Revision: Atoms and Nuclei JEE Main Atoms and Nuclei

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.

Dark spectral absorption lines are the lines seen in a continuous spectrum at the frequencies absorbed by the atoms of a rarefied gas.

When an atom absorbs a photon having precisely the same energy as that required for an electron in a lower energy state to make a transition to a higher energy state, the process is called absorption.

OR

Absorption is the process in which an atom takes in a photon whose energy exactly matches the energy needed for an electronic transition from a lower level to a higher level.

The various lines in atomic spectra are produced when electrons jump from a higher energy state to a lower energy state, and photons are emitted. These spectral lines are called emission lines.

OR

Emission lines are the spectral lines produced when electrons fall from higher energy states to lower energy states and emit photons.

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.

• 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.

A new type of neutral particle whose mass is very nearly the same as that of a proton is called neutron.

Atomic species of the same element differing in mass are called isotopes.

Nuclides with the same neutron number but different atomic number are called isotones.

All nuclides with the same mass number are called isobars.

A proton or a neutron is called a nucleon.

A different mass unit used for expressing atomic masses is the atomic mass unit (u), defined as one-twelfth of the mass of a carbon-12 atom, called atomic mass unit.

Mathematically: 1 u = \[\frac{\text{mass of one }^{12}\mathrm{C~atom}}{12}\] = 1.660539 × 10⁻²⁷ kg

The nucleus of the lightest atom of hydrogen, which has a relative abundance of 99.985%, is called the proton.

The nuclear radius is the effective distance from the centre of the nucleus to its outer boundary. Since the nucleus does not have a perfectly sharp surface, this radius is treated as an approximate effective value in physics.

Mass defect is the difference between:

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

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

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

`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.

The nuclear force is the strong attractive force between nucleons (protons and neutrons) that binds them together inside the nucleus.

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.

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

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.

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

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

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.

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.

Controlled thermonuclear fusion is the attempt to produce this fusion process under controlled conditions on Earth so that the released energy can be used safely and continuously.

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 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 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.

If the distance between the alpha-particle and the nucleus is rr, then the electrostatic force between them is given by:

F = \[\frac {1}{4πε_0}\] ⋅ \[\frac{2e\cdot Ze}{r^2}\]

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

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

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

m_p ​= 1.00727 u = 1.67262 × 10⁻²⁷ kg

m_n​ = 1.00866 u = 1.6749 × 10⁻²⁷ kg

A = Z + N

Where:

• Z = atomic number = number of protons.
• N = neutron number = number of neutrons.
• A = mass number = total number of protons and neutrons.

The experimentally observed relation is: R = R₀A^1/3

where:

• R = radius of the nucleus
• R₀​ = radius constant
• A = mass number of the nucleus.

For nuclei,

• R₀ ≈ 1.2 × 10⁻¹⁵ m.

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

Where:

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

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

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

Einstein gave the famous mass-energy equivalence relation:

Where: 

• E = energy equivalent of mass mm 
• m = mass 
• c = velocity of light in vacuum 
• c ≈ 3 × 10⁸ m s⁻¹

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

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

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

\[E=mc^2\]

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.

• Bohr accepted Rutherford’s nuclear model but modified it using quantum ideas.
• Classical mechanics and electromagnetism could not explain atomic-scale behaviour fully.
• Only certain orbits are allowed for the electron in the hydrogen atom.
• These orbits have definite total energy.
• Electron transitions between energy levels lead to photon emission.
• The energy of the hydrogen atom is negative because the electron is bound to the nucleus.

• Ground state energy of hydrogen = -13.6 eV.
• Ionisation energy of hydrogen in the ground state = 13.6 eV.
• Energy required for first excitation = 10.2 eV.
• Energy required for second excitation = 12.09 eV.
• The energy of a free electron is 0 eV.

• An atom emits radiation when it moves from a higher energy state to a lower energy state.
• The energy difference appears as a photon.
• Because the quantum numbers are integers, only discrete frequencies are emitted.
• These give rise to emission lines.
• If atoms absorb photons of the exact required energy, dark absorption lines appear in a continuous spectrum.

• 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.

• 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.

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).

• 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.