Half life of radioactive substance is `3.20` h. What is the time taken for `75%` substance to be used?

Options

A. 6.4h
B. 12 h
C. 4.18 day
D. 1.2 day

Answer

To solve the problem of finding the time taken for 75% of a radioactive substance to decay, we can follow these steps:

### Step 1: Understand the Problem
The half-life of the radioactive substance is given as 3.20 hours. We need to find the time taken for 75% of the substance to decay, which means that only 25% of the substance will remain.

### Step 2: Set Up the Equation
Let \( N_0 \) be the initial amount of the substance. After a certain time \( t \), the remaining amount \( N \) can be expressed as:
\[ N = N_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \]
where \( t_{1/2} \) is the half-life of the substance.

### Step 3: Determine the Remaining Quantity
Since we want 75% of the substance to decay, we have:
\[ N = 25\% \text{ of } N_0 = 0.25 N_0 \]

### Step 4: Substitute into the Equation
Substituting \( N = 0.25 N_0 \) into the equation gives:
\[ 0.25 N_0 = N_0 \left( \frac{1}{2} \right)^{\frac{t}{3.20}} \]

### Step 5: Simplify the Equation
Dividing both sides by \( N_0 \) (assuming \( N_0 \neq 0 \)):
\[ 0.25 = \left( \frac{1}{2} \right)^{\frac{t}{3.20}} \]

### Step 6: Rewrite 0.25
We know that \( 0.25 = \left( \frac{1}{2} \right)^{2} \). Therefore, we can rewrite the equation as:
\[ \left( \frac{1}{2} \right)^{2} = \left( \frac{1}{2} \right)^{\frac{t}{3.20}} \]

### Step 7: Set the Exponents Equal
Since the bases are the same, we can set the exponents equal to each other:
\[ 2 = \frac{t}{3.20} \]

### Step 8: Solve for \( t \)
Multiplying both sides by \( 3.20 \) gives:
\[ t = 2 \times 3.20 = 6.40 \text{ hours} \]

### Conclusion
The time taken for 75% of the substance to decay is **6.40 hours**.

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