Atoms Class 12 Notes Physics Chapter 12

Dalton’s Atomic Theory

According to Dalton's atomic theory, all matter is composed of tiny, indivisible particles called atoms. Atoms of the same element are identical in mass and properties, while atoms of different elements vary in these aspects.

Dalton also proposed that atoms cannot be created, destroyed, or transformed into other types of atoms in a chemical reaction. Instead, they combine in fixed ratios to form compounds.

Thomson’s Atomic Model 

Every atom is uniformly positive charged sphere of radius of the order of 10^-10 m, in which entire mass is uniformly distributed and negative charged electrons are embedded randomly. The atom as a whole is neutral.

Limitations of Thomson’s Atomic Model 

• Thomson's model could not explain the origin of the spectral series of hydrogen and other atoms, which are the distinct lines seen in the spectrum of light emitted by these elements.
• It also failed to account for the large angle scattering of α-particles, which was later explained by Rutherford's model of the atom.

Rutherford’s Atomic Model

On the basis of this experiment, Rutherford made following observations

• The entire positive charge and almost entire mass of the atom is concentrated at its centre in a very tiny region of the order of 10^-15 m, called nucleus.
• The negatively charged electrons revolve around the nucleus in different orbits.
• The total positive charge of the nucleus is equal to the total negative charge on electron. Therefore atom as a overall is neutral.
•  The centripetal force required by electron for revolution is provided by the electrostatic force of attraction between the electrons and the nucleus.

Distance of Closest Approach

It refers to the point at which an alpha particle comes closest to the nucleus before being repelled. At this distance, the kinetic energy of the alpha particle is converted entirely into potential energy. The formula to calculate the distance of closest approach is: r ₀ . 1 / 4π ε ₀  . 2Z e ²  / E _k 

Where:

• E _{k =} kinetic energy of the alpha particle
• Z =  atomic number of the nucleus
• e =  charge of the electron
• ε _{0 =} permittivity of free space

Impact Parameter 

The perpendicular distance of the velocity vector of alpha-particle from the central line of the nucleus, when the particle is far away from the nucleus is called impact parameter.
Impact parameter
where, Z = atomic number of the nucleus, E_k = kinetic energy of the c-particle and θ = angle of scattering.

Rutherford’s Scattering Formula

where, 

N(θ) =number of c-particles, 
N_i = total number of α-particles reach the screen. 
n = number of atoms per unit volume in the foil, 
Z = atoms number, 
E = kinetic energy of the alpha particles and
 t = foil thickness

Limitations of Rutherford Atomic Model

(i) About the Stability of Atom According to Maxwell’s electromagnetic wave theory electron should emit energy in the form of electromagnetic wave during its orbital motion. Therefore. radius of orbit of electron will decrease gradually and ultimately it will fall in the nucleus.
(ii) About the Line Spectrum Rutherford atomic model cannot explain atomic line spectrum.

Bohr’s Atomic Model

• Electron can revolve in certain non-radiating orbits called stationary or bits for which the angular momentum of electron is an integer multiple of (h / 2π).
• The formula for angular momentum is mvr = nh / 2π, where n = I, 2. 3,… called principle quantum number.
• The radiation of energy occurs only when any electron jumps from one permitted orbit to another permitted orbit. Energy of emitted photon is hv = E₂ – E_1, where E₁ and E₂are energies of electron in orbits.
• Radius of orbit of electron is given by
  r = n²h² / 4π² mK Ze² ⇒ r ∝ n² / Z
  where, n = principle quantum number, h = Planck’s constant, m = mass of an electron, K = 1 / 4 π ε, Z = atomic number and e = electronic charge.
• Velocity of electron in any orbit is given by v = 2πKZe² / nh ⇒ v ∝ Z / n
• Frequency of electron in any orbit is given by v = KZe² / nhr = 4π²Z²e⁴mK² / n³ h³
  ⇒ v prop; Z³ / n³
• Kinetic energy of electron in any orbit is given by
  E_k = 2π²me⁴Z²K² / n² h² = 13.6 Z² / n² eV
• Potential energy of electron in any orbit is given by
  E_p = – 4π²me⁴Z²K² / n² h² = 27.2 Z² / n² eV
  ⇒ E_p = ∝ Z² / n²
• Total energy of electron in any orbit is given by
  E = – 2π²me⁴Z²K² / n² h² = – 13.6 Z² / n² eV

Potential Energy of Electron

• The equation for potential energy for hydrogen atoms is E = -me⁴ / 8n² h²ε₀².
• The wavelength of radiation emitted when an electron transitions from orbit n₂ to n₁ is represented in the image below:
  

In quantum mechanics, the energies of a system are discrete or quantized. The energy of a particle of mass m is confined to a box of length L can have discrete values of energy given by the relation
E_n = n² h² / 8mL² ; n < 1, 2, 3,…

Hydrogen Spectrum Series

Each element emits a spectrum of radiation, which is characteristic of the element itself. The spectrum consists of a set of isolated parallel lines and is called the line spectrum.

