The radioactive substance `""_(92)U^(238)` has a half-life of `4.5 xx 10^(9)` years. Determine the time taken for 3/4 of the substance to disintegrate.

Answer

To determine the time taken for 3/4 of the radioactive substance \( _{92}U^{238} \) to disintegrate, we can follow these steps:

### Step 1: Understanding Half-Life
The half-life of a radioactive substance is the time taken for half of the substance to decay. For \( _{92}U^{238} \), the half-life is given as \( 4.5 \times 10^9 \) years.

### Step 2: Initial and Final Amounts
Let the initial amount of the substance be \( N_0 \). If 3/4 of the substance has disintegrated, then 1/4 of the substance remains. Therefore, the remaining amount after disintegration is:
\[ N = \frac{N_0}{4} \]

### Step 3: Relating Remaining Quantity to Half-Lives
To find the time taken for the substance to decay from \( N_0 \) to \( \frac{N_0}{4} \), we need to determine how many half-lives it takes to reach this amount.

1. After 1 half-life, the amount remaining is:
\[ N = \frac{N_0}{2} \]
2. After 2 half-lives, the amount remaining is:
\[ N = \frac{N_0}{4} \]

### Step 4: Calculate Total Time
Since it takes 2 half-lives to reduce the substance from \( N_0 \) to \( \frac{N_0}{4} \), we can calculate the total time taken as follows:
\[ \text{Total time} = 2 \times \text{half-life} \]
Substituting the value of the half-life:
\[ \text{Total time} = 2 \times 4.5 \times 10^9 \text{ years} = 9 \times 10^9 \text{ years} \]

### Final Answer
The time taken for 3/4 of the substance \( _{92}U^{238} \) to disintegrate is:
\[ 9 \times 10^9 \text{ years} \]

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