A substance is kept for 2 hours and three-fourth of that substance disintegrates during this period. The half life of the substance is

Options

A. 2 hr
B. 1 hr
C. 30 min
D. 4 hr

Explanation

To find the half-life of a substance that disintegrates over a period of time, we can follow these steps:

### Step 1: Understand the Disintegration
We know that three-fourths of the substance disintegrates in 2 hours. This means that only one-fourth of the substance remains.

### Step 2: Set Up the Initial and Remaining Amounts
Let the initial amount of the substance be \( n_0 = 1 \) (this is just a convenient choice for calculation). After 2 hours, the remaining amount \( n_t \) is:
\[ n_t = n_0 - \text{disintegrated amount} = 1 - \frac{3}{4} = \frac{1}{4} \]

### Step 3: Use the Half-Life Formula
The relationship between the remaining quantity and the half-lives can be expressed as:
\[ n_t = n_0 \left(\frac{1}{2}\right)^n \]
where \( n \) is the number of half-lives.

### Step 4: Substitute Known Values
Substituting the known values into the equation:
\[ \frac{1}{4} = 1 \left(\frac{1}{2}\right)^n \]
This simplifies to:
\[ \left(\frac{1}{2}\right)^n = \frac{1}{4} \]

### Step 5: Solve for \( n \)
We can express \( \frac{1}{4} \) as \( \left(\frac{1}{2}\right)^2 \):
\[ \left(\frac{1}{2}\right)^n = \left(\frac{1}{2}\right)^2 \]
From this, we can conclude that:
\[ n = 2 \]
This means that 2 half-lives have occurred in the 2-hour period.

### Step 6: Calculate the Half-Life
The number of half-lives \( n \) is related to the total time \( t \) by the formula:
\[ n = \frac{t}{t_{1/2}} \]
where \( t_{1/2} \) is the half-life. We know \( n = 2 \) and \( t = 2 \) hours. We can rearrange the formula to find \( t_{1/2} \):
\[ t_{1/2} = \frac{t}{n} = \frac{2 \text{ hours}}{2} = 1 \text{ hour} \]

### Final Answer
The half-life of the substance is **1 hour**.

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