JEE Mains Physics · Formula sheet

Nuclei formulas

8 formulas, 2 reference tables and 30 common traps for JEE Mains Physics Nuclei, grouped by subtopic.

Nuclear Size, Mass Defect and Binding Energy

Nuclear radius and constant density

Nuclear radius

R = R_0A^{1/3}, \qquad \rho = \frac{3m}{4\pi R_0^{3}}\ \text{(same for all A)}

$R_0$ a constant, about 1.2 fm
$A$ mass number (protons + neutrons)
$m$ mass of one nucleon

Mass defect and binding energy

Binding energy

BE = \left[Zm_p + (A - Z)m_n - M\right]c^{2}, \qquad 1\ \text{u}\,c^{2} = 931.5\ \text{MeV}

The binding-energy curve and the nuclear force

Part of the curve	Mass number	BE per nucleon	What it means
Lightest nuclei	Below about 20	Low and uneven: about 1.1 MeV for $^{2}\text{H}$ , a spike near 7 MeV for $^{4}\text{He}$	Fusing two light nuclei raises BE per nucleon and releases energy.
Flat middle	About 30 to 170	Nearly constant, about 8 MeV	The force is short ranged and saturates: each nucleon binds only to its neighbours. Flat because the force is SHORT ranged. A reason that says long range is false.
Peak	Near 56 (iron)	Highest, about 8.8 MeV	The most tightly bound nuclei; neither fission nor fusion releases energy from them.
Heavy nuclei	Above about 170	Falls slowly, to about 7.6 MeV for uranium	Coulomb repulsion grows; splitting into two middle nuclei releases energy.

Energy is released whenever the products sit higher on this curve than what you started with.

Common traps

Any statement that orders nuclear densities is false

Bismuth is not denser than lithium. The A in the mass cancels the A in the volume, so a statement ranking nuclei by density, or saying density grows with A, is wrong. The reason

R \propto A^{1/3}

is true; the claim built on it is not.

Cube the radius ratio, do not cube-root it twice

Halving the radius divides A by 8, not by 2. Go from radii to mass numbers by cubing, and from mass numbers to radii by taking the cube root.

Absorbed electrons do not change A

If a nucleus captures protons, neutrons and electrons, only the protons and neutrons raise A. Count the nucleons, then apply

A^{1/3}

A^{2/3}

Atom mass or nucleus mass

Tables give atomic masses, which include the electrons. Pair an atomic mass with

m(^{1}\text{H})

for the protons so the electrons cancel. Mixing an atomic mass with bare proton masses leaves Z electron masses in the defect.

Bound mass is the smaller one

The defect is (free nucleons) − (nucleus). Written the other way round, the binding energy comes out negative.

Per nucleon or in total

Read whether the question wants BE or BE/A. Stability is compared with BE per nucleon; the energy to break the whole nucleus is the total.

Heavier is not always more tightly bound

BE per nucleon rises only up to iron. Beyond A ≈ 56 it falls. A statement that says it grows with mass for all nuclei is false.

Isobars share A, isotopes share Z

Iso-BAR: same mass number (the bar on a balance weighs mass). Iso-TOPE: same place in the periodic table, so same Z. Isotones share N.

Stability is judged per nucleon

A bigger nucleus has a bigger total binding energy almost always. Compare BE per nucleon to say which nucleus is more stable.

Q-value, Fission and Fusion

Learn this subtopic in the notes

Q from binding energies per nucleon

Q-value from binding energy

Q = \sum_{\text{products}} A\,b - \sum_{\text{reactants}} A\,b

$b$ binding energy per nucleon of that nucleus

Q from masses, and how it is shared

Q-value from masses; alpha's share

Q = \left(\sum m_i - \sum m_f\right)c^{2}, \qquad K_\alpha = Q\,\frac{A - 4}{A}

Energy from a sample, and power

Energy from a sample

E = \frac{m}{M}N_A\,Q, \qquad \text{rate} = \frac{P}{Q}

Common traps

Multiply by A before subtracting

BE per nucleon values cannot be subtracted directly: 8.4 − 7.6 = 0.8 MeV is per nucleon, not the answer. Multiply each by its own A first, then subtract totals.

Products minus reactants, for binding energy

With binding energies, Q = after − before. With masses it is the other way round, before − after. Mixing the two flips the sign.

Count every product nucleus

Two deuterons make one helium: the reactants hold 2 × 2 = 4 nucleons. Forgetting the second deuteron halves the reactant side.

The alpha does not get all of Q

Momentum is shared equally and oppositely, so kinetic energy splits inversely with mass. The alpha gets

Q(A - 4)/A

. Giving it all of Q ignores the recoil.

Before minus after, for masses

Mass lost is released energy, so Q = (reactant masses) − (product masses). A negative answer means the reaction needs energy.

