To determine which statements are correct regarding chemical kinetics, 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.
**Explanation:**
The relationship between the equilibrium constant (Kp) and temperature (T) can be described by the van 't Hoff equation:
\[ \ln K_p = -\frac{\Delta H}{R} \cdot \frac{1}{T} + C \]
When we take the logarithm (base 10) of Kp, we can express it as:
\[ \log K_p = -\frac{\Delta H}{2.303R} \cdot \frac{1}{T} + \text{constant} \]
This equation is in the form of \( y = mx + c \), where \( y = \log K_p \), \( x = \frac{1}{T} \), and the slope \( m = -\frac{\Delta H}{2.303R} \). Thus, the plot of log Kp versus 1/T is indeed linear.
**Conclusion:** This statement is correct.
### Step 2: Analyze the second statement
**Statement:** A plot of log x versus time is linear for the first-order reaction x → p.
**Explanation:**
For a first-order reaction, the integrated rate law is given by:
\[ \ln [A] = \ln [A_0] - kt \]
Where [A] is the concentration of reactant A at time t, and k is the rate constant. If we convert this to log base 10:
\[ \log [A] = \log [A_0] - \frac{k}{2.303} t \]
This is also in the form of \( y = mx + c \) where \( y = \log [A] \), \( x = t \), and the slope \( m = -\frac{k}{2.303} \). Therefore, the plot of log x versus time is linear.
**Conclusion:** This statement is correct.
### Step 3: Analyze the third statement
**Statement:** A plot of P versus 1/T is linear at constant volume.
**Explanation:**
According to Gay-Lussac's law, for a fixed amount of gas at constant volume, pressure (P) is directly proportional to temperature (T):
\[ P \propto T \]
This can be expressed as:
\[ P = kT \]
If we rearrange this to express P in terms of 1/T, we get:
\[ P = k \cdot T \]
This does not yield a linear relationship when plotted against \( \frac{1}{T} \); instead, it results in a hyperbolic curve, not a straight line.
**Conclusion:** This statement is incorrect.
### Step 4: Analyze the fourth statement
**Statement:** A plot of P versus 1/V is linear at constant temperature.
**Explanation:**
According to Boyle's law, for a fixed amount of gas at constant temperature, pressure (P) is inversely proportional to volume (V):
\[ P \propto \frac{1}{V} \]
This can be expressed as:
\[ P = k \cdot \frac{1}{V} \]
This is indeed a linear relationship, where if we plot P against \( \frac{1}{V} \), we will get a straight line.
**Conclusion:** This statement is correct.
### Final Answer:
The correct statements are:
1. A plot of log Kp versus 1/T is linear.
2. A plot of log x versus time is linear for the first-order reaction x → p.
4. A plot of P versus 1/V is linear at constant temperature.
### Summary of Correct Statements:
- **Correct Statements:** 1, 2, and 4.
