# Exponential Nature of Nuclear Decay Radioactive decay is **exponential** in nature, meaning the rate of decay is directly proportional to the number of undecayed nuclei present at any given time. This relationship implies that substances decay more rapidly when there are more radioactive nuclei, and the rate slows down as the number of nuclei decreases. ### Mathematical Representation of Nuclear Decay **Rate of Decay:** The instantaneous rate of change in the number of nuclei $N$ with respect to time $t$ is given by: $$ \frac{dN}{dt} = -\lambda N $$ where: - $N$ is the number of undecayed nuclei at time $t$ - $\lambda$ is the **decay constant** ($\text{s}^{-1}$) - The negative sign indicates that the number of undecayed nuclei is decreasing **Decay Law:** By integrating the rate equation, the number of undecayed nuclei at any time $t$ can be expressed as: $$ N = N_0 e^{-\lambda t} $$ where: - $N_0$ is the initial number of undecayed nuclei at $t = 0$ - $e$ is the base of the natural logarithm This equation demonstrates the exponential decrease of radioactive nuclei over time. Unlike deterministic classical systems, this process is governed by probability, similar to concepts found in . ### Decay Constant ($\lambda$) The decay constant $\lambda$ is a characteristic constant for a particular radioactive isotope. It represents the probability per unit time for a single nucleus to decay. Its SI unit is inverse seconds ($\text{s}^{-1}$). From the rate equation, an approximate formula for $\lambda$ over a small time interval is given by: $$ \lambda = \frac{1}{N} \times \frac{\Delta N}{\Delta t} $$ where $N$ can be approximated as the initial number of nuclei $N_0$ for small $\Delta N$ and $\Delta t$. ### Activity ($A$) **Activity** is the rate at which nuclei decay in a radioactive sample. It is directly proportional to the number of undecayed nuclei present at that instant. The formula for activity is: $$ A = \lambda N $$ where: - $A$ is the activity - $\lambda$ is the decay constant - $N$ is the number of undecayed nuclei at that instant **Units of Activity:** The SI unit of activity is the **Becquerel (Bq)**, where $1 \text{ Bq} = 1 \text{ decay per second}$ ($\text{s}^{-1}$). Another common unit is the **Curie (Ci)**, where $1 \text{ Ci} = 3.7 \times 10^{10} \text{ Bq}$. ## Worked Examples **Example 1:** If 100 out of 700 nuclei decay in 5 minutes, calculate the decay constant. Given: $N_0 = 700$, $\Delta N = 100$, $\Delta t = 5 \text{ min} = 300 \text{ s}$ $$ \lambda = \frac{1}{N_0} \times \frac{\Delta N}{\Delta t} = \frac{1}{700} \times \frac{100}{300} = 4.76 \times 10^{-4} \text{ s}^{-1} $$ **Example 2:** If 240 out of 780 nuclei decay in 12 seconds, calculate the decay constant. Given: $N_0 = 780$, $\Delta N = 240$, $\Delta t = 12 \text{ s}$ $$ \lambda = \frac{1}{N_0} \times \frac{\Delta N}{\Delta t} = \frac{1}{780} \times \frac{240}{12} = \frac{1}{780} \times 20 = \frac{20}{780} = \frac{1}{39} \approx 0.02564 \text{ s}^{-1} $$ **Example 3:** Calculate the activity of a sample with $N_0 = 780$ nuclei and $\lambda \approx 0.02564 \text{ s}^{-1}$. $$ A = \lambda N_0 = (0.02564 \text{ s}^{-1}) \times 780 = 20 \text{ Bq} $$ | Concept | Formula | Units | | :------------- | :---------------------------------------- | :----------------- | | Rate of Decay | $\frac{dN}{dt} = -\lambda N$ | nuclei/s | | Decay Law | $N = N_0 e^{-\lambda t}$ | nuclei (dimensionless) | | Decay Constant | $\lambda = \frac{1}{N} \frac{\Delta N}{\Delta t}$ | $\text{s}^{-1}$ | | Activity | $A = \lambda N$ | Bq (or Ci) | Nuclear decay is fundamental to understanding natural radioactivity, dating techniques such as carbon-14 dating, and nuclear energy applications.