The half-life of decomposition of gaseous `CH_(3)CHO` at initial pressure of `365 mm` and `170 mm` of `Hg` were `420 sec` and `880 sec` respectively. The order of the reaction is:

Options

A. 2
B. 4
C. 6
D. 8

Explanation

To determine the order of the reaction for the decomposition of gaseous CH₃CHO based on the given half-lives at different initial pressures, we can follow these steps:

### Step 1: Understand the relationship between half-life and initial concentration
The half-life (t₁/₂) of a reaction is related to the initial concentration (or pressure for gases) and the order of the reaction (n) by the following relationship:
\[ t_{1/2} \propto P_0^{(1-n)} \]
where \( P_0 \) is the initial pressure.

### Step 2: Set up the equation for two different pressures
Given:
- \( t_{1/2,1} = 420 \, \text{sec} \) at \( P_{0,1} = 365 \, \text{mm Hg} \)
- \( t_{1/2,2} = 880 \, \text{sec} \) at \( P_{0,2} = 170 \, \text{mm Hg} \)

We can write the relationship for the two half-lives:
\[ \frac{t_{1/2,1}}{t_{1/2,2}} = \left( \frac{P_{0,1}}{P_{0,2}} \right)^{(1-n)} \]

### Step 3: Substitute the values into the equation
Substituting the known values:
\[ \frac{420}{880} = \left( \frac{365}{170} \right)^{(1-n)} \]

### Step 4: Simplify the left side
Calculating the left side:
\[ \frac{420}{880} = \frac{21}{44} \approx 0.4773 \]

### Step 5: Simplify the right side
Calculating the right side:
\[ \frac{365}{170} = 2.147 \]

### Step 6: Set up the equation
Now we have:
\[ 0.4773 = (2.147)^{(1-n)} \]

### Step 7: Take logarithms to solve for n
Taking logarithms on both sides:
\[ \log(0.4773) = (1-n) \cdot \log(2.147) \]

### Step 8: Calculate the logarithms
Calculating the logarithms:
- \( \log(0.4773) \approx -0.320 \)
- \( \log(2.147) \approx 0.332 \)

### Step 9: Substitute and solve for n
Substituting the values:
\[ -0.320 = (1-n)(0.332) \]

Now, rearranging to solve for n:
\[ 1 - n = \frac{-0.320}{0.332} \]
\[ 1 - n \approx -0.963 \]
\[ n \approx 1 + 0.963 \]
\[ n \approx 1.963 \]

### Step 10: Round to the nearest whole number
Since n must be a whole number, we round n to 2.

### Conclusion
Thus, the order of the reaction is:
**n = 2**