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Atoms and Nuclei - Radioactivity

Grade 12ICSEPhysics

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Radioactivity is a spontaneous nuclear phenomenon in which an unstable nucleus decays by emitting radiations such as α\alpha-particles, β\beta-particles, and γ\gamma-rays to attain stability.

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The Law of Radioactive Decay states that the number of nuclei disintegrating per unit time is directly proportional to the total number of nuclei present at that instant: −dNdt∝N-\frac{dN}{dt} \propto N.

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An α\alpha-decay reduces the atomic number ZZ by 22 and the mass number AA by 44: ZAX→Z−2A−4Y+24He^{A}_{Z}X \rightarrow ^{A-4}_{Z-2}Y + ^{4}_{2}He.

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In β−\beta^{-}-decay, a neutron converts into a proton, emitting an electron and an antineutrino: ZAX→Z+1AY+e−+νˉ^{A}_{Z}X \rightarrow ^{A}_{Z+1}Y + e^{-} + \bar{\nu}. The mass number remains unchanged.

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γ\gamma-emission occurs when a nucleus in an excited state transitions to a lower energy state, emitting high-energy photons without changing AA or ZZ.

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Half-life (T1/2T_{1/2}) is the time required for the number of radioactive nuclei to reduce to half of its initial value.

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Mean life (τ\tau) is the average lifetime of all the nuclei in a radioactive sample, given by the reciprocal of the decay constant λ\lambda.

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Mass Defect (Δm\Delta m) is the difference between the sum of the masses of the nucleons and the actual mass of the nucleus: Δm=[Zmp+(A−Z)mn]−Mnucleus\Delta m = [Z m_p + (A-Z)m_n] - M_{nucleus}.

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Binding Energy (BEBE) is the energy required to break a nucleus into its constituent nucleons, calculated as BE=Δm⋅c2BE = \Delta m \cdot c^2.

📐Formulae

N=N0e−λtN = N_0 e^{-\lambda t}

R=∣dNdt∣=λNR = \left| \frac{dN}{dt} \right| = \lambda N

T1/2=ln⁡(2)λ≈0.693λT_{1/2} = \frac{\ln(2)}{\lambda} \approx \frac{0.693}{\lambda}

τ=1λ=1.44T1/2\tau = \frac{1}{\lambda} = 1.44 T_{1/2}

N=N0(12)n where n=tT1/2N = N_0 \left( \frac{1}{2} \right)^n \text{ where } n = \frac{t}{T_{1/2}}

BE=[Zmp+(A−Z)mn−M]×931.5 MeVBE = [Z m_p + (A-Z)m_n - M] \times 931.5 \text{ MeV}

Binding Energy per nucleon=BEA\text{Binding Energy per nucleon} = \frac{BE}{A}

💡Examples

Problem 1:

The half-life of a radioactive substance is 3030 days. Calculate the time taken for 78\frac{7}{8} of the original mass to disintegrate.

Solution:

Nrem=N0−78N0=18N0N_{rem} = N_0 - \frac{7}{8}N_0 = \frac{1}{8}N_0 Using the formula N=N0(12)nN = N_0 \left( \frac{1}{2} \right)^n: 18N0=N0(12)n  ⟹  (12)3=(12)n  ⟹  n=3\frac{1}{8}N_0 = N_0 \left( \frac{1}{2} \right)^n \implies \left( \frac{1}{2} \right)^3 = \left( \frac{1}{2} \right)^n \implies n = 3 Since t=n×T1/2t = n \times T_{1/2}: t=3×30=90 dayst = 3 \times 30 = 90 \text{ days}

Explanation:

To find the time taken, we first determine the remaining fraction of the substance. 78\frac{7}{8} disintegration means 18\frac{1}{8} remains. We find the number of half-lives (nn) required to reach this fraction and multiply by the half-life duration.

Problem 2:

Find the energy equivalent of 1 amu1 \text{ amu} in MeV\text{MeV}.

Solution:

1 amu=1.66×10−27 kg1 \text{ amu} = 1.66 \times 10^{-27} \text{ kg} Using Einstein's equation E=mc2E = mc^2: E=(1.66×10−27)×(3×108)2 JoulesE = (1.66 \times 10^{-27}) \times (3 \times 10^8)^2 \text{ Joules} E≈1.494×10−10 JE \approx 1.494 \times 10^{-10} \text{ J} Converting to eV\text{eV} (1 eV=1.6×10−19 J1 \text{ eV} = 1.6 \times 10^{-19} \text{ J}): E=1.494×10−101.6×10−13 MeV≈931.5 MeVE = \frac{1.494 \times 10^{-10}}{1.6 \times 10^{-13}} \text{ MeV} \approx 931.5 \text{ MeV}

Explanation:

Mass is converted into energy using the mass-energy equivalence principle. One atomic mass unit is multiplied by the square of the speed of light and then converted from Joules to Mega-electron volts.