Revision: Atoms and Nuclei JEE Main Atoms and Nuclei

The atom emits a photon with energy equal to the difference between the two energy levels, a phenomenon known as luminescence.

Completely removing an electron from atom is called ionisation.

• 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.

Both the proton and neutron together constitute the nucleus. They are called nucleons.

The minimum energy required to make an electron free from the nucleus is called the Binding Energy of an electron.

The ratio of the binding energy \[E_n\]_​ of a nucleus to the number of nucleons A in that nucleus is called Binding Energy Per Nucleon.

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 equivalent to that of mass defect, i.e., the energy required for holding the nucleons together in a nucleus, is called the Binding Energy of the nucleus.

In a graph plotting binding energy per nucleon (Bₙ) against mass number (A) for all known nuclei, the resulting curve is called binding energy curve.

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 minimum amount of energy required to be given to an electron in the ground state of an atom to set the electron free is called the Ionization Energy of that atom.

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.

In a graph plotting binding energy per nucleon (Bₙ) against mass number (A) for all known nuclei, the resulting curve is called binding energy curve.

The minimum energy required to make an electron free from the nucleus is called the Binding Energy of an electron.

The energy equivalent to that of mass defect, i.e., the energy required for holding the nucleons together in a nucleus, is called the Binding Energy of the nucleus.

The minimum amount of energy required to be given to an electron in the ground state of an atom to set the electron free is called the Ionization Energy of that atom.

The ratio of the binding energy \[E_n\]_​ of a nucleus to the number of nucleons A in that nucleus is called Binding Energy Per Nucleon.

`1/12`th of the mass of an atom of ₆C¹² isotope.

1 AMU is the average of proton rest mass and the neutron rest mass. Thus can be expressed as

1 AMU = 1.67377 × 10^-27 kg

= 1.67377 × 10^-24 gram

and C-12 is considered a reference for all atomic mass calculations.

The attractive force which holds the nucleons together in the nucleus is called nuclear force.

Radioactivity is a nuclear phenomenon. It is the process of spontaneous emission of α or β and γ radiations from the nucleus of atoms during their decay.

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.

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.

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.

The energy released due to loss in mass during the processes of nuclear fission and fusion is called nuclear (or atomic) energy.

OR

The energy released when nuclei undergo a nuclear reaction (change in structure, forming new nuclei) is called nuclear energy.

OR

The energy released during the transformation of nuclei is called Nuclear Energy.

The difference between the sum of the masses of the nucleons composing a nucleus and the rest mass of the nucleus is called the mass defect.

Nuclear fission is the process in which a heavy nucleus splits into two lighter nuclei of nearly the same size, when bombarded with slow neutrons. In each fission reaction, a tremendous amount of energy (≈ 190 MeV) is released.

OR

The process of splitting of a heavy nucleus (₉₂U²³⁵ or ₉₂U²³⁹) into two lighter nuclei of comparable masses along with the release of a large amount of energy after being bombarded by slow neutrons is called Nuclear Fission.

• Nuclear fusion is the process in which two light nuclei combine to form a heavy nucleus. In this process also, huge amount of energy is released.
• The phenomenon in which two light nuclei fuse to form a larger nucleus and energy is released is called Nuclear Fusion.

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

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.

Radius of the n-th Bohr Orbit (General):

\[r_n=\frac{\varepsilon_0n^2h^2}{\pi mZe^2}\]

\[\mathrm{i.e.,}r_n\propto n^2\mathrm{and}r_n\propto\frac{1}{Z}\]

Radius of n-th orbit for Hydrogen-like atom:

\[r_n=0.53\left(\frac{n^2}{Z}\right)\mathrm{\r{A}}\]

\[v_n=\frac{nh}{2\pi mr_n}\]

\[r=\frac{n^2h^2}{4\pi^2mkZe^2}\]

\[v=\frac{2\pi kZe^2}{nh}\]

For hydrogen atom (Z = 1):

\[v=\frac{2\pi ke^2}{nh}=\alpha\frac{c}{n}\]

where α is the fine structure constant and \[\alpha=\frac{1}{137}.\]

Total Energy of Electron in n-th Orbit (General):

• \[E_n=\frac{-Z^2me^4}{8\varepsilon_0^2n^2h^2}\]

Total Energy (Alternate form):

• \[E_n=-\frac{2\pi^2mk^2Z^2e^4}{n^2h^2}\]

Total Energy for Hydrogen-like Atom (Simplified):

\[\bar{v}=\frac{1}{\lambda}=RZ^2\left[\frac{1}{n_1^2}-\frac{1}{n_2^2}\right]\mathrm{m}^{-1}\]

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

\[E_\text{ionisation}=13.6Z^2\mathrm{~eV}\]

\[E_n=-Rhc\left(\frac{1}{n^2}\right)\]