Keep the electrons balanced

Atomic masses carry electrons. In alpha decay the parent atom's electrons equal the daughter's plus helium's, so atomic masses work. In beta-plus decay they do not balance; take care there.

Grams over grams per mole

Moles = mass ÷ molar mass with both in grams. Using kilograms for one and grams for the other is off by a thousand.

MeV is not joules

Power is in watts, joules per second. Turn Q into joules (× 1.6 × 10⁻¹³) before dividing a power by it.

Several nuclei per reaction

If three helium nuclei make one carbon, the number of reactions is a third of the number of helium nuclei.

Decay Modes and Half-Lives

Learn this subtopic in the notes

Counting half-lives

Half-life law

N = N_0\left(\tfrac{1}{2}\right)^{t/T_{1/2}}

Alpha, beta and gamma decay: what changes

Decay	What leaves the nucleus	Change in A	Change in Z	Example
Alpha ( $\alpha$ )	A helium nucleus, $^{4}_{2}\text{He}$	Falls by 4	Falls by 2	$^{226}_{88}\text{Ra} \rightarrow {}^{222}_{86}\text{Rn} + \alpha$
Beta-minus ( $\beta^-$ )	An electron and an antineutrino	No change	Rises by 1	$^{14}_{6}\text{C} \rightarrow {}^{14}_{7}\text{N} + e^- + \bar{\nu}$ A neutron becomes a proton and the partner is an ANTI-neutrino.
Beta-plus ( $\beta^+$ )	A positron and a neutrino	No change	Falls by 1	$^{22}_{11}\text{Na} \rightarrow {}^{22}_{10}\text{Ne} + e^+ + \nu$ Happens only inside a nucleus: a free proton is lighter than a neutron.
Gamma ( $\gamma$ )	A photon, as an excited nucleus drops to a lower level	No change	No change	$^{60}\text{Ni}^{*} \rightarrow {}^{60}\text{Ni} + \gamma$

Only alpha decay changes A, so count alphas from A and then betas from Z.

Common traps

Count the alphas first

Betas do not change A, so A alone fixes the number of alphas. Starting from Z mixes two unknowns.

Beta-minus raises Z

It feels like a minus should lower Z, but losing a negative electron leaves the nucleus one charge more positive. Beta-minus: Z + 1. Beta-plus: Z − 1.

Neutrino or antineutrino

Beta-minus emits an electron and an antineutrino; beta-plus emits a positron and a neutrino. The lepton number must balance.

Decayed is not left

If 15/16 has decayed, 1/16 is left, so 4 half-lives have passed. Taking the decayed fraction as the fraction left gives a time far too short.

Halve, do not subtract halves

After two half-lives a quarter is left, not zero. Each half-life halves what remains; it does not remove half of the original.

Find n from a ratio, then the time

When a count or activity falls by a power of 2, write the ratio as

2^{n}

first. Then time = n × half-life, or half-life = time ÷ n.

Decay Constant, Activity and Two-Isotope Decay

Learn this subtopic in the notes

Decay constant, mean life and activity

Decay law

N = N_0e^{-\lambda t}, \qquad T_{1/2} = \frac{\ln 2}{\lambda}, \qquad \tau = \frac{1}{\lambda}, \qquad A = \lambda N

Two isotopes, two routes and a decay chain

Two routes add their decay constants

\lambda = \lambda_1 + \lambda_2, \qquad T = \frac{T_1T_2}{T_1 + T_2}

Common traps

λ must be in per second for becquerel

A becquerel is one decay per second. If the half-life is in days, turn it into seconds before finding λ, or the activity is off by 86 400.

Mean life is longer than half-life

\tau = T_{1/2}/0.693

, so the mean life is about 1.44 half-lives. Multiplying by 0.693 instead gives a value that is too small.

Decay ignores conditions

Heating, pressure or a chemical reaction does not change λ. A statement saying it does is false.

Add decay constants, not half-lives

Two routes make decay faster, so the effective half-life is SHORTER than either. Adding the half-lives gives a longer one, which is always wrong.

Equal masses are not equal numbers

If two samples have the same mass but different molar masses, the lighter nuclide has more nuclei. Convert to numbers before applying the decay law.

Read where B starts

If no B is present at first, its curve starts at zero, rises and falls. If some B is present, the curve starts above zero; whether it rises first depends on how much A feeds it.

Nuclei formulas

Nuclear Size, Mass Defect and Binding Energy

Nuclear radius and constant density

Mass defect and binding energy

The binding-energy curve and the nuclear force

Common traps

Q-value, Fission and Fusion

Q from binding energies per nucleon

Q from masses, and how it is shared

Energy from a sample, and power

Common traps

Decay Modes and Half-Lives

Counting half-lives

Alpha, beta and gamma decay: what changes

Common traps

Decay Constant, Activity and Two-Isotope Decay

Decay constant, mean life and activity

Two isotopes, two routes and a decay chain

Common traps

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