Which of the following curves obtained at constant temperature and for a given mass of a gas is not a straight line ?

Options

A. P vs V
B. P vs `( 1//V )`
C. V vs ( `1//P ` )
D. (PV ) vs P

Answer

To solve the question of which curve obtained at constant temperature and for a given mass of gas is not a straight line, we will analyze each option based on the ideal gas law and the relationships between pressure (P), volume (V), and temperature (T).

### Step-by-Step Solution:

1. **Understand the Ideal Gas Law**:
The ideal gas law is given by the equation:
\[ PV = nRT \]
where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature. For a given mass of gas at constant temperature, \(n\), \(R\), and \(T\) are constants.

2. **Rearranging the Ideal Gas Law**:
Since \(PV = nRT\) is constant, we can express this relationship in different forms:
- \(P = \frac{nRT}{V}\) (Pressure vs Volume)
- \(PV = \text{constant}\)

3. **Analyzing Each Option**:
- **Option 1: \(P\) vs \(V\)**:
From \(PV = \text{constant}\), if we plot \(P\) against \(V\), we get:
\[ P = \frac{k}{V} \quad \text{(where \(k = nRT\))} \]
This is a hyperbolic curve, not a straight line.

- **Option 2: \(P\) vs \(\frac{1}{V}\)**:
If we plot \(P\) against \(\frac{1}{V}\):
\[ P = k \cdot \frac{1}{V} \]
This is a straight line passing through the origin.

- **Option 3: \(V\) vs \(\frac{1}{P}\)**:
If we plot \(V\) against \(\frac{1}{P}\):
\[ V = k \cdot \frac{1}{P} \]
This is also a straight line passing through the origin.

- **Option 4: \(PV\) vs \(P\)**:
Here, we have \(PV = \text{constant}\). If we plot \(PV\) against \(P\):
Since \(PV\) is constant, this will yield a horizontal line (constant value) on the graph.

4. **Conclusion**:
The only curve that is not a straight line is the one corresponding to **Option 1: \(P\) vs \(V\)**, which is a hyperbola.

### Final Answer:
The correct answer is **Option 1: \(P\) vs \(V\)** is not a straight line.