Three-fourth of a radioactive material decays in 2.5 days. How long will it take for `15//16th` of the material to decay?
a. 2 days b. 5 days c. 7.5 days d. 10 days

Options

A. a. 2 days
B. b. 5 days
C. c. 7.5 days
D. c. 7.5 days

Explanation

To solve the problem of how long it will take for \( \frac{15}{16} \) of a radioactive material to decay, given that three-fourths of the material decays in 2.5 days, we can follow these steps:

### Step 1: Understand the decay process
Radioactive decay follows first-order kinetics, which means we can use the decay constant \( \lambda \) to relate the initial and final quantities of the substance over time.

### Step 2: Calculate the decay constant \( \lambda \)
Given that three-fourths of the material decays in 2.5 days, we can express this mathematically. If \( A_0 \) is the initial amount of the material, then after 2.5 days, the remaining amount \( A_f \) is:

\[ A_f = A_0 - \frac{3}{4} A_0 = \frac{1}{4} A_0 \]

Using the formula for the decay constant:

\[ \lambda = \frac{2.303}{T} \log\left(\frac{A_0}{A_f}\right) \]

Substituting the values:

\[ \lambda = \frac{2.303}{2.5} \log\left(\frac{A_0}{\frac{1}{4} A_0}\right) = \frac{2.303}{2.5} \log(4) \]

### Step 3: Set up the equation for \( \frac{15}{16} \) decay
Now we need to find the time \( t \) it takes for \( \frac{15}{16} \) of the material to decay. After this decay, the remaining amount \( A_f \) will be:

\[ A_f = A_0 - \frac{15}{16} A_0 = \frac{1}{16} A_0 \]

Using the same decay constant \( \lambda \):

\[ \lambda = \frac{2.303}{t} \log\left(\frac{A_0}{A_f}\right) = \frac{2.303}{t} \log\left(\frac{A_0}{\frac{1}{16} A_0}\right) = \frac{2.303}{t} \log(16) \]

### Step 4: Equate the two expressions for \( \lambda \)
Since \( \lambda \) is the same in both cases, we can set the two expressions equal to each other:

\[ \frac{2.303}{2.5} \log(4) = \frac{2.303}{t} \log(16) \]

### Step 5: Solve for \( t \)
We can cancel \( 2.303 \) from both sides:

\[ \frac{1}{2.5} \log(4) = \frac{1}{t} \log(16) \]

Cross-multiplying gives:

\[ t \cdot \log(4) = 2.5 \cdot \log(16) \]

Now, we know that \( \log(16) = \log(4^2) = 2 \log(4) \):

\[ t \cdot \log(4) = 2.5 \cdot 2 \log(4) \]

Dividing both sides by \( \log(4) \):

\[ t = 5 \]

### Conclusion
Thus, the time it will take for \( \frac{15}{16} \) of the material to decay is **5 days**. The correct answer is **b. 5 days**.

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