For a first order reaction `A(g)rarr2B(g)+C(g)` at constant volume and 300 K, the total pressure at the beginning (t=0) and at time t ae `P_(0) and P_(t)` respectively. Initially, only A is present with concentration `[A]_(0) and t_(1//3)` is the time required for the partial pressure of A to reach 1/3rd of its initial value. The correct options (s) is (are) :
(Assume that all these behave as ideal gases)

Options

A.
B.
C.
D.

Explanation

To solve the problem, we will analyze the first-order reaction given and derive the necessary relationships step by step.

### Step 1: Understand the Reaction
The reaction is given as:
\[ A(g) \rightarrow 2B(g) + C(g) \]
Initially, only A is present with an initial pressure \( P_0 \).

### Step 2: Define Initial Conditions
At \( t = 0 \):
- Partial pressure of A, \( P_A(0) = P_0 \)
- Partial pressures of B and C are both \( 0 \).

### Step 3: Define Conditions at Time \( t \)
At time \( t \):
- Let \( x \) be the change in pressure due to the reaction.
- The partial pressure of A will be \( P_A(t) = P_0 - x \).
- The partial pressure of B will be \( P_B(t) = 2x \) (since 2 moles of B are produced).
- The partial pressure of C will be \( P_C(t) = x \).

### Step 4: Total Pressure at Time \( t \)
The total pressure at time \( t \) is given by:
\[ P_t = P_A(t) + P_B(t) + P_C(t) = (P_0 - x) + 2x + x = P_0 + 2x \]

### Step 5: Condition for \( t_{1/3} \)
The time \( t_{1/3} \) is defined as the time required for the partial pressure of A to reach \( \frac{1}{3} P_0 \):
\[ P_A(t_{1/3}) = P_0 - x = \frac{1}{3} P_0 \]
From this, we can find \( x \):
\[ P_0 - x = \frac{1}{3} P_0 \]
\[ x = P_0 - \frac{1}{3} P_0 = \frac{2}{3} P_0 \]

### Step 6: Substitute \( x \) into Total Pressure Equation
Substituting \( x \) into the total pressure equation:
\[ P_t = P_0 + 2x = P_0 + 2 \left( \frac{2}{3} P_0 \right) = P_0 + \frac{4}{3} P_0 = \frac{7}{3} P_0 \]

### Step 7: Relate to Rate Constant
For a first-order reaction, the rate constant \( k \) can be expressed as:
\[ k = \frac{1}{t} \ln \left( \frac{P_0}{P_0 - x} \right) \]
Using \( x = \frac{2}{3} P_0 \):
\[ k = \frac{1}{t_{1/3}} \ln \left( \frac{P_0}{\frac{1}{3} P_0} \right) = \frac{1}{t_{1/3}} \ln(3) \]

### Step 8: Conclusion
The options that do not depend on concentration are correct. The correct options are those that reflect the derived relationships, specifically those that are constants and do not vary with concentration.