Radioactive Decay Laws — Half-Life Problems

How Radioactive Decay Actually Works

Imagine a bag of $1000$ unstable atoms. Some decay this second, some decay tomorrow, some decay in a thousand years. We can’t predict which atom decays when — but we can predict, with great accuracy, what fraction decays in any given time. That fraction is governed by a clean exponential law, and the entire chapter rests on it.

Decay is a statistical process. The probability that any single atom decays in a small time $dt$ is $\lambda \, dt$ , where $\lambda$ is the decay constant. Multiply by the number of atoms and you get the number decaying per second. That gives us the differential equation $dN/dt = -\lambda N$ , whose solution is the famous exponential decay law.

This hub walks through the laws, the half-life and mean-life concepts, the activity formulas, the cascade-decay logic, and 8 practice problems graded by difficulty. The same ideas drive carbon dating, medical isotope dosing, and nuclear reactor calculations.

Key Terms & Definitions

Decay constant ( $\lambda$ ): Probability per unit time that a given atom decays. Units: $\text{s}^{-1}$ .

Half-life ( $T_{1/2}$ ): Time for half the atoms to decay. $T_{1/2} = \ln 2/\lambda \approx 0.693/\lambda$ .

Mean life ( $\tau$ ): Average lifetime of an atom. $\tau = 1/\lambda$ . Note: $\tau = T_{1/2}/\ln 2$ , so mean life is longer than half-life.

Activity ( $A$ ): Number of decays per second. $A = \lambda N$ . Units: becquerel (1 Bq = 1 decay/s) or curie (1 Ci = $3.7 \times 10^{10}$ Bq).

Daughter nucleus: The product of decay; if it’s also radioactive, the chain continues.

The Decay Law

N(t) = N_0 e^{-\lambda t}

In terms of half-life:

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

Both forms are equivalent; the first is good for calculus problems, the second is faster when $t$ is a clean multiple of $T_{1/2}$ .

Activity Formula

A = \lambda N = A_0 e^{-\lambda t}

So activity decays at the same rate as the number of atoms. If half the atoms remain after one half-life, half the activity remains too.

Half-Life vs Mean-Life Relationships

$T_{1/2} = \dfrac{0.693}{\lambda}, \quad \tau = \dfrac{1}{\lambda}, \quad \tau = \dfrac{T_{1/2}}{0.693} \approx 1.44 \, T_{1/2}$ .

Memorize these — they show up in almost every JEE/NEET radioactivity question.

Worked Example 1 — Counting Atoms

A radioactive sample has $T_{1/2} = 5 \text{ years}$ and currently contains $1.6 \times 10^{20}$ atoms. How many remain after $20 \text{ years}$ ?

$20$ years = $4$ half-lives.

$N = N_0 \times (1/2)^4 = 1.6 \times 10^{20} \times 1/16 = 10^{19}$ atoms.

Worked Example 2 — Activity Decay

A sample has initial activity $A_0 = 8000 \text{ Bq}$ and half-life $T_{1/2} = 10 \text{ minutes}$ . Find the activity after $30 \text{ minutes}$ .

$30$ min = $3$ half-lives. $A = 8000/2^3 = 1000 \text{ Bq}$ .

Worked Example 3 — Finding $\lambda$

A sample’s activity drops from $1000 \text{ Bq}$ to $250 \text{ Bq}$ in $20 \text{ days}$ . Find $\lambda$ and $T_{1/2}$ .

$A/A_0 = 1/4 = (1/2)^2$ , so $20 \text{ days} = 2 T_{1/2}$ , giving $T_{1/2} = 10 \text{ days}$ .

$\lambda = 0.693/10 = 0.0693 \text{ day}^{-1}$ .

Carbon Dating

Living organisms maintain a constant ratio of $\text{C}^{14}$ to $\text{C}^{12}$ through atmospheric exchange. After death, $\text{C}^{14}$ decays with $T_{1/2} = 5730 \text{ years}$ , while $\text{C}^{12}$ stays. Measuring the present ratio tells us how long ago the organism died.

If the present $\text{C}^{14}$ activity is $1/8$ of the living-tissue activity, then $3$ half-lives have passed: age $= 3 \times 5730 = 17{,}190 \text{ years}$ .

Solved Examples

Example A — Easy (CBSE)

The half-life of a radioactive sample is $5 \text{ days}$ . What fraction remains after $15 \text{ days}$ ?

$15 = 3 T_{1/2}$ . Fraction = $(1/2)^3 = 1/8$ .

Example B — Medium (NEET 2023)

A radioactive nucleus has decay constant $\lambda = 0.1 \text{ s}^{-1}$ . Find its mean life and half-life.

$\tau = 1/\lambda = 10 \text{ s}$ . $T_{1/2} = 0.693 \times 10 = 6.93 \text{ s}$ .

Example C — Medium (JEE Main 2024)

Two radioactive samples A and B have decay constants $\lambda$ and $2\lambda$ , both starting with $N_0$ atoms. After what time do their decay rates become equal?

A’s rate: $R_A = \lambda N_0 e^{-\lambda t}$ . B’s rate: $R_B = 2\lambda N_0 e^{-2\lambda t}$ .

Equating: $e^{-\lambda t} = 2 e^{-2\lambda t} \implies e^{\lambda t} = 2 \implies t = \ln 2/\lambda = T_{1/2,A}$ .

So A and B have equal activity exactly at A’s half-life.

