JEE Mains Physics · Nuclei

Decay Constant, Activity and Two-Isotope Decay

Each nucleus decays with a fixed chance λ per second, so N = N₀e^(−λt), the half-life is ln 2/λ, the mean life is 1/λ and the activity is λN; two decay routes add their λ's.

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Why this matters

Twenty PYQs, four of them numerical. Ten are from 2021, only one is later than 2023 (January 2025), and none is from 2026, which fits radioactivity leaving the syllabus in 2023-24. Nine use λ directly: activity from a mass, λ from two count rates, mean life against half-life. Eleven compare two decays: two isotopes side by side, one nucleus with two routes, or a chain A → B → C.

Concept 1 of 2: Decay constant, mean life and activity

Every nucleus has the same small chance λ of decaying in each second, whatever its age and whatever the temperature or chemistry. So the number decaying per second, the activity, is λ times the number present, and the sample shrinks exponentially. The half-life and the mean life are two ways of writing 1/λ.

Definition

$N = N_0e^{-\lambda t}$ , activity $A = -\dfrac{dN}{dt} = \lambda N$ , and $A = A_0e^{-\lambda t}$ .
Half-life $T_{1/2} = \dfrac{\ln 2}{\lambda} \approx \dfrac{0.693}{\lambda}$ ; mean life $\tau = \dfrac{1}{\lambda} = \dfrac{T_{1/2}}{\ln 2} \approx 1.44\,T_{1/2}$ . The mean life is longer.
Activity from a mass: $N = \dfrac{m}{M}N_A$ , λ in $\text{s}^{-1}$ , then $A = \lambda N$ in becquerel. $1\ \text{Ci} = 3.7 \times 10^{10}$ Bq.
λ from two activities: $\lambda = \dfrac{\ln(A_1/A_2)}{t_2 - t_1}$ .
Time between two amounts: $t_2 - t_1 = \dfrac{1}{\lambda}\ln\dfrac{N_1}{N_2}$ .
A graph of $\ln N$ against t is a straight line of slope $-\lambda = -1/\tau$ .
Decay does not depend on temperature, pressure or chemical state.

Decay law

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

Worked example

A nuclide has decay constant

2 \times 10^{-6}\ \text{s}^{-1}

and molar mass 100 g/mol. Find the activity of 5 μg of it, in Bq and in Ci. (Avogadro's number

6.0 \times 10^{23}

)

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 10 April 2023 · Q27Moderate

Example 1 · Nuclei · Decay Constant, Activity and Two-Isotope Decay

The decay constant for a radioactive nuclide is

1.5 \times10^{- 5}{\text{ }s}^{- 1}

. Atomic mass of the substance is

60\text{ }g

mole

\ ^{- 1},\left( {\text{ }N}_{A}= 6 \times10^{23} \right)

. The activity of

1.0\mu g

of the substance is___

\times10^{10}\text{ }Bq

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.

Drill 8 more on decay constant, mean life and activity

Concept 2 of 2: Two isotopes, two routes and a decay chain

Two isotopes decay on their own clocks, so compare them by writing each one's law and dividing. One nucleus with two ways to decay is like a tank with two drains: the rates add, so λ adds and the effective half-life is shorter than either. In a chain A → B → C, B is fed by A and drained by its own decay, so it rises, peaks and falls.

Definition

Two isotopes: $\dfrac{N_1}{N_2} = \dfrac{N_1(0)}{N_2(0)}e^{-(\lambda_1 - \lambda_2)t}$ . In half-lives: $\dfrac{N_1}{N_2} = \dfrac{N_1(0)}{N_2(0)}\,2^{-(n_1 - n_2)}$ .
Given masses, change to numbers first: $N = \dfrac{m}{M}N_A$ .
Two routes (parallel decay): $\lambda = \lambda_1 + \lambda_2$ , so $\dfrac{1}{T} = \dfrac{1}{T_1} + \dfrac{1}{T_2}$ and $T = \dfrac{T_1T_2}{T_1 + T_2}$ .
Chain A → B → C: $\dfrac{dN_B}{dt} = \lambda_AN_A - \lambda_BN_B$ . $N_B$ grows while A feeds it faster than it decays, peaks when $\lambda_AN_A = \lambda_BN_B$ , then falls to zero.
Chain with equal λ: $N_B = \left[N_B(0) + \lambda N_A(0)t\right]e^{-\lambda t}$ . Starting with no B, $N_B$ starts at zero.

Two routes add their decay constants

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

Worked example

A nucleus can decay by two routes with half-lives 6 h and 12 h. Find its effective half-life and the fraction left after 8 h.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 26 June 2022 · Q108Moderate

Example 2 · Nuclei · Decay Constant, Activity and Two-Isotope Decay

A radioactive nucleus can decay by two different processes. Half-life for the first process is 3.0 hours while it is 4.5 hours for the second process. The effective half- life of the nucleus will be:

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.

Drill 10 more on two isotopes, two routes and a decay chain

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

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

Watch out for (6)

λ must be in per second for becquerel
→ Decay constant, mean life and activity
Mean life is longer than half-life
→ Decay constant, mean life and activity
Decay ignores conditions
→ Decay constant, mean life and activity
Add decay constants, not half-lives
→ Two isotopes, two routes and a decay chain
Equal masses are not equal numbers
→ Two isotopes, two routes and a decay chain
Read where B starts
→ Two isotopes, two routes and a decay chain

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