The half life of a first order reaction is 6 hours. How long will it take
for the concentration of reactant to change from 0.8 M to 0.25 M ?

Options

A. 1.07 hour
B. 5.1 hour
C. 2.7 hour
D. 10.07 hour

Answer

To solve the problem of how long it will take for the concentration of a reactant to change from 0.8 M to 0.25 M in a first-order reaction with a half-life of 6 hours, we can follow these steps:

### Step 1: Convert the half-life to minutes
The half-life (T_half) is given as 6 hours. To convert this into minutes:
\[ T_{half} = 6 \text{ hours} \times 60 \text{ minutes/hour} = 360 \text{ minutes} \]

**Hint:** Remember that 1 hour equals 60 minutes when converting time units.

### Step 2: Calculate the rate constant (k)
For a first-order reaction, the relationship between half-life and the rate constant (k) is given by:
\[ T_{half} = \frac{0.693}{k} \]
Rearranging this formula to find k:
\[ k = \frac{0.693}{T_{half}} = \frac{0.693}{360 \text{ minutes}} \approx 0.001925 \text{ min}^{-1} \]

**Hint:** Use the formula for half-life to find the rate constant, ensuring you use the correct units.

### Step 3: Use the first-order reaction formula
The formula for the time (T) taken for a first-order reaction is:
\[ T = \frac{2.303}{k} \log\left(\frac{A_0}{A_t}\right) \]
Where:
- \(A_0\) = initial concentration = 0.8 M
- \(A_t\) = final concentration = 0.25 M

Substituting the values:
\[ T = \frac{2.303}{0.001925} \log\left(\frac{0.8}{0.25}\right) \]

**Hint:** Make sure to correctly substitute the initial and final concentrations into the logarithmic equation.

### Step 4: Calculate the logarithm
First, calculate the ratio:
\[ \frac{0.8}{0.25} = 3.2 \]
Now calculate the logarithm:
\[ \log(3.2) \approx 0.505 \]

**Hint:** Use a scientific calculator or logarithm tables to find the logarithm of the ratio.

### Step 5: Substitute back to find T
Now substitute back into the equation for T:
\[ T = \frac{2.303}{0.001925} \times 0.505 \approx 1196.3 \text{ minutes} \]

**Hint:** Ensure you multiply correctly and keep track of significant figures.

### Step 6: Convert time to hours
Finally, convert the time from minutes to hours:
\[ T = \frac{1196.3 \text{ minutes}}{60 \text{ minutes/hour}} \approx 19.94 \text{ hours} \]

**Hint:** When converting from minutes to hours, divide by 60.

### Final Answer
The time taken for the concentration of the reactant to change from 0.8 M to 0.25 M is approximately **19.94 hours**.