A graph plotted between `(P)/(d) vs d` (where `p` is osmotic pressure of solution of a solute of mol. Wt. `m` and `d` is its density temperature `T`. Pick out the correct statements about the plots :

Options

A. `[(P)/(d)]_(drarr0)=(ST)/(m)`
B. The intercept of the plot `(ST)/(m)`
C. The intercept of the plot `=[(P)/(d)]_(drarr0)`
D. `[(P)/(d)]_(drarr0)` is independent of temperature

Explanation

To solve the problem, we need to analyze the relationship between osmotic pressure (P), density (d), and the molar mass (m) of a solute in a solution. We will derive the relationship and then interpret the graph plotted between \( \frac{P}{d} \) and \( d \).

### Step-by-Step Solution:

1. **Understanding Osmotic Pressure**:
The osmotic pressure (\( \Pi \)) of a solution is given by the formula:
\[ \Pi = CRT \]
where \( C \) is the concentration of the solute, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

2. **Relating Concentration to Density**:
The concentration \( C \) can be expressed in terms of density \( d \) and molar mass \( m \):
\[ C = \frac{n}{V} = \frac{m}{M \cdot V} \]
where \( n \) is the number of moles, \( V \) is the volume, and \( M \) is the molar mass of the solute.

3. **Using the Ideal Gas Law**:
From the ideal gas law, we have:
\[ PV = nRT \]
Rearranging gives us:
\[ P = \frac{nRT}{V} \]

4. **Substituting for Density**:
We know that density \( d \) is defined as:
\[ d = \frac{m}{V} \]
Therefore, we can express \( V \) in terms of \( d \):
\[ V = \frac{m}{d} \]

5. **Substituting Back into the Pressure Equation**:
Substituting this expression for \( V \) back into the equation for \( P \):
\[ P = \frac{nRT}{\frac{m}{d}} = \frac{nRTd}{m} \]

6. **Expressing \( \frac{P}{d} \)**:
Now, we can express \( \frac{P}{d} \):
\[ \frac{P}{d} = \frac{nRT}{m} \]
This shows that \( \frac{P}{d} \) is directly proportional to the temperature \( T \) and inversely proportional to the molar mass \( m \).

7. **Analyzing the Graph**:
When we plot \( \frac{P}{d} \) against \( d \), we can see that as \( d \) increases, \( \frac{P}{d} \) remains constant for a given temperature \( T \) and molar mass \( m \). The intercept of this plot will be related to \( \frac{RT}{m} \).

8. **Conclusion**:
Based on the analysis, we can conclude that:
- The intercept of the plot is \( \frac{RT}{m} \).
- The relationship shows that \( \frac{P}{d} \) is dependent on temperature and molar mass.

### Correct Statements:
- The intercept of the plot is \( \frac{RT}{m} \).
- The plot indicates that \( \frac{P}{d} \) is directly proportional to temperature.