For a first order homogenous gaseous reaction, `Ato2B+C`
then initial pressure was `P_(i)` while total pressure after time 't' was `P_(t)`. The right expression for the rate constants k in terms of `P_(i),P_(t)` and t is :

Options

A. `k=(2.303)/(t)"log"((2P_(i))/(3P_(i)-P_(t)))`
B. `k=(2.303)/(t)"log"((2P_(i))/(2P_(t)-P_(i)))`
C. `k=(2.303)/(t)"log"((P_(i))/(P_(i)-P_(t)))`
D. None of these

Answer

To derive the expression for the rate constant \( k \) in terms of the initial pressure \( P_i \), total pressure at time \( t \) \( P_t \), and time \( t \) for the reaction \( A \rightarrow 2B + C \), we can follow these steps:

### Step 1: Understand the Reaction Stoichiometry
The reaction shows that for every 1 mole of \( A \) that reacts, 2 moles of \( B \) and 1 mole of \( C \) are produced.

### Step 2: Set Up Initial Conditions
At time \( t = 0 \):
- The initial pressure of \( A \) is \( P_i \).
- The initial pressure of products \( B \) and \( C \) is 0.

### Step 3: Define Changes in Pressure
Let \( P \) be the change in pressure of \( A \) that has reacted at time \( t \). Thus:
- The pressure of \( A \) at time \( t \) will be \( P_i - P \).
- The pressure of \( B \) will be \( 2P \) (since 2 moles of \( B \) are produced for every mole of \( A \)).
- The pressure of \( C \) will be \( P \) (since 1 mole of \( C \) is produced for every mole of \( A \)).

### Step 4: Write the Total Pressure Expression
The total pressure at time \( t \) can be expressed as:
\[ P_t = (P_i - P) + 2P + P \]
This simplifies to:
\[ P_t = P_i + 2P \]

### Step 5: Solve for \( P \)
Rearranging the equation gives:
\[ P_t - P_i = 2P \implies P = \frac{P_t - P_i}{2} \]

### Step 6: Substitute \( P \) into the Rate Constant Expression
For a first-order reaction, the rate constant \( k \) is given by:
\[ k = \frac{2.303}{t} \log \left( \frac{P_i}{P_i - P} \right) \]
Substituting \( P \) from the previous step:
\[ k = \frac{2.303}{t} \log \left( \frac{P_i}{P_i - \frac{P_t - P_i}{2}} \right) \]

### Step 7: Simplify the Logarithmic Expression
Now, simplify the expression inside the logarithm:
\[ P_i - \frac{P_t - P_i}{2} = P_i - \frac{P_t}{2} + \frac{P_i}{2} = \frac{3P_i - P_t}{2} \]
Thus, the expression for \( k \) becomes:
\[ k = \frac{2.303}{t} \log \left( \frac{P_i}{\frac{3P_i - P_t}{2}} \right) \]

### Step 8: Final Expression for \( k \)
This can be rewritten as:
\[ k = \frac{2.303}{t} \log \left( \frac{2P_i}{3P_i - P_t} \right) \]

### Final Answer
The expression for the rate constant \( k \) in terms of \( P_i \), \( P_t \), and \( t \) is:
\[ k = \frac{2.303}{t} \log \left( \frac{2P_i}{3P_i - P_t} \right) \]