Decay Constant — Revision Notes

⚡ 30-Second Revision

Decay Constant ($\lambda$) — Probability of decay per unit time. Unit: $\text{s}^{-1}$.
Radioactive Decay Law — $\frac{dN}{dt} = -\lambda N \implies N = N_0 e^{-\lambda t}$
Activity (A) — $A = \lambda N \implies A = A_0 e^{-\lambda t}$
Half-Life ($T_{1/2}$) — Time for half nuclei to decay. $T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}$
Mean Life ($\tau$) — Average lifetime of a nucleus. $\tau = \frac{1}{\lambda}$
Relationship — $\tau = \frac{T_{1/2}}{\ln 2} \approx 1.44 T_{1/2}$
After 'n' half-lives — $N = N_0 \left(\frac{1}{2}\right)^n$, $A = A_0 \left(\frac{1}{2}\right)^n$

2-Minute Revision

The decay constant ($\lambda$) is a fundamental measure of how quickly a radioactive substance decays, representing the probability per unit time for a single nucleus to disintegrate. It's an intrinsic property of a specific radionuclide, unaffected by external conditions.

The core principle is the Law of Radioactive Decay, which states that the rate of decay is proportional to the number of radioactive nuclei present, leading to the exponential decay equation $N = N_0 e^{-\lambda t}$.

Similarly, the activity ($A = \lambda N$) also decays exponentially as $A = A_0 e^{-\lambda t}$. The decay constant is inversely related to the half-life ($T_{1/2}$), the time for half the nuclei to decay, by $T_{1/2} = \frac{\ln 2}{\lambda}$.

It is also the reciprocal of the mean life ($\tau$), the average lifetime of a nucleus, given by $\tau = \frac{1}{\lambda}$. Remember that $\tau \approx 1.44 T_{1/2}$. For NEET, focus on applying these formulas to calculate remaining nuclei, activity, or time, and understand the conceptual independence of $\lambda$ from environmental factors.

5-Minute Revision

The decay constant ($\lambda$) is the cornerstone of understanding radioactive decay. It's defined as the probability per unit time that a single radioactive nucleus will decay. This value is unique for each radionuclide and is independent of temperature, pressure, or chemical state.

Its unit is inverse time (e.g., $\text{s}^{-1}$). The fundamental equation governing decay is the Law of Radioactive Decay: $\frac{dN}{dt} = -\lambda N$. Integrating this gives the exponential decay law for the number of nuclei: $N = N_0 e^{-\lambda t}$, where $N_0$ is the initial number of nuclei and $N$ is the number at time $t$.

Similarly, the activity ($A = \lambda N$), which is the rate of decay, also follows an exponential law: $A = A_0 e^{-\lambda t}$.

Crucially, $\lambda$ is linked to two other important time constants: half-life ($T_{1/2}$) and mean life ($\tau$). The half-life is the time for half the nuclei to decay, related by $T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}$. The mean life is the average lifetime of a nucleus, given by $\tau = \frac{1}{\lambda}$. Note that $\tau \approx 1.44 T_{1/2}$.

Worked Example: A radioactive substance has a half-life of 5 hours. Calculate its decay constant and mean life.

Solution:

1
Decay Constant ($\lambda$) — $T_{1/2} = \frac{\ln 2}{\lambda} \implies \lambda = \frac{\ln 2}{T_{1/2}} = \frac{0.693}{5 \text{ h}} = 0.1386 \text{ h}^{-1}$.
2
Mean Life ($\tau$) — $\tau = \frac{1}{\lambda} = \frac{1}{0.1386 \text{ h}^{-1}} \approx 7.215 \text{ h}$.

For NEET, practice converting between these quantities and applying the exponential decay laws. Remember that after 'n' half-lives, the remaining fraction is $(1/2)^n$.

Prelims Revision Notes

1
Decay Constant ($\lambda$) — Intrinsic property of a radionuclide, representing the probability of decay per unit time. Unit: $\text{s}^{-1}$, $\text{min}^{-1}$, etc.
2
Law of Radioactive Decay — $\frac{dN}{dt} = -\lambda N$. The negative sign indicates decrease in nuclei.
3
Exponential Decay Law (Nuclei) — $N = N_0 e^{-\lambda t}$. $N_0$ = initial nuclei, $N$ = nuclei at time $t$.
4
Exponential Decay Law (Activity) — $A = A_0 e^{-\lambda t}$. $A_0$ = initial activity, $A$ = activity at time $t$. Activity $A = \lambda N$.
5
Half-Life ($T_{1/2}$) — Time for half of the nuclei to decay. $T_{1/2} = \frac{\ln 2}{\lambda}$. Use $\ln 2 \approx 0.693$.
6
Mean Life ($\tau$) — Average lifetime of a radioactive nucleus. $\tau = \frac{1}{\lambda}$.
7
Relationship between $T_{1/2}$ and $\tau$ — $\tau = \frac{T_{1/2}}{\ln 2} \approx 1.44 T_{1/2}$. Mean life is always greater than half-life.
8
Decay after 'n' half-lives — If $t = n T_{1/2}$, then $N = N_0 \left(\frac{1}{2}\right)^n$ and $A = A_0 \left(\frac{1}{2}\right)^n$.
9
Independence — $\lambda$ is independent of temperature, pressure, chemical state, or amount of substance. It's a nuclear property.
10
Units Consistency — Always ensure units of time for $\lambda$, $T_{1/2}$, $\tau$, and $t$ are consistent in calculations.
11
Logarithm Rules — Remember $\ln(x/y) = \ln x - \ln y$, $\ln(1/x) = -\ln x$, $\ln(e^x) = x$.
12
Activity Units — Becquerel (Bq = 1 decay/s) and Curie (Ci = $3.7 \times 10^{10}$ Bq).

Vyyuha Quick Recall

Lambda's Life is Half-Baked: Lambda ($\lambda$) is Inverse to Life (Mean Life, $\tau$), and Half-life ($T_{1/2}$) is Based on Lambda ($\ln 2 / \lambda$).

Think: $\lambda$ is the 'rate', $\tau$ is 'total time', $T_{1/2}$ is 'half time'. $\tau = 1/\lambda$ (Simple inverse) $T_{1/2} = 0.693/\lambda$ (Need the 'ln 2' factor for half)