The mean life time of a radionuclide, if the activity decrease by `4%` for every `1 h`, would b e(product is non-radioactive, i.e., stable )

Answer

To find the mean lifetime of a radionuclide given that its activity decreases by 4% every hour, we can follow these steps:

### Step 1: Understand the decay percentage
The problem states that the activity of the radionuclide decreases by 4% every hour. This means that after one hour, 96% of the original activity remains.

### Step 2: Relate the decay to mean lifetime
The mean lifetime (τ) of a radionuclide is related to its decay constant (λ) by the formula:
\[ \tau = \frac{1}{\lambda} \]
The decay constant can be found using the formula for exponential decay:
\[ N(t) = N_0 e^{-\lambda t} \]
where \(N(t)\) is the remaining quantity after time \(t\), \(N_0\) is the initial quantity, and \(e\) is the base of the natural logarithm.

### Step 3: Calculate the decay constant
Since we know that the activity decreases by 4% in one hour, we can express this mathematically:
\[ N(1) = N_0 \times (1 - 0.04) = N_0 \times 0.96 \]
Substituting into the exponential decay equation for \(t = 1\) hour:
\[ N_0 \times 0.96 = N_0 e^{-\lambda \cdot 1} \]
Dividing both sides by \(N_0\) (assuming \(N_0 \neq 0\)):
\[ 0.96 = e^{-\lambda} \]

### Step 4: Solve for the decay constant
Taking the natural logarithm of both sides:
\[ \ln(0.96) = -\lambda \]
Thus,
\[ \lambda = -\ln(0.96) \]

### Step 5: Calculate the mean lifetime
Now, substituting the value of λ into the mean lifetime formula:
\[ \tau = \frac{1}{\lambda} = \frac{1}{-\ln(0.96)} \]

### Step 6: Calculate the numerical value
Calculating \( \ln(0.96) \):
\[ \ln(0.96) \approx -0.0408 \]
Therefore,
\[ \tau \approx \frac{1}{0.0408} \approx 24.51 \text{ hours} \]

### Conclusion
The mean lifetime of the radionuclide is approximately **24.51 hours**.

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