The disintegration rate of a certain radioactive sample at any instant is 5400 dpm. After 5 min the rate becomes 2700 dpm. The half life of the sample in min is approximately

Explanation

To find the half-life of the radioactive sample, we can follow these steps:

### Step 1: Understand the given information
We know the initial disintegration rate (R₀) is 5400 dpm and the disintegration rate after 5 minutes (Rₜ) is 2700 dpm.

### Step 2: Use the formula for radioactive decay
The relationship between the initial and final disintegration rates can be expressed using the formula:

\[ \frac{R₀}{Rₜ} = 2^{(t/T_{1/2})} \]

Where:
- \( R₀ \) = initial disintegration rate
- \( Rₜ \) = disintegration rate after time \( t \)
- \( T_{1/2} \) = half-life of the sample
- \( t \) = time elapsed

### Step 3: Substitute the known values into the formula
Substituting the values we have:

\[ \frac{5400}{2700} = 2^{(5/T_{1/2})} \]

This simplifies to:

\[ 2 = 2^{(5/T_{1/2})} \]

### Step 4: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal:

\[ 1 = \frac{5}{T_{1/2}} \]

### Step 5: Solve for the half-life
Now, we can solve for \( T_{1/2} \):

\[ T_{1/2} = 5 \text{ minutes} \]

### Conclusion
The half-life of the sample is approximately **5 minutes**.

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