For the homogenous gaseous reaction `Ato3B`, if pressure after time t was `P_(t)` and after completion of reaction, pressure was `P_(oo)` then select correct relation

Options

A. `k=(1)/(t)"in"((P_(oo))/(3(P_(oo)-P_(t))))`
B. `k=(1)/(t)"in"((2P_(oo))/((P_(oo)-P_(T))))`
C. `k=(1)/(t)"in"((3P_(oo))/(2P_(oo)-P_(t)))`
D. `k=(1)/(t)"in"((P_(oo))/(3(P_(oo)-P_(T))))`

Answer

To solve the problem regarding the homogeneous gaseous reaction \( A \rightarrow 3B \), we need to establish the relationships between the pressures at different stages of the reaction. Here’s a step-by-step breakdown of the solution:

### Step 1: Define Initial Conditions
- Let the initial pressure of gas A be \( P_0 \).
- At the start of the reaction (when \( t = 0 \)), the pressure of B is \( 0 \).

### Step 2: Establish Pressure Changes Over Time
- As the reaction progresses, let \( P \) be the change in pressure due to the conversion of A to B.
- At time \( t \), the pressure of A will be \( P_0 - P \).
- The pressure of B will be \( 3P \) (since 1 mole of A produces 3 moles of B).

### Step 3: Total Pressure at Time \( t \)
- The total pressure at time \( t \) can be expressed as:
\[ P_t = (P_0 - P) + 3P = P_0 + 2P \]

### Step 4: Pressure at Completion of Reaction
- When the reaction is complete (as \( t \rightarrow \infty \)), all of A is converted to B.
- Thus, the pressure at completion \( P_\infty \) will be:
\[ P_\infty = 3P_0 \]

### Step 5: Relate \( P \) and \( P_\infty \)
- From the expression for \( P_t \), we can rearrange to find \( P \):
\[ P = \frac{P_t - P_0}{2} \]

### Step 6: Substitute \( P \) in Terms of \( P_\infty \)
- Since \( P_\infty = 3P_0 \), we can express \( P_0 \) as:
\[ P_0 = \frac{P_\infty}{3} \]

### Step 7: Substitute \( P_0 \) into the Equation for \( P \)
- Substitute \( P_0 \) into the equation for \( P \):
\[ P = \frac{P_t - \frac{P_\infty}{3}}{2} \]

### Step 8: Substitute \( P \) back into the expression for \( P_t \)
- Now we can substitute \( P \) back into the equation for \( P_t \):
\[ P_t = P_0 + 2\left(\frac{P_t - \frac{P_\infty}{3}}{2}\right) \]
- This simplifies to:
\[ P_t = P_0 + P_t - \frac{P_\infty}{3} \]
- Rearranging gives us:
\[ P_t = P_0 + P_t - \frac{P_\infty}{3} \]

### Step 9: Final Relation
- After simplification, we find the relationship:
\[ P_t = \frac{2P_\infty}{3} \]

### Summary of the Solution
The correct relation derived from the problem is:
\[ P_t = \frac{2P_\infty}{3} \]