The isomerization of cyclopropane to form propane is a first order reaction. At `760 K, 85%` of a sample of cyclopropane change to propane in `79 min`. Calculate the value of the rate constant.

Options

A. `A. 3.66 xx 10^(-2) min^(-1)`
B. `B. 1.04 xx 10^(-2) min^(-1)`
C. `C. 2.42 min^(-1)`
D. `D. 2.40 xx 10^(-2) min^(-1)`

Explanation

To calculate the rate constant for the isomerization of cyclopropane to propane, we can follow these steps:

### Step 1: Understand the Reaction
The isomerization of cyclopropane to propane is a first-order reaction. We know that 85% of cyclopropane converts to propane in 79 minutes at 760 K.

### Step 2: Determine Initial and Remaining Concentrations
Assume the initial concentration of cyclopropane is 100 units. If 85% has reacted, the remaining concentration of cyclopropane is:
\[ \text{Remaining concentration} = 100 - 85 = 15 \]

### Step 3: Use the First-Order Rate Constant Formula
For a first-order reaction, the rate constant \( k \) can be calculated using the formula:
\[ k = \frac{2.303}{t} \log \left( \frac{[C]_0}{[C]} \right) \]
where:
- \( t \) is the time in minutes,
- \( [C]_0 \) is the initial concentration,
- \( [C] \) is the remaining concentration.

### Step 4: Substitute Values into the Formula
Substituting the values we have:
- \( t = 79 \) minutes,
- \( [C]_0 = 100 \),
- \( [C] = 15 \).

The equation becomes:
\[ k = \frac{2.303}{79} \log \left( \frac{100}{15} \right) \]

### Step 5: Calculate the Logarithm
Now, calculate \( \log \left( \frac{100}{15} \right) \):
\[ \frac{100}{15} = \frac{20}{3} \]

Using properties of logarithms:
\[ \log \left( \frac{20}{3} \right) = \log(20) - \log(3) \]

### Step 6: Calculate \( \log(20) \) and \( \log(3) \)
Using known logarithm values:
- \( \log(20) = \log(2 \times 10) = \log(2) + \log(10) = 0.3010 + 1 = 1.3010 \)
- \( \log(3) = 0.4770 \)

Thus,
\[ \log \left( \frac{20}{3} \right) = 1.3010 - 0.4770 = 0.8240 \]

### Step 7: Substitute Back to Find \( k \)
Now substitute back into the equation for \( k \):
\[ k = \frac{2.303}{79} \times 0.8240 \]

### Step 8: Calculate \( k \)
Now perform the calculation:
\[ k = \frac{2.303 \times 0.8240}{79} \approx \frac{1.897}{79} \approx 0.0240 \text{ per minute} \]

### Final Answer
Thus, the value of the rate constant \( k \) is:
\[ k \approx 0.0240 \text{ per minute} \]
or in scientific notation:
\[ k \approx 2.4 \times 10^{-2} \text{ per minute} \]