The following statement (s) is (are) correct ?

Options

A. A plot of log `K_p` vs. 1/T is linear
B. A plot of log [x] vs. time is linear for a first order reaction `X to P`
C. A plot of log P vs . 1/T is linear at constant volume
D. A plot of P vs . 1/V is linear at constant temperature

Answer

To determine which of the given statements regarding chemical equilibrium are correct, we will analyze each statement step by step.

### Step 1: Analyze the first statement
**Statement:** A plot of log Kp versus 1/T is linear.

**Solution:**
1. We start with the Arrhenius equation:
\[ K = A \cdot e^{-\frac{E_a}{RT}} \]
2. Taking the natural logarithm of both sides gives:
\[ \ln K = \ln A - \frac{E_a}{RT} \]
3. Rearranging this, we can express it as:
\[ \ln K = \ln A - \frac{E_a}{R} \cdot \frac{1}{T} \]
4. If we convert this to base 10 logarithm:
\[ \log K = \log A - \frac{E_a}{2.303R} \cdot \frac{1}{T} \]
5. This is in the form of \( Y = C - MX \), which represents a straight line where:
- \( Y = \log K \)
- \( X = \frac{1}{T} \)
- \( C = \log A \)
- \( M = \frac{E_a}{2.303R} \)

**Conclusion:** The statement is correct.

### Step 2: Analyze the second statement
**Statement:** A plot of log X versus time is linear for a first-order reaction.

**Solution:**
1. For a first-order reaction, the integrated rate law is:
\[ \ln [A_t] = \ln [A_0] - kt \]
2. If we let \( X = [A_0] - [A_t] \), then:
\[ \ln [A_t] = \ln [A_0] - kt \implies \log [A_t] = \log [A_0] - \frac{k}{2.303} t \]
3. Rearranging gives:
\[ \log [A_t] = \log [A_0] - \frac{k}{2.303} t \]
4. This is again in the form of \( Y = C - MX \), indicating a straight line.

**Conclusion:** The statement is correct.

### Step 3: Analyze the third statement
**Statement:** A plot of log P versus 1/T at constant volume is linear.

**Solution:**
1. From the ideal gas law, we have:
\[ PV = nRT \implies P = \frac{nRT}{V} \]
2. If V is constant, then:
\[ P = kT \quad (where \, k = \frac{nR}{V}) \]
3. Taking logarithms:
\[ \log P = \log k + \log T \]
4. This implies:
\[ \log P = \log k + \log T \implies \log P = \log k + \log(1/T) \]
5. This does not represent a straight line; it represents a logarithmic relationship.

**Conclusion:** The statement is incorrect.

### Step 4: Analyze the fourth statement
**Statement:** A plot of P versus 1/V is linear at constant temperature.

**Solution:**
1. From the ideal gas law:
\[ PV = nRT \implies P = \frac{nRT}{V} \]
2. Rearranging gives:
\[ P = k \cdot \frac{1}{V} \quad (where \, k = nRT) \]
3. This is in the form \( Y = MX \), indicating a linear relationship.

**Conclusion:** The statement is correct.

### Final Conclusion
The correct statements are:
- Statement 1: Correct
- Statement 2: Correct
- Statement 3: Incorrect
- Statement 4: Correct

Thus, the final answer is that statements A, B, and D are correct.