Revision: Atoms, Molecules and Nuclei Physics HSC Science (General) 12th Standard Board Exam Maharashtra State Board

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.

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

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.

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.

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.

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.

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

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 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 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.

• 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.

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):

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

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

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

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

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

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

\[E=mc^2\]

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

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

BE per nucleon ​= \[\frac {E.E.}{A}\]

E_b = ΔM ⋅ c²

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

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

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

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.

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

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

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

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

• 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 \[-13.6\frac{Z^2}{n^2}\mathrm{~eV}.\] .

• 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.

• 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 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.

• 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.

• de Broglie (1924) Hypothesis: If radiation (waves) shows particle behaviour, then particles of matter should also show wave behaviour. This concept is called Matter Waves or de Broglie Waves.
• Nature's symmetry: electrons, protons, and neutrons can behave as waves under suitable conditions.

de Broglie Wave Equation:

For a particle of mass m moving with velocity v: \[\lambda=\frac{h}{p}=\frac{h}{mv}\]

Also written as: \[\lambda=\frac{h}{\sqrt{2mK.E}}\]

• Larger mass or velocity → smaller wavelength → harder to detect wave nature.
• For large (macroscopic) bodies, wavelengths are so tiny they cannot be measured — hence, no observable wave nature.

Experimental Proof of Matter Waves:

• Davisson and Germer experiment: Electrons showed diffraction patterns — direct proof of wave nature.
• G.P. Thomson's experiment also confirmed electron diffraction.
• Electrons have mass and move with definite velocity → can display wave-like behaviour.

Acceptance of Duality:

• Bohr's Law of Complementarity: Matter can be observed as either a particle or a wave, but not both simultaneously.
• Particle and wave aspects are complementary.

1.  The experiment verified the de-Broglie hypothesis.
2. In this experiment, the wave nature of electron particles was studied with the help of a nickel crystal.
3. Electrons undergo interference and diffraction phenomena and produce alternate bright and dark rings.
4. When accelerating potential V = 54 V:
   λ = 0.165 nm (Experimental value)
   λ = 0.167 nm (Theoretical value from de-Broglie hypothesis)