Which of the following diagram correctly describes the behaviour of a fixed mass of an ideal gas ? (T is measured in K)

Options

A.
B.
C.
D.

Answer

To determine which diagram correctly describes the behavior of a fixed mass of an ideal gas, we can analyze the relationships between pressure (P), volume (V), and temperature (T) using the ideal gas law, which is given by:

\[ PV = nRT \]

Where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles of gas (constant for fixed mass)
- \( R \) = ideal gas constant
- \( T \) = temperature in Kelvin

### Step-by-Step Solution:

1. **Understanding the Ideal Gas Law**:
- The ideal gas law states that for a fixed mass of gas, the product of pressure and volume is proportional to the temperature.

2. **Analyzing the First Graph (P vs. V at Constant T)**:
- If temperature (T) is constant, then from the ideal gas equation, we can express it as:
\[ P \propto \frac{1}{V} \]
- This indicates that pressure is inversely proportional to volume, which means the graph of P vs. V should be a hyperbola, not a straight line. Therefore, the first graph is incorrect.

3. **Analyzing the Second Graph (P vs. T at Constant V)**:
- If volume (V) is constant, we can rearrange the ideal gas law to show:
\[ P \propto T \]
- This means that pressure increases linearly with temperature. The graph of P vs. T at constant volume should be a straight line. However, if the graph is not a straight line, this option is incorrect.

4. **Analyzing the Third Graph (PV vs. P at Constant T)**:
- Here, if we consider the product PV at constant temperature, we see:
\[ PV = nRT \]
- Since PV is constant, this means that the graph of PV vs. P should also be constant. If the graph shows any variation, this option is also incorrect.

5. **Analyzing the Fourth Graph (V vs. T at Constant P)**:
- If pressure (P) is constant, we can rearrange the ideal gas law to show:
\[ V \propto T \]
- This indicates a direct linear relationship between volume and temperature. Thus, the graph of V vs. T at constant pressure should be a straight line. If this graph is indeed a straight line, then this option is correct.

### Conclusion:
The correct diagram that describes the behavior of a fixed mass of an ideal gas is the fourth graph, which shows a linear relationship between volume (V) and temperature (T) at constant pressure (P).