Overview: Structure of Atoms and Nuclei

The smallest indivisible particles of matter proposed by early philosophers are called atoms.

The central, tiny, positively charged part of an atom that contains almost all of its mass is called the nucleus.

The spontaneous disintegration of an unstable nucleus accompanied by emission of radiation is called radioactive decay.

The negatively charged particles discovered by J. J. Thomson are called electrons.

The atomic model proposed by J. J. Thomson in which an atom is considered as a uniformly positively charged sphere with electrons embedded in it is called Thomson’s atomic model. Thomson’s atomic model is also called the plum-pudding model.

An atom having equal total positive and negative charges and therefore having no net charge is called an electrically neutral atom.

• In this experiment, alpha particles were fired at a thin gold foil to study the structure of the atom.
• Most alpha particles passed straight through, but some were deflected at different angles.
• A very few particles were deflected through large angles, showing that most of the mass and positive charge of the atom is concentrated in a small region.

• Rutherford said that almost all the mass and positive charge of an atom are concentrated in a tiny central nucleus.
• The nucleus is very small compared to the atom, so most of the atom is empty space.
• Electrons move around the nucleus in circular orbits like planets around the Sun.
• Large deflection of some alpha particles was due to their close approach to the dense nucleus.
• The model failed because revolving electrons should lose energy and fall into the nucleus, but atoms are stable and do not continuously emit radiation.

The spectrum obtained when a hot gas emits radiation at only certain specific wavelengths is called a line spectrum.

The bright lines seen in the spectrum due to emission of radiation at specific wavelengths are called emission lines.

The groups of spectral lines arranged according to a pattern are called spectral series (such as Lyman series, Balmer series, Paschen series, etc.).

\[\frac{1}{\lambda}=R\left(\frac{1}{n^2}-\frac{1}{m^2}\right)\]

The circular paths in which electrons revolve around the nucleus without emitting radiation are called stable orbits (stationary states).

The condition that the angular momentum of an electron in a stable orbit is an integral multiple of \[\frac {h}{2π}\] is called Bohr’s quantization condition.

The process in which an electron moves from one orbit to another with emission or absorption of a photon is called an electronic transition.

mvr = \[\frac {nh}{2π}\]

where

• m = mass of electron
• v = velocity
• r = radius of orbit
• n =1,2,3,… (principal quantum number)
• h = Planck’s constant

The positive integer nnn that determines the orbit of an electron is called the principal quantum number.

The radius of the first orbit of the hydrogen atom (n = 1) is called the Bohr radius.

\[r_n=a_0\frac{n^2}{Z}\]

The lowest energy state of an atom (n = 1) is called the ground state.

The higher energy states (n > 1) are called excited states. The minimum energy required to raise an electron from the ground state to a higher energy state is called the excitation energy.

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

The constant appearing in the spectral formula of hydrogen is called the Rydberg constant.

R_H = \[\frac{m_{\mathrm{e}}e^{4}}{8c\varepsilon_{0}h^{3}}=1.097\times10^{7}\mathrm{m}^{-1}\]

• Bohr’s model could explain only the hydrogen spectrum and failed to explain the spectra of other atoms. It also could not explain the fine structure of hydrogen lines.
• It could not explain why some spectral lines are brighter (more intense) than others.
• The idea of fixed stable orbits was assumed without proper theoretical proof, so the model was not fully satisfactory.

• De Broglie said every moving particle has a wave nature.
  λ = \[\frac {h}{mv}\]
• Electrons in atoms behave like standing waves, not just particles.
• Stable orbits form only when the orbit length equals whole multiples of the wavelength:
  2πr = nλ
• Only certain orbits are allowed, because the wavelength must fit exactly in the orbit.
• This explains Bohr’s rule:
  mvr = \[\frac {nh}{2π}\] (angular momentum is quantised).

Protons and neutrons together are called nucleons.

Atoms having the same mass number (A) but different atomic numbers (Z).

Atoms having the same number of neutrons (N) but different atomic numbers (Z).

• Three units are used to measure atomic masses:
  kg, atomic mass unit (u), and eV/c².
• Atomic mass unit (u):
  1 u = \[\frac {1}{12}\]​ of carbon-12 atom mass
  1 u = 1.66 × 10⁻²⁷
  1 u = 931.5, MeV/c².
• Mass–energy relation:
  From E = mc², mass can be expressed in energy units (eV/c²).
  Proton and neutron ≈ 1 u each; electron mass is much smaller.

\[\rho=\frac{mA}{\frac{4}{3}\pi R^3}\]

After substituting R = R₀A^1/3:

\[\rho=\frac{3m}{4\pi R_0^3}\]

The force that binds protons and neutrons (nucleons) inside the nucleus is called the nuclear force or strong force.

