At high pressure suppose all the constant temperature curve varies linearly with pressure according to the following equation : `Z=1+(Pb)/(RT)` (`R=2 cal "mol"^(-1) K^(-1)`)
Plot at Boyle's temperature for the gas will be

Options

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Explanation

To solve the problem, we need to analyze the given equation and understand the implications of Boyle's temperature for a gas. Let's break it down step by step.

### Step 1: Understand the Equation
The equation provided is:
\[ Z = 1 + \frac{Pb}{RT} \]
where:
- \( Z \) is the compressibility factor,
- \( P \) is the pressure,
- \( b \) is a constant specific to the gas,
- \( R = 2 \, \text{cal} \, \text{mol}^{-1} \, \text{K}^{-1} \) is the universal gas constant,
- \( T \) is the temperature.

### Step 2: Define Boyle's Temperature
Boyle's temperature (\( T_B \)) is the temperature at which a real gas behaves ideally. At this temperature, the compressibility factor \( Z \) approaches 1 at high pressures.

### Step 3: Analyze the Equation at Boyle's Temperature
At Boyle's temperature, we can set \( T = T_B \):
\[ Z = 1 + \frac{Pb}{R T_B} \]

At \( P = 0 \) (the ideal gas limit), we have:
\[ Z = 1 \]

As pressure increases, \( Z \) will increase according to the equation. The relationship indicates that as \( P \) increases, \( Z \) will linearly increase due to the term \( \frac{Pb}{RT} \).

### Step 4: Plotting the Graph
To plot the graph of \( Z \) versus \( P \):
1. At \( P = 0 \), \( Z = 1 \).
2. As \( P \) increases, \( Z \) increases linearly based on the equation \( Z = 1 + \frac{Pb}{RT_B} \).

The graph will start at the point (0, 1) and will have a slope of \( \frac{b}{RT_B} \).

### Step 5: Conclusion
The plot at Boyle's temperature will be a straight line starting from the point (0, 1) and increasing linearly with pressure.

### Final Answer
The plot at Boyle's temperature for the gas will be a straight line starting from \( Z = 1 \) at \( P = 0 \) and increasing linearly with pressure.

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