For a homogeneous gaseous reaction `A rarr B + C + D` , the initial pressure was `P_0` white pressure after time 't' was P. if `(P gt P_0)` The expression for the constant K is

Options

A. `K=2.303/tlog.((2P_0)/(3P_0-P))`
B. `K=2.303/tlog.((3P_0)/(2P_0-P))`
C. `K=2.303/tlog.((P_0)/(P_0-P))`
D. `K=2.303/tlog.((P_0)/(4P_0-P))`

Answer

To find the expression for the rate constant \( K \) for the given homogeneous gaseous reaction \( A \rightarrow B + C + D \), we can follow these steps:

### Step 1: Understand the Reaction and Initial Conditions
The reaction involves one reactant \( A \) producing three products \( B \), \( C \), and \( D \). The initial pressure of the system is given as \( P_0 \).

### Step 2: Define the Change in Pressure
Let the change in pressure of \( A \) after time \( t \) be \( P_1 \). Therefore, at time \( t \):
- The pressure of \( A \) will be \( P_0 - P_1 \).
- The pressure of \( B \), \( C \), and \( D \) will each increase by \( P_1 \).

### Step 3: Write the Total Pressure at Time \( t \)
The total pressure \( P \) at time \( t \) can be expressed as:
\[ P = (P_0 - P_1) + P_1 + P_1 + P_1 \]
This simplifies to:
\[ P = P_0 + 2P_1 \]
This is our Equation 1.

### Step 4: Solve for \( P_1 \)
From Equation 1, we can rearrange to find \( P_1 \):
\[ 2P_1 = P - P_0 \quad \Rightarrow \quad P_1 = \frac{P - P_0}{2} \]

### Step 5: Substitute \( P_1 \) into the Expression for \( K \)
The rate constant \( K \) can be expressed in terms of concentrations or pressures. Using the formula:
\[ K = \frac{2.303}{T} \log \left( \frac{[A]_{initial}}{[A]_{t}} \right) \]
In terms of pressure, this becomes:
\[ K = \frac{2.303}{T} \log \left( \frac{P_0}{P_0 - P_1} \right) \]
Substituting \( P_1 \) from Step 4:
\[ K = \frac{2.303}{T} \log \left( \frac{P_0}{P_0 - \frac{P - P_0}{2}} \right) \]

### Step 6: Simplify the Expression
Now, simplify the expression inside the logarithm:
\[ P_0 - \frac{P - P_0}{2} = P_0 - \frac{P}{2} + \frac{P_0}{2} = \frac{3P_0 - P}{2} \]
Thus, we have:
\[ K = \frac{2.303}{T} \log \left( \frac{P_0}{\frac{3P_0 - P}{2}} \right) \]
This can be rewritten as:
\[ K = \frac{2.303}{T} \log \left( \frac{2P_0}{3P_0 - P} \right) \]

### Final Expression for \( K \)
The final expression for the rate constant \( K \) is:
\[ K = \frac{2.303}{T} \log \left( \frac{2P_0}{3P_0 - P} \right) \]

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