(a) Differentiate between half-life and average life of a radioactive substance. (b) A radioactive substance d

Question

Q. (a) Differentiate between half-life and average life of a radioactive substance.

(b) A radioactive substance decays for an interval of time equal to its mean life. Find the fraction of the amount of the substance which is left undecayed after this time interval.

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Concept understanding — Decay Constant

Decay Constant — The First Meeting

Imagine you have a room full of 1000 identical, unstable nuclei. Each one is like a tiny time bomb, but with no clock — it could pop in the next second, or a million seconds from now. You cannot predict which one will go next, but you can measure how likely any single nucleus is to decay in a given interval of time.

That likelihood is the decay constant, denoted by the Greek letter $\lambda$ (lambda).

The Intuition

If $\lambda = 0.1\ \text{s}^{-1}$ , it means: each nucleus has a 10% chance of decaying in the next second. Not that 10% of the nuclei will decay in exactly one second — but that the probability per unit time is 0.1.

This is a rate, not a count. It stays constant for a given isotope, no matter how many nuclei are left. Whether you have a billion nuclei or just ten, each one still "rolls the dice" with the same probability per second.

Note

The decay constant does not depend on how many nuclei are present. It is an intrinsic property of the isotope — like its fingerprint.

The Precise Statement

For a sample containing $N$ radioactive nuclei at time $t$ , the rate at which they decay (the activity) is:

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

The minus sign means $N$ is decreasing. The equation says: the number of decays per second is proportional to the number of nuclei present, and $\lambda$ is the constant of proportionality.

From this, we get the exponential decay law:

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

where $N_0$ is the initial number of nuclei.

Connecting to Half-Life and Mean Life

The decay constant is the most fundamental of the three related quantities. The others are derived from it.

Half-life $T_{1/2}$ is the time after which half the nuclei remain. Set $N = N_0/2$ :

\frac{N_0}{2} = N_0 e^{-\lambda T_{1/2}} \quad\Rightarrow\quad \frac{1}{2} = e^{-\lambda T_{1/2}}

Take natural logs:

\ln\left(\frac{1}{2}\right) = -\lambda T_{1/2} \quad\Rightarrow\quad -\ln 2 = -\lambda T_{1/2}

So:

\boxed{\lambda = \frac{\ln 2}{T_{1/2}} = \frac{0.693}{T_{1/2}}}

Mean life $\tau$ is the average lifetime of a nucleus. It turns out to be:

\tau = \frac{1}{\lambda}

Important

Three forms of the same idea:

$\lambda = \frac{0.693}{T_{1/2}} = \frac{1}{\tau}$

If you know any one of $\lambda$ , $T_{1/2}$ , or $\tau$ , you know all three.

A Concrete Example

Carbon-14 has a half-life of about 5730 years. Its decay constant is:

\lambda = \frac{0.693}{5730\ \text{years}} \approx 1.21 \times 10^{-4}\ \text{year}^{-1}

That is a very small number — each C-14 nucleus has only a 0.012% chance of decaying in a year. That is why carbon dating works: the decay is slow enough to measure over thousands of years, but fast enough to be detectable.

Common Misconception …

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Half-life is the time for half the nuclei to decay; average (mean) life is the average lifespan of a all nuclei. After one mean life, the fraction remaining is $1/e$ , about 37%.

Why This Matters: The Heart of Radioactive Decay

Radioactive decay is a random, probabilistic process — you can never predict which nucleus will decay next, only the average behaviour of a huge population. This leads to two natural time scales: the half-life (how long until half are gone) and the average life (the mean time a nucleus survives before decaying). They are related by a simple factor, and understanding the difference is a classic exam trap.

(a) Half-life vs Average Life

Property	Half-life ( $T_{1/2}$ )	Average/Mean life ( $\tau$ )
Definition	Time for the number of undecayed nuclei to reduce to half its initial value.	The arithmetic mean of the lifetimes of all individual nuclei.
Formula	$T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}$	$\tau = \frac{1}{\lambda}$
Relation	$\tau = \frac{T_{1/2}}{\ln 2} \approx 1.44 \, T_{1/2}$	$T_{1/2} = \tau \ln 2 \approx 0.693 \, \tau$
Physical meaning	After one half-life, exactly 50% remains.	After one mean life, the fraction remaining is $1/e \approx 37\%$ .

Watch out

A common mistake is to think that after one mean life, all nuclei have decayed. That’s false — decay is exponential, so a fraction always remains. The mean life is the average survival time, not the time for complete decay.

(b) Fraction Remaining After One Mean Life

We start from the exponential decay law:

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

where $\lambda$ is the decay constant and $N_0$ is the initial number of nuclei.

Step 1: Identify the time interval.

The problem says “interval equal to its mean life”. The mean life $\tau$ is given by $\tau = 1/\lambda$ .

Step 2: Substitute $t = \tau$ into the decay law.

N(\tau) = N_0 e^{-\lambda \cdot \tau} = N_0 e^{-\lambda \cdot (1/\lambda)} = N_0 e^{-1}

Step 3: Write the fraction remaining.

$\frac{N(\tau)}{N_0} = e^{-1} = \frac{1}{e}$ …

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

CBSE 2024Set A1 markMCQ
$Q. S.I. unit of decay constant is (A) metre (B) hertz (C) per metre (D) metre^2$
›Reveal solutionSolution

The SI unit of the decay constant is per second (s⁻¹) = hertz → option (B).

The decay law dN/dt = −λN shows that λ has dimensions of (rate)/(number) = per unit time. Its SI unit is therefore per second (s⁻¹), which is the same as the hertz. Among the given choices, hertz is th …

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CBSE 2021Set A1 markMCQ
$Q. Which of the following is correct for radioactive element? (A) T_a = λ/0.6931 (B) T_a = 1/λ (C) T_a = (0.6931)λ (D) T_a = 1/λ²$
›Reveal solutionSolution

Mean (average) life T_a = 1/λ.

Radioactive decay follows $N = N_0 e^{-\lambda t}$ , where $\lambda$ is the decay constant. The average (mean) life is the average time an atom survives before decaying, obtained by weighting each lifetime by the number of atoms decaying then:

$T_a = \frac{\int_0^\infty t\,\lambda N_0 e^{-\lambda t}\,dt}{N_0} = \frac{1}{\lambda}.$

…

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CBSE 2019Set ANNUAL1 mark
$Q. Write the relation between average life and decay constant of a radioactive atom.$
›Reveal solutionSolution

Average life is the reciprocal of the decay constant: $\tau = 1/\lambda$ .

For a sample undergoing radioactive decay, the number of undecayed nuclei falls as

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

where $\lambda$ is the decay constant (probability of decay per unit time per nucleus).

The average (mean) life $\tau$ is the average time for which a nucleus exists before decaying, obtained by averaging $t$ over the decay distribution:

$\tau = \dfrac{\displaystyle\int_0^\infty t\,N_0\lambda e^{-\lambda t}\,dt}{\displaystyle\int_0^\infty N_0\lambda e^{-\lambda t}\,dt} = \dfrac{1}{\lambda}$

…

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Q.(a) Differentiate between half-life and average life of a radioactive substance.

Why This Matters: The Heart of Radioactive Decay

(a) Half-life vs Average Life

(b) Fraction Remaining After One Mean Life

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