Revision: 12th Std >> Structure of Atoms and Nuclei MAH-MHT CET (PCM/PCB) Structure of Atoms and Nuclei

The mass number of an atom is equal to the total number of nucleons (i.e., the sum of the number of protons and the number of neutrons) in its nucleus.

The total number of neutrons and protons in the nucleus is called the mass number of the element and is denoted by A.

The atomic number of an atom is equal to the number of protons in its nucleus (which is same as the number of electrons in a neutral atom).

The number of protons in the nucleus is known as the atomic number of the element and is denoted by Z.

The number of protons in the nucleus of an atom, which is characteristic of a chemical element and determines its place in the periodic table. Atomic number is also equal to the number of electrons in an atom.

When white light from a source passes through an atomic gas or vapour, the gas absorbs radiation of certain wavelengths. As a result, dark lines appear in the otherwise continuous spectrum. This type of spectrum is called an absorption spectrum.

When an atomic gas or vapour at low pressure is excited, it emits radiation of certain wavelengths. The emitted radiation, when analysed with a spectroscope, shows a series of bright lines on a dark background. This type of spectrum is called an emission line spectrum.

The definite amount of energies associated with the electrons in different orbits of an atom are called the Energy Levels (of that atom).

The energy required to take an electron from the ground state to an excited state is called the Excitation Energy of the electron in that state.

The nucleus is made up of protons and neutrons, with protons having a positive charge and neutrons being neutral. Nucleons are made up of protons and neutrons.

The binding energy of a nucleus is the minimum energy required to separate the nucleus completely into its constituent protons and neutrons.

Mass defect is the difference between:

• the sum of the masses of the constituent protons and neutrons, and
• the actual mass of the nucleus.

Binding energy per nucleon is the average binding energy associated with each nucleon in the nucleus.

The rate of decay, i.e., the number of decays per unit time \[\left(-\frac{dN(t)}{dt}\right)\], is called Activity A(t).

The time in which half the substance (radioactive) is disintegrated is called the Half-Life Period of a radioactive substance.

When a nucleus in an excited state spontaneously decays to its ground state and a photon is emitted with energy equal to the difference in the two energy levels of the nucleus, this is called γ-Decay.

The difference in the energy equivalent of the mass of the parent atom and that of the sum of masses of the products is called the Q-Value of the decay.

The arithmetic average of the lives of all the nuclei present initially is called the Average Life of a radioactive element.

The spontaneous emission of an electron (β⁻-decay) or a positron (β⁺-decay) from a radioactive nucleus is called β-Decay.

The half-life of a reaction is the time it takes for a reactant’s concentration to decrease to half of its initial value.

The phenomenon of emission of a nucleus of helium (₂He⁴) from a radioactive nucleus is called α-Decay.

The nuclear phenomenon in which an unstable nucleus undergoes decay with the emission of some particles (α, β) and electromagnetic radiation (γ-rays) is called Radioactive Decay.

One Becquerel (Bq) is defined as the activity of a quantity of radioactive samples in which one nucleus decays per second. It is the SI unit of the activity.

Energy is released when nuclei undergo reactions (fission or fusion) that move them from a state of lower total binding energy to a state of higher total binding energy.

The combination of light nuclei to form a heavier nucleus, releasing energy.

The splitting of a heavy nucleus into lighter nuclei, releasing energy.

Nuclear fission is a neutron-induced nuclear reaction in which a heavy nucleus, such as uranium-235, breaks into two intermediate-mass nuclear fragments.

Because both nuclei carry a positive charge, they repel each other through electrostatic force. This repulsion is called the Coulomb barrier in the context of fusion.

Nuclear fusion is the process in which light nuclei combine to form a more tightly bound heavier nucleus, and energy is released during the process.

Thermonuclear fusion is fusion initiated by extremely high temperature, which gives nuclei enough kinetic energy to approach one another closely.

• The collection of different spectral lines obtained due to transition of an electron in hydrogen atom from upper energy levels to lower energy levels is called the Hydrogen Spectrum.
• The hydrogen spectrum consists of specific wavelengths of light emitted by hydrogen atoms. When transition of an electron in a hydrogen atom occurs between energy levels, it emits or absorbs photons of certain wavelengths, creating a series of lines known as the hydrogen spectrum.

The spectrum consisting of bright lines on a dark background, emitted when an atomic gas is excited at low pressure by passing an electric current through it, is called the Emission Line Spectrum.

Radioactivity is the spontaneous emission of high-energy radiation from the nucleus of an unstable atom.

A radioactive substance is a substance whose nuclei are unstable and therefore emit radiation on their own.

As nucleus is positively charged it strongly attracts the negative charged electrons. The electron orbit close to the nucleus are tightly bound by strong attractive force of nucleus. These electrons are known as bound electrons.

The phenomenon of spontaneous disintegration of an unstable nucleus of a naturally occurring isotope accompanied by emission of active radiations, α particles, β particles and γ radiations is called radioactivity.

