A radioactive sample contains $N_0 = 3.2\times10^{18}$ atoms of an isotope with half-life $T_{1/2} = 8$ days. Find (a) the initial activity of the sample, and (b) its activity after $16$ days.

First find the decay constant in per-second units. With $T_{1/2}=8$ days $=8\times86400=691200\ \text{s}$: $$\lambda = \frac{0.693}{691200} \approx 1.003\times10^{-6}\ \text{s}^{-1}$$ (a) The initial activity is $\mathcal{A}_0=\lambda N_0$: $$\mathcal{A}_0 = (1.003\times10^{-6})\times(3.2\times10^{18}) \approx 3.21\times10^{12}\ \text{Bq}$$ (b) Since $16$ days is exactly TWO half-lives ($16/8=2$), the activity after $16$ days is $\mathcal{A}_0$ halved twice: $$\mathcal{A} = \frac{\mathcal{A}_0}{4} = \frac{3.21\times10^{12}}{4} \approx 8.0\times10^{11}\ \text{Bq}$$ > [!ANSWER] > $\mathcal{A}_0 \approx 3.21\times10^{12}\ \text{Bq}$; $\mathcal{A}$ after $16$ days $\approx8.0\times10^{11}\ \text{Bq}$.

A radioactive sample contains N₀ = 3.2×10¹⁸ atoms of an isotope with half-life T1/2 = 8 days. Find (a) the ini

Numerical · Q21

N_0 = 3.2\times10^{18}

(a) the initial activity of the sample, and

(b) its activity after

16

days.

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Concept understanding — Activity of a Radioactive Sample

Activity of a Radioactive Sample

Imagine you have a jar of fireflies. Each firefly glows at a random moment, and once it glows, it stops forever. You don't care about any single firefly — you care about how many flashes you see per second. That rate of flashing is the activity of your jar.

A radioactive sample is exactly that jar. Inside are unstable nuclei. Each nucleus will "flash" (decay) at some unpredictable instant, and after that it becomes a stable atom. The activity $A$ is simply the number of decays happening each second.

The precise definition

If you have $N$ unstable nuclei at a given moment, and each has a constant probability $\lambda$ of decaying per second (the decay constant), then the number of decays per second is:

A = \lambda N

$\lambda$ is a fixed property of the isotope — it tells you how "eager" each nucleus is to decay. A large $\lambda$ means each nucleus is very likely to decay soon, so the activity is high even with few nuclei.

A = \lambda N

Why activity itself decays

As nuclei decay, $N$ drops. Since $A$ is proportional to $N$ , the activity must drop too. The mathematics is straightforward: the rate at which $N$ decreases is exactly the activity (each decay removes one nucleus), so

\frac{dN}{dt} = -A = -\lambda N

This differential equation gives the familiar exponential decay:

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

Multiply both sides by $\lambda$ and you get the activity:

A(t) = \lambda N_0 e^{-\lambda t} = A_0 e^{-\lambda t}

The activity itself decays exponentially with the same decay constant $\lambda$ .

Watch out

A common mistake is to think activity is constant. It is not — it falls off exactly like the number of undecayed nuclei, because every decay reduces the remaining stock.

Units

The SI unit is the becquerel (Bq): 1 Bq = 1 decay per second. That is a tiny number. Historically, the curie (Ci) was used: 1 Ci = $3.7 \times 10^{10}$ Bq, roughly the activity of one gram of radium-226.

The key insight

Activity is not a property of a single nucleus — it is a bulk observable. You cannot say "this nucleus has activity 5 Bq". Activity only makes sense for a collection. It is the macroscopic consequence of a microscopic probability, and it is the quantity you actually measure with a Geiger counter or scintillator.

When you hear "the source has an activity of 100 Bq", it means: right now, 100 nuclei in that sample are disintegrating every second. And if you wait one half-life, that number will be 50 Bq.

Activity of a radioactive sample is a definition-and-formula CBSE Class 12 Physics NCERT topic, frequently searched as activity formula radioactive decay class 12 or becquerel and curie unit difference. Because activity decays with the same decay constant as the parent nuclei, this concept is a common numerical topic in board exams and JEE Main/NEET physics.

