22.4.2 Decay Constant and Half-Life Relation | Nuclear Physics | Physics 12

This topic builds on the exponential nature of radioactive decay to establish the precise mathematical relationship between the decay constant ( $λ$ ) and the half-life ( $T_{1/2}$ ), and to apply the exponential decay equation quantitatively.

1. Activity and Decay Constant

The activity ( $A$ ) of a radioactive source is defined as the number of nuclear disintegrations per second:

$A = λ N$

where:

$A$ = activity (SI unit: Becquerel, Bq = 1 decay per second)
$λ$ = decay constant (SI unit: $s^{- 1}$ )
$N$ = number of undecayed nuclei present at that instant

The decay constant $λ$ represents the probability of decay of a single nucleus per unit time, or equivalently, the fraction of the total number of atoms that decay per unit time.

A larger $λ$ means a higher probability of decay per unit time — the substance is more radioactive and less stable.

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

The half-life of a radioactive substance is the time interval during which half of the unstable nuclei in a sample undergo decay.

After each successive half-life, the number of undecayed nuclei is halved:

Time elapsed	Fraction remaining
$0$	$1$
$T_{1/2}$	$\frac{1}{2}$
$2 T_{1/2}$	$\frac{1}{4}$
$3 T_{1/2}$	$\frac{1}{8}$
$n \cdot T_{1/2}$	$(\frac{1}{2})^{n}$

The number of undecayed nuclei after $n$ half-lives is:

$N = N_{0} (\frac{1}{2})^{n} where n = \frac{t}{T _{1/2}}$

3. Deriving the Relationship Between $T_{1/2}$ and $λ$

From the Law of Radioactive Decay, the number of undecayed nuclei at time $t$ is:

$N = N_{0} e^{- λ t}$

At $t = T_{1/2}$ , by definition $N = \frac{N _{0}}{2}$ . Substituting:

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

$\frac{1}{2} = e^{- λ T_{1/2}}$

Taking the natural logarithm of both sides:

$ln (\frac{1}{2}) = - λ T_{1/2}$

$- ln 2 = - λ T_{1/2}$

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

This is the fundamental relationship between half-life and decay constant.

Key implications:

$T_{1/2}$ and $λ$ are inversely proportional.
A large $λ$ → short $T_{1/2}$ → highly unstable, rapidly decaying nucleus.
A small $λ$ → long $T_{1/2}$ → more stable nucleus.
The product $λ \times T_{1/2} = ln 2 \approx 0.693$ (always).

4. The Exponential Decay Equation

The number of undecayed nuclei at any time $t$ is given by:

$N = N_{0} e^{- λ t}$

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

$A = A_{0} e^{- λ t}$

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

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

5. Worked Examples

Example 1 — Finding half-life from decay constant

A radioactive isotope has a decay constant $λ = 0.0231 s^{- 1}$ . Find its half-life.

$T_{1/2} = \frac{0.693}{λ} = \frac{0.693}{0.0231} \approx 30 s$

Example 2 — Amount remaining after multiple half-lives

A $400 g$ sample of uranium has a half-life of 4.5 billion years. How much remains after 3 half-lives?

$N = N_{0} (\frac{1}{2})^{3} = 400 \times \frac{1}{8} = 50 g$

Example 3 — Using the exponential equation

A sample initially contains $N_{0} = 8.0 \times 1 0^{20}$ nuclei with $λ = 1.5 \times 1 0^{- 3} s^{- 1}$ . How many nuclei remain after $t = 200 s$ ?

$N = N_{0} e^{- λ t} = 8.0 \times 1 0^{20} \times e^{- (1.5 \times 1 0^{- 3}) (200)}$ $N = 8.0 \times 1 0^{20} \times e^{- 0.30} \approx 8.0 \times 1 0^{20} \times 0.741 \approx 5.9 \times 1 0^{20} nuclei$

Summary Table

Quantity	Symbol	Formula	SI Unit
Activity	$A$	$A = λ N$	Becquerel (Bq)
Decay constant	$λ$	—	$s^{- 1}$
Half-life	$T_{1/2}$	$T_{1/2} = 0.693/ λ$	s
Undecayed nuclei	$N$	$N = N_{0} e^{- λ t}$	—
After $n$ half-lives	$N$	$N = N_{0} (1/2)^{n}$	—

1. Activity and Decay Constant

The activity ( $A$ ) of a radioactive source is defined as the number of nuclear disintegrations per second:

$A = λ N$

where:

$A$ = activity (SI unit: Becquerel, Bq = 1 decay per second)
$λ$ = decay constant (SI unit: $s^{- 1}$ )
$N$ = number of undecayed nuclei present at that instant

The decay constant $λ$ represents the probability of decay of a single nucleus per unit time, or equivalently, the fraction of the total number of atoms that decay per unit time.

A larger $λ$ means a higher probability of decay per unit time — the substance is more radioactive and less stable.

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

The half-life of a radioactive substance is the time interval during which half of the unstable nuclei in a sample undergo decay.

After each successive half-life, the number of undecayed nuclei is halved:

Time elapsed	Fraction remaining
$0$	$1$
$T_{1/2}$	$\frac{1}{2}$
$2 T_{1/2}$	$\frac{1}{4}$
$3 T_{1/2}$	$\frac{1}{8}$
$n \cdot T_{1/2}$	$(\frac{1}{2})^{n}$

The number of undecayed nuclei after $n$ half-lives is:

$N = N_{0} (\frac{1}{2})^{n} where n = \frac{t}{T _{1/2}}$

3. Deriving the Relationship Between $T_{1/2}$ and $λ$

From the Law of Radioactive Decay, the number of undecayed nuclei at time $t$ is:

$N = N_{0} e^{- λ t}$

At $t = T_{1/2}$ , by definition $N = \frac{N _{0}}{2}$ . Substituting:

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

$\frac{1}{2} = e^{- λ T_{1/2}}$

Taking the natural logarithm of both sides:

$ln (\frac{1}{2}) = - λ T_{1/2}$

$- ln 2 = - λ T_{1/2}$

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

This is the fundamental relationship between half-life and decay constant.

Key implications:

$T_{1/2}$ and $λ$ are inversely proportional.
A large $λ$ → short $T_{1/2}$ → highly unstable, rapidly decaying nucleus.
A small $λ$ → long $T_{1/2}$ → more stable nucleus.
The product $λ \times T_{1/2} = ln 2 \approx 0.693$ (always).

4. The Exponential Decay Equation

The number of undecayed nuclei at any time $t$ is given by:

$N = N_{0} e^{- λ t}$

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

$A = A_{0} e^{- λ t}$

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

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

Quantity	Symbol	Formula	SI Unit
Activity	$A$	$A = λ N$	Becquerel (Bq)
Decay constant	$λ$	—	$s^{- 1}$
Half-life	$T_{1/2}$	$T_{1/2} = 0.693/ λ$	s
Undecayed nuclei	$N$	$N = N_{0} e^{- λ t}$	—
After $n$ half-lives	$N$	$N = N_{0} (1/2)^{n}$	—