General Formula for all Spectral Series:

\[\frac{1}{\lambda}=R\left[\frac{1}{n_1^2}-\frac{1}{n_2^2}\right]\]

where \[R=1.097\times10^7\mathrm{m}^{-1},n_1=\text{final state},n_2=\text{initial state},n_2>n_1\]

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

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

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

\[q=+1.6\times10^{-19}\text{C}\]

\[m_p=1.6726\times10^{-27}\text{kg}\]

\[R=R_0A^{1/3}\]

where \[\mathrm R_{0}=1.4\times10^{-15}\mathrm{m}\]

\[\frac{R_1}{R_2}=\left(\frac{A_1}{A_2}\right)^{1/3}\]

BE per nucleon ​= \[\frac {E.E.}{A}\]

\[\text{Binding Energy per Nucleon}=\frac{\text{Binding Energy}}{\text{Nucleon Number}}\]

Binding Energy = \[(\Delta m)\cdot c^2=(\text{Mass defect})\cdot c^2\]

E_b = ΔM ⋅ c²

E_b ​= [(Zm_p​ + (A − Z)m_n​) − M] × c²

\[\text{Binding Energy per Nucleon}=\frac{\text{Binding Energy}}{\text{Nucleon Number}}\]

BE per nucleon ​= \[\frac {E.E.}{A}\]

E_b = ΔM ⋅ c²

E_b ​= [(Zm_p​ + (A − Z)m_n​) − M] × c²

Binding Energy = \[(\Delta m)\cdot c^2=(\text{Mass defect})\cdot c^2\]

\[BE=\Delta m\cdot c^2\]

\[\Delta m_a=Am_p+Bm_n+Am_e-M_{ar}\]

\[\Delta m=[ZM_p+(A-Z)M_n]-M_\mathrm{nucleus}\]

\[E=mc^2\]

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.

\[\text{Fission Reaction of Uranium-235:}\_{92}\mathrm{U}^{235}+_0n^1\longrightarrow\left[_{92}\mathrm{U}^{236}\right]\longrightarrow_{56}\mathrm{Ba}^{144}+_{36}\mathrm{Kr}^{89}+3_0n^1+200\mathrm{~MeV}\]

\[_1\mathrm{H}^2+_1\mathrm{H}^2\longrightarrow_2\mathrm{He}^3+_0n^1+3.27\mathrm{~MeV}\]

\[_1\mathrm{H}^2+_1\mathrm{H}^2\longrightarrow_1\mathrm{H}\mathrm{e}^3+_1\mathrm{H}^1+4.03\mathrm{~MeV}\]

\[_1\mathrm{H}^2+_1\mathrm{H}^3\longrightarrow_2\mathrm{H}\mathrm{e}^4+_0n^1+17.59\mathrm{~MeV}\]

\[_1\mathrm{H}^2+_2\mathrm{He}^3\longrightarrow_2\mathrm{He}^4+_1\mathrm{H}^1+18.3\mathrm{~MeV}\]

• Most alpha particles passed through the gold foil without any deflection, proving the atom is mostly empty space.
• Around 0.14% of incident alpha particles are scattered by more than 1°.
• Around 1 in 8000 alpha particles are deflected by more than 90°.
• Large-angle deflections indicated a small, dense, positively charged nucleus at the centre of the atom.
• The gold foil used had a thickness of 2.1 × 10⁻⁷ m; the alpha particles had an energy of 5.5 MeV.
• Electrons have negligible mass and do not affect the trajectory of incident alpha particles.
• This experiment disproved Thomson's plum-pudding model and established the nuclear structure of the atom.

• The radius of Bohr's orbit is proportional to \[n^{2}\] and inversely proportional to Z.
• For hydrogen (Z = 1), the ground state (n = 1) radius is 0.53 Å, known as Bohr's radius.
• The velocity of an electron decreases as the orbital number (n) increases.
• For hydrogen, orbital speed of electron equals \[\alpha\frac{c}{n}\]​, where \[\alpha=\frac{1}{137}\]​.
• The total energy of an electron in any orbit is negative, indicating a bound state.
• For hydrogen-like atoms, the energy of an electron in the n-th orbit is \[-13.6\frac{Z^2}{n^2}\mathrm{~eV}.\] .

• For hydrogen (Z = 1): ground state energy = −13.6 eV; at n = ∞, energy = 0 eV.
• Energy levels for hydrogen: n=1: −13.6 eV, n=2: −3.4 eV, n=3: −1.511 eV, n=4: −0.850 eV, n=5: −0.544 eV.
• In normal conditions, electrons are in the ground state, occupying orbitals closest to the nucleus.
• Beyond ionisation potential, the electron is no longer bound — energy levels form a continuum (starts at 13.6 eV above ground in hydrogen).
• Electrons in orbitals close to the nucleus are stable (need more energy to remove); electrons farther away are less stable.
• If energy supplied ≥ ionisation energy, ionisation occurs.
• Spectral Series (from energy level transitions): Lyman, Balmer, Paschen, Bracket, Pfund series.