Electrons in outer orbits are weakly bound with the nucleus. In solids these weakly bound electrons leave their individual atom and become a part of it. These electrons are known as free electrons.

\[L=mvr=\frac{nh}{2\pi},\quad n=1,2,3\ldots\]

\[h\nu=E_2-E_1=\frac{hc}{\lambda}\]

\[\Delta M=[Zm_p+(A-Z)m_n]-M\]

Where:

• ΔM = mass defect
• m_p​ = mass of one proton
• m_n​ = mass of one neutron
• M = actual mass of the nucleus

E_b ​= ΔMc²

Where:

• E_b​ = binding energy
• ΔM = mass defect
• c = speed of light

\[E_{bn}=\frac{E_b}{A}\]

Where:

• E_bn = binding energy per nucleon
• E_b​ = total binding energy of the nucleus
• A = mass number

Q = [M_parent​ − M_products​]c²

Q = [m_X​ − m_Y​ − m_He​]c²

Q = [m_X ​− m_Y ​− m_e​]c²

\[\Delta E=h\nu=E_i-E_f\]

\[\frac{1}{\lambda_{\mathrm{vac}}}=R_H\left[\frac{1}{n_1^2}-\frac{1}{n_2^2}\right]\]

where \[R_{H}=1.097\times10^{7}\mathrm{m}^{-1}\] (Rydberg constant)

Bohr's First Postulate:
An atom consists of a small, massive central core called the nucleus, around which planetary electrons revolve. The centripetal force required for their rotation is provided by the electrostatic attraction between the electrons and the nucleus.

Bohr's Second Postulate (Quantum Condition):
The electrons are permitted to circulate only in those orbits in which the angular momentum of an electron is an integral multiple of \[\frac{h}{2\pi}\]; h being Planck's constant.

Bohr's Third Postulate:
While revolving in the permissible orbits, an electron does not radiate energy. These non-radiating orbits are called stationary orbits.

Bohr's Fourth Postulate:
An atom can emit or absorb radiation in the form of discrete energy photons only when an electron jumps from a higher to a lower orbit or from a lower to a higher orbit, respectively.

1. The law states that the rate at which a radioactive substance undergoes decay is directly proportional to the number of undecayed nuclei present in the sample.
2. Mathematically: \[\frac {dN}{dt}\] ∝ N, which gives \[\frac {dN}{dt}\] = −λN, where λ is the decay constant.
3. On solving, the number of undecayed nuclei at time t is:
   N(t) = N₀e^−λt
   where N₀ is the number of nuclei present initially.
4. The time taken for the number of parent radioactive nuclei to reduce to half its value is called the half-life of the species, and the average life of a radioactive species is the average time a nucleus survives before it decays.

• The structure of an atom and its nucleus was developed from the discovery of electrons by J.J. Thomson and alpha particle scattering experiments by Rutherford.
• An atom consists of electrons, protons, and neutrons, with protons and neutrons in the nucleus and electrons revolving in stationary orbits.
• The maximum number of electrons in a shell is given by 2n², and the shells are named K, L, M, N, O, P, and Q.

• Proposed by Ernest Rutherford in 1911 based on the gold foil (α-particle scattering) experiment.
• Most α-particles passed straight through, showing that the atom is mostly empty space.
• Some α-particles were deflected, indicating the presence of a positively charged centre.
• Very few α-particles were deflected at large angles or bounced back, proving a dense nucleus.
• All the positive charge and most of the mass are concentrated in a tiny nucleus (~10⁻¹⁵ m).
• Electrons revolve around the nucleus in circular orbits.
• The electrostatic force of attraction between nucleus and electrons keeps them in orbit.
• Limitation: Could not explain stability of atom and line spectra of hydrogen.

• Bohr modified Rutherford's model - electrons move in fixed orbital shells, each with fixed energy levels.
• The centripetal force for electron revolution is provided by electrostatic attraction between the electron and the nucleus.
• An electron does not radiate energy while revolving in a stationary orbit.
• Energy is emitted or absorbed only during electron transitions between orbits.
• Limitations of Bohr's Model:
• Fails to explain the Zeeman Effect (effect of high magnetic fields on atomic spectra).
• Contradicts the Heisenberg Uncertainty Principle.
• Unable to explain the spectra of larger/multi-electron atoms.

• Could not explain the fine structure (splitting) of the spectral lines of hydrogen.
• Failed to explain the spectra of multi-electron atoms.
• Could not explain the splitting of spectral lines in a magnetic field (Zeeman effect) and an electric field (Stark effect).
• Failed to explain the formation of molecules and chemical bonding.
• Inconsistent with Heisenberg’s Uncertainty Principle.
• Could not explain the intensity of spectral lines.

de Broglie Wavelengths for Charged Particles (accelerated through potential V)

de Broglie Wavelengths for Uncharged Particles:

Key Derivation Logic:

• Planck's quantum theory: photon energy E = hν, de Broglie wavelength λ = h/p
• If a photon has energy E = hν, treating it as mass m by relativity: E = mc², so p = mc = h/λ
• For a material particle: momentum p = mv, so de Broglie wavelength λ = h/(mv)
• Kinetic energy \[K=p^2/2m\to\lambda=h/\sqrt{2mK}\]

Dalton's atomic theory laid the foundation of modern chemistry with four core postulates:

1. All matter is made up of extremely small particles called atoms.
2. Atoms of the same element are identical to each other in mass and properties; atoms of different elements differ.
3. Atoms can neither be created nor destroyed — they are indestructible.
4. Atoms combine in fixed, simple whole-number ratios to form compound atoms (molecules).

Note: Modern discoveries have refined some postulates (e.g., isotopes show atoms of the same element can differ in mass), but the core framework remains foundational.

• Lyman series — transitions to n = 1; region: ultraviolet
• Balmer series — transitions to n = 2; region: visible
• Paschen series — transitions to n = 3; region: infrared
• Brackett series — transitions to n = 4; region: infrared
• Pfund series — transitions to n = 5; region: infrared
• The spectrum of hydrogen is important as most of the universe is made of hydrogen.
• Balmer series involves transitions starting/ending with the first excited state (n = 2) of hydrogen.