Which amongst the given plots is the correct plot for pressure (p) vs density (d) for an ideal gas ?

Options

A.
B.
C.
D.

Explanation

To determine the correct plot for pressure (p) versus density (d) for an ideal gas, we can start by using the ideal gas law, which is given by the equation:

\[ PV = nRT \]

Where:
- \( P \) = Pressure
- \( V \) = Volume
- \( n \) = Number of moles
- \( R \) = Universal gas constant
- \( T \) = Temperature

We can also express density (\( d \)) in terms of mass (\( m \)) and volume (\( V \)):

\[ d = \frac{m}{V} \]

From the ideal gas law, we can rearrange it to express pressure in terms of density:

1. Substitute \( V \) in terms of density:
\[ V = \frac{m}{d} \]

2. Substitute \( V \) back into the ideal gas law:
\[ P \left(\frac{m}{d}\right) = nRT \]

3. Rearranging gives:
\[ P = \frac{nRTd}{m} \]

4. If we consider \( n \) and \( m \) as constants for a given gas, we can simplify this to:
\[ P = k \cdot d \]
where \( k = \frac{nRT}{m} \) is a constant that depends on the temperature and the specific gas.

This equation shows that pressure is directly proportional to density. Therefore, the graph of pressure (P) versus density (d) will be a straight line passing through the origin with a positive slope.

Now, considering the effect of temperature:
- If the temperature increases, the value of \( k \) increases, which means the slope of the line will be steeper.
- Therefore, for different temperatures \( T_1 < T_2 < T_3 \), the plots will be straight lines through the origin with increasing slopes.

### Conclusion:
The correct plot for pressure vs density for an ideal gas will be a straight line passing through the origin, with the slope increasing with temperature.