• Lyman series — n₁=1, n₂: 2→∞; converges toward 91–122 nm; UV region
• Balmer series — n₁=2, n₂: 3→∞; converges toward 365–657 nm; visible region (Hα = Red, Hβ = Blue-green, Hγ = Blue)
• Paschen series — n₁=3, n₂: 4→∞; converges toward 821–1876 nm; infrared (IR)
• Brackett series — n₁=4, n₂: 5→∞; converges toward 1459–4053 nm; IR region
• Pfund series — n₁=5, n₂: 6→∞; converges toward 2280–7462 nm; IR region
• Humphreys series — n₁=6, n₂: 7→∞; converges toward 3283 nm–∞; IR region
• Setting n₁=1 and n₂ from 2 to ∞ gives the Lyman series converging to 91 nm

• 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.

• All atomic nuclei are made up of elementary particles called protons and neutrons.
• Protons are positively charged particles with charge 1.6 × 10⁻¹⁹ C.
• The mass of a neutron is slightly greater than that of a proton.
• Neutrons are electrically neutral (uncharged) particles.
• The number of protons in the nucleus of an element equals the number of electrons in the neutral atom.
• All nuclei of a given element may not have the same number of neutrons.

• Nuclear size is measured using the Rutherford scattering experiment.
• Alpha particles deflected at distances ~10⁻¹⁴ m.
• Fast electrons and neutron scattering are also used for measurement.
• Nuclear radius is expressed using the radius parameter \[R_0\]
• Nucleus size is extremely small compared to an atom.
• Nuclear density remains constant for all nuclei.

• The greater the binding energy per nucleon, the more stable the nucleus.
• Iron-56 (Fe⁵⁶) and Nickel-62 are among the most stable nuclei, lying at the peak of the binding energy curve.
• Light nuclei (A < 20): Binding energy per nucleon increases rapidly with mass number.
• Intermediate nuclei (A ≈ 20–60): Highest binding energy per nucleon — most stable region.
• Heavy nuclei (A > 60): Binding energy per nucleon gradually decreases — less tightly bound.
• Very heavy nuclei can become unstable and may undergo fission, splitting into smaller, more stable nuclei, releasing energy.
• If nucleons are separated, the energy required to separate them gets converted into mass.

• The greater the binding energy per nucleon, the more stable the nucleus.
• Iron-56 (Fe⁵⁶) and Nickel-62 are among the most stable nuclei, lying at the peak of the binding energy curve.
• Light nuclei (A < 20): Binding energy per nucleon increases rapidly with mass number.
• Intermediate nuclei (A ≈ 20–60): Highest binding energy per nucleon — most stable region.
• Heavy nuclei (A > 60): Binding energy per nucleon gradually decreases — less tightly bound.
• Very heavy nuclei can become unstable and may undergo fission, splitting into smaller, more stable nuclei, releasing energy.
• If nucleons are separated, the energy required to separate them gets converted into mass.

Mass defect refers to the difference between the mass of a nucleus and the sum of the masses of its individual protons and neutrons (nucleons).

• Mass of ₆C¹² is exactly 12 amu; 1 amu = 1.660565 × 10⁻²⁷ kg.
• 1 amu of mass, when converted to energy, gives 931.5 MeV.
• Mass defect arises because some mass is converted into binding energy that holds the nucleus together.
• Atomic mass = Number of protons + Number of neutrons.
• There are three fundamental particles of an atom: protons, neutrons, and electrons.
• Protons and neutrons are big-sized particles present in the nucleus of an atom.
• The density of the nucleus is independent of the mass number of the atom.

• In a fission reaction, a heavy atomic nucleus is split into smaller nuclei, other particles and radiation.
• Uranium-235 absorbs a neutron and splits into barium and krypton, emitting neutrons and radiation.
• Each fission of U²³⁵ releases approximately 200 MeV of energy.
• 3 neutrons are released per fission, which can trigger further fissions — leading to a chain reaction.
• Nuclear power plants exploit the process of fission to create energy.
• If an incoming neutron strikes a uranium nucleus, fragments produced are chemical elements like barium or krypton, while some are free neutrons.

• In a fusion reaction, two or more light atomic nuclei fuse to form a single heavier nucleus.
• The mass change in the process is the source of nuclear energy.
• Fusion within the cores of the sun and other stars generates their radiating energy by fusing two hydrogen atoms to produce a helium atom.
• The product nucleus has less mass than the total mass of the combining nuclei — the difference is released as energy.
• Fusion of deuterium (²H) and tritium (³H) produces helium-4 and releases 17.59 MeV — the most energy-rich reaction listed.
• Fusion releases far more energy per unit mass than fission.

• 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.