Hydrogen spectrum contains five series
(i) Lyman Series When electron jumps from n = 2, 3,4, …orbit to n = 1 orbit, then a line of Lyman series is obtained.
This series lies in ultra violet region.
(ii) Balmer Series When electron jumps from n = 3, 4, 5,… orbit to n = 2 orbit, then a line of Balmer series is obtained.
This series lies in visual region.
(iii) Paschen Series When electron jumps from n = 4, 5, 6,… orbit to n = 3 orbit, then a line of Paschen series is obtained.
This series lies in infrared region 
(iv) Brackett Series When electron jumps from n = 5,6, 7…. orbit to n = 4 orbit, then a line of Brackett series is obtained.
This series lies in infrared region. 
(v) P fund Series When electron jumps from n = 6,7,8, … orbit to n = 5 orbit, then a line of Pfund series is obtained.
This series lies in infrared region.

Spectral Series

Wave Model

• A wave model is a theoretical framework used to describe the behavior of particles, such as electrons, in terms of wave phenomena. This model is particularly relevant in the field of quantum mechanics, where particles exhibit both particle-like and wave-like properties.
• In the wave model, particles are considered to be associated with wave functions, which provide information about the probability of finding a particle in a particular location at a given time. The wave function is a mathematical representation of the particle's wave-like behavior and is central to understanding phenomena such as interference and diffraction.
• One of the key principles underlying the wave model is the concept of wave-particle duality, which posits that particles can exhibit characteristics of both waves and particles depending on the experimental conditions. This duality is a fundamental aspect of quantum mechanics and has been confirmed through various experiments, such as the double-slit experiment.
• The wave model also incorporates the idea of quantization, where certain physical properties, such as energy, are restricted to discrete values. This is in contrast to classical physics, where these properties can take on a continuous range of values. The quantization of energy levels in atoms, for example, leads to the discrete spectral lines observed in atomic emission and absorption spectra.

Quantum Numbers

• Principal quantum number (n): This quantum number determines the energy level and size of the orbital. It can take positive integer values such as 1, 2, 3, and so on, theoretically up to infinity. Higher values of n correspond to orbitals that are farther from the nucleus and have higher energy.
• Orbital angular momentum quantum number (l): Also known as the azimuthal quantum number, l determines the shape of the orbital. It can take integer values from 0 to n-1 for a given principal quantum number n. For example, if n = 3, l can be 0, 1, or 2. The values of l correspond to different orbital shapes: s (l = 0), p (l = 1), d (l = 2), and f (l = 3).
• Magnetic quantum number (m): This quantum number specifies the orientation of the orbital in space. It can take integer values from -l to +l, including zero. For instance, if l = 1, m can be -1, 0, or +1. This quantum number is particularly important in determining how orbitals are oriented in a magnetic field.
• Magnetic spin angular momentum quantum number (m_s): This quantum number describes the intrinsic spin of an electron within an orbital. Electrons have a property called spin, which can be thought of as a type of intrinsic angular momentum. The spin quantum number can have one of two values: +1/2 or -1/2, indicating the two possible orientations of an electron's spin. This quantum number is crucial for understanding the Pauli exclusion principle, which states that no two electrons in an atom can have the same set of four quantum numbers.

Some Solved Examples:

Example 1: In H-atom, a transition takes place from n = 3 to n = 2 orbit. Calculate the wavelength of the emitted photon, will the photon be visible? To which spectral series will this photon belong?
(Take, R = 1.097 × 10⁷ m⁻¹)

Sol. The wavelength of the emitted photon is given by:

When the transition takes place from n = 3 to n = 2, then

Since $\backslash lambda$λ falls in the visible (red) part of the spectrum, hence the photon will be visible. This photon is the first member of the Balmer series.

Example 2:  In a head-on collision between an α-particle and gold nucleus, the closest distance of approach is $4\; \backslash times\; 10^\{-14\}$4×10⁻¹⁴ m. Calculate the initial kinetic energy of α-particle.

Sol. Here, closest distance of approach, $r\mathrm{<sub\; style="word-break:\; break-word;\; line-height:\; 1.8;\; font-family:\; Lato,\; sans-serif;\; font-size:\; 20px;\; letter-spacing:\; inherit;\; color:\; black;">0</sub>}=$4×10⁻¹⁴ m, atomic number,  Z=79, KE = ?

Example 3: It is found experimentally that 13.6 eV energy is required to separate a H-atom into a proton and an electron. Compute the orbital radius and velocity of the electron in a H-atom.

Sol. Total energy of the electron in H-atom,
TE = −13.6 eV = −13.6 × 1.6 × 10⁻¹⁹J
$\hspace{0.17em}J\backslash therefore\; TE\; =\; -2.2\; \backslash times\; 10^\{-18\}\; \backslash ,\; \backslash text\{J\}$∴ TE = −2.2 × 10⁻¹⁸ J

Total energy is: 

Velocity of the revolving electron,