22.4 Half-Life and Rate of Decay | Nuclear Physics | Physics 12

Activity and Decay Constant

Radioactive decay is a spontaneous, random nuclear process. The rate of decay (also called activity, $A$ ) of a sample is the number of disintegrations occurring per second.

Decay Constant ( $λ$ )

The decay constant $λ$ is the probability that a given nucleus will decay per unit time. It represents the fraction of the total number of atoms that decay per unit time.

$\frac{d N}{d t} = - λ N$

where $N$ is the number of undecayed nuclei at time $t$ . The negative sign indicates that $N$ is decreasing.

SI unit of $λ$ : $s^{- 1}$ (per second)
A large $λ$ means rapid decay (highly unstable nucleus).
A small $λ$ means slow decay (more stable nucleus).

Activity ( $A$ )

$A = λ N$

SI unit of Activity: Becquerel (Bq), where $1 Bq = 1$ disintegration per second.
Another common unit is the Curie (Ci): $1 Ci = 3.7 \times 1 0^{10} Bq$ .

Since $N$ decreases over time, the activity $A$ also decreases over time.

Exponential Nature of Radioactive Decay

Solving the differential equation $\frac{d N}{d t} = - λ N$ gives the Law of Radioactive Decay:

$N = N_{0} e^{- λ t}$

where:

$N_{0}$ = initial number of undecayed nuclei at $t = 0$
$N$ = number of undecayed nuclei at time $t$
$λ$ = decay constant
$e$ = base of natural logarithm ( $\approx 2.718$ )

The term $e^{- λ t}$ represents the fraction of nuclei remaining undecayed at time $t$ .

Since $A = λ N$ , the activity also decays exponentially:

$A = A_{0} e^{- λ t}$

where $A_{0} = λ N_{0}$ is the initial activity.

Key point: As the number of undecayed nuclei decreases exponentially, the rate of decay (activity) also decreases exponentially.

Half-Life ( $T_{1/2}$ )

The half-life $T_{1/2}$ is the time interval during which half of the unstable nuclei in a radioactive sample undergo decay.

After one half-life, $N = \frac{N _{0}}{2}$ . Substituting into the decay equation:

$\frac{N _{0}}{2} = N_{0} e^{- λ T_{1/2}}$

$\frac{1}{2} = e^{- λ T_{1/2}}$

Taking the natural logarithm of both sides:

$ln (\frac{1}{2}) = - λ T_{1/2}$

$T_{1/2} = \frac{ln 2}{λ} \approx \frac{0.693}{λ}$

Key Properties of Half-Life

Property	Detail
Relationship to $λ$	$T_{1/2} \propto \frac{1}{λ}$ (inversely proportional)
Large $λ$	Short $T_{1/2}$ → highly unstable
Small $λ$	Long $T_{1/2}$ → more stable
Independence	$T_{1/2}$ does not change with time or sample size

Decay After $n$ Half-Lives

After $n$ half-lives, the number of undecayed nuclei is:

$N = N_{0} (\frac{1}{2})^{n}$

where $n = \frac{t}{T _{1/2}}$ .

Worked Example

Problem: A sample of uranium has an initial mass of $400 g$ . How much remains after 3 half-lives?

Solution: $N = 400 \times (\frac{1}{2})^{3} = 400 \times \frac{1}{8} = 50 g$

Problem: A radioactive sample has $T_{1/2} = 10$ minutes. What percentage has decayed after 20 minutes?

Solution: $n = \frac{20}{10} = 2$ half-lives. $Fraction remaining = (\frac{1}{2})^{2} = \frac{1}{4} = 25%$ $Fraction decayed = 100% - 25% = 75%$

Summary of Key Equations

Quantity	Formula
Decay law	$N = N_{0} e^{- λ t}$
Activity	$A = λ N$
Half-life	$T_{1/2} = \frac{0.693}{λ}$
After $n$ half-lives	$N = N_{0} (\frac{1}{2})^{n}$

Activity and Decay Constant

Radioactive decay is a spontaneous, random nuclear process. The rate of decay (also called activity, $A$ ) of a sample is the number of disintegrations occurring per second.

