What fraction of a reactant remains unreacted after 40 min if `t_(1//2)` is of the reaction is 20 min? consider the reaction is first order

Options

A. `1//4`
B. `1//2`
C. `1//8`
D. `1//6`

Answer

To solve the problem of finding the fraction of a reactant that remains unreacted after 40 minutes for a first-order reaction with a half-life of 20 minutes, we can follow these steps:

### Step 1: Understand the relationship for first-order reactions
For a first-order reaction, the relationship between the amount of reactant remaining at time \( t \) and the initial amount is given by the equation:

\[ \frac{A_t}{A_0} = \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} \]

Where:
- \( A_t \) = amount of reactant remaining at time \( t \)
- \( A_0 \) = initial amount of reactant
- \( t \) = time elapsed
- \( t_{1/2} \) = half-life of the reaction

### Step 2: Substitute the known values
In this case, we know:
- \( t = 40 \) minutes
- \( t_{1/2} = 20 \) minutes

Substituting these values into the equation gives:

\[ \frac{A_t}{A_0} = \left(\frac{1}{2}\right)^{\frac{40}{20}} \]

### Step 3: Simplify the exponent
Now, simplify the exponent:

\[ \frac{A_t}{A_0} = \left(\frac{1}{2}\right)^{2} \]

### Step 4: Calculate the fraction
Calculating the right-hand side:

\[ \frac{A_t}{A_0} = \frac{1}{4} \]

### Step 5: Determine the fraction of reactant remaining unreacted
The fraction of the reactant that remains unreacted is given by:

\[ \text{Fraction unreacted} = 1 - \frac{A_t}{A_0} \]

Substituting the value we found:

\[ \text{Fraction unreacted} = 1 - \frac{1}{4} = \frac{3}{4} \]

### Conclusion
Thus, the fraction of the reactant that remains unreacted after 40 minutes is \( \frac{1}{4} \).