First find the decay constant in per-second units. With $T_{1/2}=8$ days $=8\times86400=691200\ \text{s}$ :

\lambda = \frac{0.693}{691200} \approx 1.003\times10^{-6}\ \text{s}^{-1}

The initial activity is $\mathcal{A}_0=\lambda N_0$ : $\mathcal{A}_0 = (1.003\times10^{-6})\times(3.2\times10^{18}) \approx 3.21\times10^{12}\ \text{Bq}$
Since $16$ days is exactly TWO half-lives ( $16/8=2$ ), the activity after $16$ days is $\mathcal{A}_0$ halved twice: $\mathcal{A} = \frac{\mathcal{A}_0}{4} = \frac{3.21\times10^{12}}{4} \approx 8.0\times10^{11}\ \text{Bq}$
✓Final answer
$\mathcal{A}_0 \approx 3.21\times10^{12}\ \text{Bq}$

Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2023Set MODEL1 markMCQ
$Q. The unit of Activity of a radioactive material is:$
(a) Becquerel
(b) Rutherford
(c) Curie
(d) All of these
›Reveal solutionSolution

Becquerel, Rutherford and Curie have all historically been used as units of radioactivity (activity).

The activity of a radioactive sample (decays per unit time) can be expressed in: Becquerel (SI unit, 1 decay/s), Rutherford (1 Rd $=10^6$ decays/s), and Curie (1 Ci $=3.7\times10^{10}$ decays/s). All three are valid units of activity.

✓Final answer
$(d) All of these$
CBSE 2023Set F1 markMCQ
$Q. The unit of radioactivity is (A) MeV (B) curie (C) a.m.u. (D) joule$
›Reveal solutionSolution

Radioactivity (activity) is measured in curie (or becquerel).

Activity is the number of nuclear disintegrations per second. Its SI unit is the becquerel (Bq = 1 decay s⁻¹); the traditional practical unit is the curie:

$1\ \text{Ci} = 3.7\times10^{10}\ \text{decays s}^{-1}.$

MeV and a.m.u. are units of energy/mass, and joule is energy — none measures the rate of decay. The unit of radioactivity is the curie.

✓Final answer
$(B) curie.$
CBSE 2020Set ANNUAL1 mark
$Q. What is SI unit of activity of radioactive substance?$
›Reveal solutionSolution

Activity (decay rate) of a radioactive sample has SI unit becquerel (Bq) = 1 disintegration per second.

Activity $A = -\dfrac{dN}{dt} = \lambda N$ , giving the number of nuclei decaying per unit time. Its SI unit is the becquerel (1 Bq = 1 decay/second). An older, non-SI unit still commonly used is the curie (1 Ci = 3.7×10¹⁰ Bq).

✓Final answer
$The SI unit of activity of a radioactive substance is the becquerel (Bq).$
CBSE 2019Set ANNUAL1 mark
$Q. What is SI unit of activity of Radioactive Substance?$
›Reveal solutionSolution

Activity (rate of radioactive decay) is measured in becquerel, $1\ \text{Bq} = 1$ decay per second.

The activity $A = -\dfrac{dN}{dt} = \lambda N$ of a radioactive sample is the number of nuclei decaying per second. Its SI unit is the becquerel (Bq), where $1\ \text{Bq} = 1$ disintegration/second. (An older, non-SI unit still used is the curie, $1\ \text{Ci} = 3.7\times10^{10}\ \text{Bq}$ .)

✓Final answer
$Becquerel (Bq).$

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Q.A radioactive sample contains N0=3.2×1018N_0 = 3.2\times10^{18}N0​=3.2×1018 atoms of an isotope with half-life T1/2=8T_{1/2} = 8T1/2​=8 days. Find

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Q.A radioactive sample contains $N_0 = 3.2\times10^{18}$ atoms of an isotope with half-life $T_{1/2} = 8$ days. Find

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