Decay Constant ( $λ$ )

The decay constant $λ$ is the probability that a given nucleus will decay per unit time. It represents the fraction of the total number of atoms that decay per unit time.

$\frac{d N}{d t} = - λ N$

where $N$ is the number of undecayed nuclei at time $t$ . The negative sign indicates that $N$ is decreasing.

SI unit of $λ$ : $s^{- 1}$ (per second)
A large $λ$ means rapid decay (highly unstable nucleus).
A small $λ$ means slow decay (more stable nucleus).

Activity ( $A$ )

$A = λ N$

SI unit of Activity: Becquerel (Bq), where $1 Bq = 1$ disintegration per second.
Another common unit is the Curie (Ci): $1 Ci = 3.7 \times 1 0^{10} Bq$ .

Since $N$ decreases over time, the activity $A$ also decreases over time.

Exponential Nature of Radioactive Decay

Solving the differential equation $\frac{d N}{d t} = - λ N$ gives the Law of Radioactive Decay:

$N = N_{0} e^{- λ t}$

where:

$N_{0}$ = initial number of undecayed nuclei at $t = 0$
$N$ = number of undecayed nuclei at time $t$
$λ$ = decay constant
$e$ = base of natural logarithm ( $\approx 2.718$ )

The term $e^{- λ t}$ represents the fraction of nuclei remaining undecayed at time $t$ .

Since $A = λ N$ , the activity also decays exponentially:

$A = A_{0} e^{- λ t}$

where $A_{0} = λ N_{0}$ is the initial activity.

Key point: As the number of undecayed nuclei decreases exponentially, the rate of decay (activity) also decreases exponentially.

Half-Life ( $T_{1/2}$ )

The half-life $T_{1/2}$ is the time interval during which half of the unstable nuclei in a radioactive sample undergo decay.

After one half-life, $N = \frac{N _{0}}{2}$ . Substituting into the decay equation:

$\frac{N _{0}}{2} = N_{0} e^{- λ T_{1/2}}$

$\frac{1}{2} = e^{- λ T_{1/2}}$

Taking the natural logarithm of both sides:

$ln (\frac{1}{2}) = - λ T_{1/2}$

$T_{1/2} = \frac{ln 2}{λ} \approx \frac{0.693}{λ}$

Key Properties of Half-Life

Property	Detail
Relationship to $λ$	$T_{1/2} \propto \frac{1}{λ}$ (inversely proportional)
Large $λ$	Short $T_{1/2}$ → highly unstable
Small $λ$	Long $T_{1/2}$ → more stable
Independence	$T_{1/2}$ does not change with time or sample size

Decay After $n$ Half-Lives

After $n$ half-lives, the number of undecayed nuclei is:

$N = N_{0} (\frac{1}{2})^{n}$

where $n = \frac{t}{T _{1/2}}$ .

Worked Example

Problem: A sample of uranium has an initial mass of $400 g$ . How much remains after 3 half-lives?

Solution: $N = 400 \times (\frac{1}{2})^{3} = 400 \times \frac{1}{8} = 50 g$

Problem: A radioactive sample has $T_{1/2} = 10$ minutes. What percentage has decayed after 20 minutes?

Solution: $n = \frac{20}{10} = 2$ half-lives. $Fraction remaining = (\frac{1}{2})^{2} = \frac{1}{4} = 25%$ $Fraction decayed = 100% - 25% = 75%$

Summary of Key Equations

Quantity	Formula
Decay law	$N = N_{0} e^{- λ t}$
Activity	$A = λ N$
Half-life	$T_{1/2} = \frac{0.693}{λ}$
After $n$ half-lives	$N = N_{0} (\frac{1}{2})^{n}$