At `518^(@)C` the rate of decomposition of a sample of gaseous acetaldehyde initially at a pressure of 363 Torr, was `1.00 Torr s^(–1)` when `5%` had reacted and `0.5 Torr s^(–1)` when 33% had reacted. The order of the reaction is

Options

A. 0
B. 2
C. 3
D. 1

Answer

To determine the order of the reaction for the decomposition of gaseous acetaldehyde, we can follow these steps:

### Step 1: Write the reaction and identify the variables
The decomposition reaction of acetaldehyde (CH₃CHO) can be represented as:
\[ \text{CH}_3\text{CHO} \rightarrow \text{CH}_4 + \text{CO} \]

Let:
- \( R_1 = 1.00 \, \text{Torr/s} \) (rate when 5% has reacted)
- \( R_2 = 0.5 \, \text{Torr/s} \) (rate when 33% has reacted)
- Initial pressure \( P_0 = 363 \, \text{Torr} \)

### Step 2: Determine the unreacted pressures
When 5% has reacted:
- Reacted pressure \( x_1 = 0.05 \times 363 = 18.15 \, \text{Torr} \)
- Unreacted pressure \( P_1 = 363 - 18.15 = 344.85 \, \text{Torr} \)

When 33% has reacted:
- Reacted pressure \( x_2 = 0.33 \times 363 = 119.79 \, \text{Torr} \)
- Unreacted pressure \( P_2 = 363 - 119.79 = 243.21 \, \text{Torr} \)

### Step 3: Write the rate equations
The rate of reaction can be expressed in terms of the unreacted pressure raised to the power of the order of the reaction \( m \):
\[ R_1 = k (P_1)^m \]
\[ R_2 = k (P_2)^m \]

### Step 4: Set up the ratio of the rates
Taking the ratio of the two rates:
\[ \frac{R_1}{R_2} = \frac{k (P_1)^m}{k (P_2)^m} \implies \frac{R_1}{R_2} = \left(\frac{P_1}{P_2}\right)^m \]

### Step 5: Substitute the known values
Substituting the values of \( R_1 \) and \( R_2 \):
\[ \frac{1.00}{0.5} = \left(\frac{344.85}{243.21}\right)^m \]
This simplifies to:
\[ 2 = \left(\frac{344.85}{243.21}\right)^m \]

### Step 6: Calculate the ratio of unreacted pressures
Calculating the ratio:
\[ \frac{344.85}{243.21} \approx 1.42 \]

### Step 7: Solve for \( m \)
Now we have:
\[ 2 = (1.42)^m \]
Taking logarithm on both sides:
\[ \log(2) = m \cdot \log(1.42) \]
Thus,
\[ m = \frac{\log(2)}{\log(1.42)} \]

Using logarithm values:
\[ m \approx \frac{0.301}{0.152} \approx 1.98 \approx 2 \]

### Conclusion
The order of the reaction is \( m = 2 \).

### Final Answer
The order of the reaction is **2**.

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