Example D — Hard (JEE Advanced)

A sample contains $10^{10}$ atoms of a nucleus with $T_{1/2} = 1 \text{ hour}$ . The daughter is stable. Find the number of daughter atoms produced in $3 \text{ hours}$ .

After $3 \text{ hours}$ , parent atoms remaining: $10^{10}/8 = 1.25 \times 10^9$ .

Daughter atoms produced = $10^{10} - 1.25 \times 10^9 = 8.75 \times 10^9$ .

Cascade Decay (Series)

If A → B → C, and B is also radioactive with constants $\lambda_A$ and $\lambda_B$ :

$dN_A/dt = -\lambda_A N_A$ .

$dN_B/dt = \lambda_A N_A - \lambda_B N_B$ .

In secular equilibrium ( $\lambda_A \ll \lambda_B$ and after long time): $\lambda_A N_A = \lambda_B N_B$ , so activities are equal. This shows up in uranium-radium-radon chains.

Exam-Specific Tips

JEE Main: 1 question per year, usually a half-life numerical or activity ratio. Memorize $T_{1/2} = 0.693/\lambda$ and $\tau = T_{1/2}/0.693$ .

NEET: 1-2 questions per year. Carbon dating, half-life chains, and radio-isotope therapy questions appear regularly. NEET 2024 had a direct $A = A_0 e^{-\lambda t}$ plug-in.

CBSE Boards: Derivation of decay law from $dN/dt = -\lambda N$ is a 3-mark question. Activity numerical for 2 marks.

When $t$ is a clean multiple of $T_{1/2}$ (like 1, 2, 3, 4 half-lives), use $(1/2)^n$ directly — much faster than $e^{-\lambda t}$ . Reach for the exponential only when $t$ is “messy”.

Common Mistakes to Avoid

Mistake 1: Confusing half-life and mean-life. $\tau = T_{1/2}/0.693 \neq T_{1/2}$ .

Mistake 2: Forgetting that activity also decays exponentially with the same $\lambda$ . Activity isn’t constant.

Mistake 3: Adding half-lives in series decay (A → B → C). The math is more involved — use rate equations.

Mistake 4: Using $\lambda$ in units of $\text{year}^{-1}$ when $t$ is in seconds. Always check unit consistency.

Mistake 5: Treating $\text{C}^{14}/\text{C}^{12}$ ratio as the activity directly. The ratio decays at the $\text{C}^{14}$ rate; the absolute activity also depends on sample mass.

Practice Questions

Q1. A sample has $T_{1/2} = 8 \text{ days}$ . What fraction remains after $24 \text{ days}$ ?

$24/8 = 3$ half-lives. Fraction = $1/8$ .

Q2. Decay constant $\lambda = 0.05 \text{ s}^{-1}$ . Half-life?

$T_{1/2} = 0.693/0.05 = 13.86 \text{ s}$ .

Q3. Initial activity $5000 \text{ Bq}$ , $T_{1/2} = 2 \text{ hours}$ . Activity after $6 \text{ hours}$ ?

$3$ half-lives. $A = 5000/8 = 625 \text{ Bq}$ .

Q4. Sample has $10^{20}$ atoms, $\lambda = 10^{-3} \text{ s}^{-1}$ . Initial activity?

$A_0 = \lambda N_0 = 10^{-3} \times 10^{20} = 10^{17} \text{ Bq}$ .

Q5. A bone has $\text{C}^{14}$ activity $1/16$ of fresh sample. Age?

$1/16 = (1/2)^4$ , so $4$ half-lives. Age = $4 \times 5730 = 22{,}920$ years.

Q6. Mean life $\tau = 100 \text{ s}$ . Half-life?

$T_{1/2} = 0.693 \tau = 69.3 \text{ s}$ .

Q7. $90\%$ of a sample decays. How many half-lives?

$N/N_0 = 0.1 = (1/2)^n$ . $n = \log_2(10) \approx 3.32$ half-lives.

Q8. A nuclide has $T_{1/2} = 1 \text{ year}$ . After how long is $99\%$ decayed?

$N/N_0 = 0.01 = (1/2)^n$ , $n = \log_2 100 \approx 6.64$ . Time $\approx 6.64$ years.

FAQs

Q: Why is the decay law exponential?

Because each atom has a constant decay probability per unit time, independent of age. Probability times population gives $dN/dt = -\lambda N$ , which integrates to $N = N_0 e^{-\lambda t}$ .

Q: Does temperature or pressure affect half-life?

For most nuclei, no. Nuclear decay is a quantum-mechanical process unaffected by atomic-scale conditions. (Tiny exceptions exist for electron-capture decays under extreme ionization, but these are research-level edge cases.)

Q: What’s the difference between alpha, beta, and gamma decay?

Alpha: emits $\text{He}^4$ nucleus, $A$ drops by 4, $Z$ drops by 2. Beta: a neutron becomes a proton (or vice versa), $A$ unchanged, $Z$ ±1. Gamma: nucleus releases excess energy as a photon, $A$ and $Z$ unchanged.

Q: Can we ever predict when a specific atom will decay?

No. Quantum mechanics says only the probability is knowable. We can predict ensemble behaviour with enormous precision but never individual events.

Q: What is “secular equilibrium”?

When the parent’s half-life is much longer than the daughter’s, the daughter’s amount adjusts so that its activity equals the parent’s. This is how radon stays in equilibrium with radium in old uranium ores.

Q: How is half-life used in medicine?

Therapeutic isotopes (like I-131) are dosed based on activity, which decays predictably. Doctors calculate how much to administer so the activity drops to safe levels by a known time after treatment.