or to characterize electrons as they occupy orbitals in different energy levels. Quantum numbers describe the main energy level, orientation of orbitals, and electron spin. There are four types of quantum numbers:

The principal quantum number specifies the location and energy of an electron. It describes the main energy level (or shell) of an electron. This number is associated with the orbital's size and energy. The principal quantum number, often denoted as n, has integer values of n = 1, 2, 3, …, where n corresponds to shells K, L, M, N, etc. The energy associated with a shell can be expressed as:

En = - 13.6 / n2 eV

where En is the energy of the electron at a particular principal quantum number n.

The azimuthal (or subsidiary) quantum number specifies the shape of the orbital. It determines the type of sublevel (orbital shape), and it is associated with the angular momentum of the electron. The value of l is an integer, and it can range from 0 to n-1, where n is the principal quantum number. l values correspond to the following orbital shapes:

• l = 0 (s orbital)
• l = 1 (p orbital)
• l = 2 (d orbital)
• l = 3 (f orbital)

The angular momentum of the electron is given by:

L = √(l(l + 1)) ℏ

where ℏ is the reduced Planck constant.

The magnetic quantum number specifies the orientation of the orbital. It defines the number of orbitals within a particular sublevel. The value of the magnetic quantum number m ranges from -l to +l, including zero. For example, for l = 1 (p orbitals), m can be -1, 0, +1, corresponding to three different p orbitals. The number of orbitals in a subshell is given by:

Number of orbitals = 2l + 1

The spin quantum number describes the direction in which an electron spins. There are two possible values for the spin quantum number:

• s = +1/2 (spin clockwise)
• s = -1/2 (spin counterclockwise)

This spin is an intrinsic property of the electron.

Quantum numbers are applied to determine the number of sublevels, orbitals, and total number of electrons for a given principal quantum number n. The process of finding the number of sublevels, orbitals, and electrons can be explained using tree diagrams or tabular forms. Example 1: Find the total number of electrons in the principal quantum number n = 2. Tree Diagram:

• Sublevels: 2 (s, p)
• Orbitals: 4 (1s, 2s, 2p)
• Electrons: 8 (since each orbital can hold 2 electrons)

Example 2: Find the total number of electrons in the principal quantum number n = 3.

Tabular Diagram:

• Sublevels: 3 (s, p, d)
• Orbitals: 9 (3 orbitals from 3s, 3 from 3p, 3 from 3d)
• Electrons: 18

The modern theory of electron behavior is based on the following two assumptions:

• Dual Nature of Electrons: Electrons exhibit both particle-like and wave-like behavior.
• Uncertainty Principle: It is impossible to determine both the position and momentum of an electron with certainty at the same time. This is described by Heisenberg's Uncertainty Principle:

Δx Δp ≥ ℏ / 2

where Δx is the uncertainty in position and Δp is the uncertainty in momentum.

1. Orbital: A region in space around the nucleus where there is a high probability of finding an electron. An orbital can hold a maximum of two electrons with opposite spins.
2. Energy Levels of Atoms: These are specific regions around the nucleus where electrons are likely to be found. The energy of an electron in these levels is quantized.
3. Shell: A complete set of orbitals with the same principal quantum number n.
4. Quantum: A discrete packet of energy that can be absorbed or emitted by an electron.

Aufbau Principle

Electrons fill orbitals starting from the lowest energy level to the highest. This is known as the Aufbau principle, and the order of orbital filling is:

1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f

The order of filling is determined by the sum of the principal quantum number and the azimuthal quantum number (n + l).

Pauli’s Exclusion Principle

This principle states that no two electrons in an atom can have the same set of four quantum numbers. Therefore, each orbital can hold a maximum of two electrons, and they must have opposite spins.

Hund’s Rule

When filling orbitals of equal energy (degenerate orbitals), electrons fill them singly first with parallel spins before pairing up. This minimizes electron repulsion. Hund’s rule is also known as the Hund’s rule of maximum multiplicity. Explanation Two electrons with parallel spins, tend to be as far apart as possible to minimize the electrostatic repulsion. Therefore, the electrons prefer to occupy the orbitals singly as far as possible. When all the orbitals get singly occupied, then the incoming electron has two choices either to pair up with the other electron or to go to the next higher orbital. When vacant orbital of suitable energy is not available, then the incoming electron will have no choice except to pair up with another electron.