Radioactive decay follows which order kinetics?

Options

A. zero
B. 1
C. 2
D. 3

Answer

To determine which order kinetics radioactive decay follows, we can analyze the decay process mathematically. Here’s a step-by-step solution:

### Step 1: Understanding Radioactive Decay
Radioactive decay is a process where unstable nuclei lose energy by emitting radiation. The rate of decay is proportional to the number of undecayed nuclei present at any time.

### Step 2: Writing the Rate Law
The rate of decay can be expressed mathematically as:
\[ -\frac{dN}{dt} = \lambda N \]
where:
- \(N\) is the number of radioactive nuclei at time \(t\),
- \(\lambda\) is the decay constant.

### Step 3: Rearranging the Equation
Rearranging the equation gives:
\[ \frac{dN}{N} = -\lambda dt \]

### Step 4: Integrating the Equation
Integrate both sides:
\[ \int \frac{dN}{N} = -\lambda \int dt \]
This results in:
\[ \ln N = -\lambda t + C \]
where \(C\) is the integration constant.

### Step 5: Solving for the Constant
To find the constant, we can use the initial condition. At \(t = 0\), let \(N = N_0\) (the initial number of nuclei):
\[ \ln N_0 = C \]
Thus, the equation becomes:
\[ \ln N = -\lambda t + \ln N_0 \]

### Step 6: Exponentiating the Equation
Exponentiating both sides gives:
\[ N = N_0 e^{-\lambda t} \]

### Step 7: Identifying the Order of Kinetics
The equation \(N = N_0 e^{-\lambda t}\) resembles the first-order kinetics equation:
\[ A_t = A_0 e^{-kt} \]
where \(A_t\) is the concentration at time \(t\), \(A_0\) is the initial concentration, and \(k\) is the rate constant.

### Conclusion
From the derived equation, we can conclude that radioactive decay follows **first-order kinetics**.

### Final Answer
The correct answer is **first-order kinetics**.

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