The energy required to completely separate the nucleons of a nucleus and take them to infinity.

The difference between the total mass of individual nucleons and the actual mass of the nucleus.

ΔM = Zm_p + Nm_n − M

EB ​= ΔMc²

Binding Energy in terms of Protons and Neutrons:

EB​ = (Zm_p​ + Nm_n​ − M)c²

Binding Energy using Atomic Masses:

EB ​= [Zm_H​ + Nm_n ​− M]c²

The spontaneous transformation of an unstable nucleus into a more stable nucleus by emission of particles or radiation is called radioactive decay.

The original unstable nucleus which undergoes decay is called the parent nucleus.

The nucleus formed after radioactive decay is called the daughter nucleus.

The emission of an alpha particle (helium nucleus) from a heavy nucleus is called alpha decay.

\[{}_Z^AX\to{}_{Z-2}^{A-4}Y+{}_2^4He\]

Q = [m_X ​− m_Y​ − m_He​]c²

The emission of an electron due to the conversion of a neutron into a proton inside the nucleus is called beta minus (β⁻) decay.

\[{}_Z^AX\to{}_{Z+1}^AY+e^-+\text{antineutrino}\]

Q = [m_X ​− m_Y ​− m_e​]c²

The emission of a positron due to conversion of a proton into a neutron inside the nucleus is called beta plus (β⁺) decay.

\[_Z^AX\to_{Z-1}^AY+e^++\mathrm{neutrino}\]

The emission of high-energy photons from an excited nucleus without a change in mass number or atomic number is called gamma decay.

\[_Z^AX^*\to_Z^AX+\gamma\]

The difference between the mass of the parent atom and the total mass of the decay products, expressed in energy terms, is called the Q-value of the decay.

Statement

The rate of decay of a radioactive substance at any instant is directly proportional to the number of undecayed nuclei present at that instant.

Proof / Mathematical Derivation

Let N (t) be the number of undecayed nuclei at time t.

The rate of decay is proportional to N (t):

\[\frac {dN}{dt}\] ∝ -N (t)

Introducing proportionality constant λ (decay constant):

\[\frac{dN}{dt}=-\lambda N(t)\]

Separating variables:

\[\frac{dN}{N}=-\lambda dt\]

Integrating from N₀​ at t = 0 to N at time t:

\[\int_{N_0}^N\frac{dN}{N}=-\lambda\int_0^tdt\]

\[\ln\left(\frac{N}{N_0}\right)=-\lambda t\]

\[N=N_0e^{-\lambda t}\]

This is the Radioactive Decay Law.

The time required for the number of radioactive nuclei to reduce to half of its initial value is called the half-life T_1/2​.

\[\frac{N_0}{2}=N_0e^{-\lambda T_{1/2}}\]

\[T_{1/2}=\frac{\ln2}{\lambda}=\frac{0.693}{\lambda}\]

The average time for which a radioactive nucleus exists before decay is called the mean (average) life τ.

τ = \[\frac {1}{λ}\]

• Nuclear reactions release much more energy (in MeV) than chemical reactions (in eV).
• For the same amount of fuel, nuclear energy is nearly a million times greater than chemical energy.
• Nuclear energy is produced by fission (splitting heavy nuclei) and fusion (joining light nuclei).
• The energy comes from the mass defect and the binding energy of the nucleus.

The process in which a heavy nucleus splits into two lighter nuclei with the release of energy is called nuclear fission.

A self-sustaining sequence of fission reactions in which neutrons produced in one fission cause further fissions is called a chain reaction.

A device in which nuclear fission is carried out in a controlled manner to produce energy is called a nuclear reactor.

Q = [m_parent ​− m_products​]c²

Example form used in fission:

Q = [m_U​ − m₁ ​− m₂ ​− (neutron masses)]c²

The process in which two light nuclei combine to form a heavier nucleus with the release of energy is called nuclear fusion.

Q = [Mass of reactants − Mass of products]c²

Example used in fusion inside the Sun:

Q = [4m_p ​− m_α​ − 2m_e​]c²