For a gaseous reaction : 2A(g)`rarr`3B(g)+4C(g), occurring in a rigid vessel, if initially pressure is 3 atm and afer 10 minutes and 20 minutes the pressure is `6.75` atm and `10.5` respectively, then what will be the order of reaction?

Options

A. 0
B. 1
C. 2
D. `(1)/(2)`

Answer

To determine the order of the reaction for the given gaseous reaction \( 2A(g) \rightarrow 3B(g) + 4C(g) \), we will analyze the pressure changes over time and apply the principles of chemical kinetics.

### Step 1: Understand the Reaction and Initial Conditions
The reaction starts with an initial pressure of \( P_0 = 3 \, \text{atm} \). The reaction produces gases B and C, which will increase the total pressure in the vessel.

### Step 2: Calculate the Change in Pressure
After 10 minutes, the pressure is \( P_1 = 6.75 \, \text{atm} \). After 20 minutes, the pressure is \( P_2 = 10.5 \, \text{atm} \).

To find the change in pressure:
- After 10 minutes:
\[ \Delta P_{10} = P_1 - P_0 = 6.75 \, \text{atm} - 3 \, \text{atm} = 3.75 \, \text{atm} \]
- After 20 minutes:
\[ \Delta P_{20} = P_2 - P_0 = 10.5 \, \text{atm} - 3 \, \text{atm} = 7.5 \, \text{atm} \]

### Step 3: Relate Pressure Change to Reaction Progress
Let \( x \) be the change in moles of A that reacted. According to the stoichiometry of the reaction:
- For every 2 moles of A that react, 3 moles of B and 4 moles of C are produced.
- Therefore, the change in pressure due to the reaction can be expressed as:
\[ \Delta P = \frac{5}{2} x \]
This is because 2 moles of A produce a total of 7 moles of products (3 moles of B and 4 moles of C), leading to a net increase of 5 moles.

### Step 4: Set Up the Equations
From the pressure changes:
- After 10 minutes:
\[ 3.75 = \frac{5}{2} x_{10} \implies x_{10} = \frac{3.75 \times 2}{5} = 1.5 \]
- After 20 minutes:
\[ 7.5 = \frac{5}{2} x_{20} \implies x_{20} = \frac{7.5 \times 2}{5} = 3.0 \]

### Step 5: Calculate the Average Rate of Reaction
The average rate of reaction can be calculated as:
- For the first 10 minutes:
\[ \text{Rate}_{10} = \frac{x_{10}}{10} = \frac{1.5}{10} = 0.15 \, \text{atm/min} \]
- For the next 10 minutes (from 10 to 20 minutes):
\[ \text{Rate}_{20} = \frac{x_{20} - x_{10}}{10} = \frac{3.0 - 1.5}{10} = 0.15 \, \text{atm/min} \]

### Step 6: Determine the Order of Reaction
Since the rate of reaction remains constant over time, this suggests that the reaction is zero-order. For a zero-order reaction, the rate is independent of the concentration of the reactants.

### Conclusion
The order of the reaction is **zero-order**.

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