Which of the following relationship is correct ?

Options

A. `K_(b)=(RT_(b)^(2))/(1000xxDeltaV_(v))`
B. `K_(b)=(RT_(b)^(2))/(1000L_(v))`
C. `K_(f)=(RT_(f)^(2))/(1000DeltaH_(f))`
D. `K_(f)=(RT_(f)^(2)xxM)/(1000xxL_(f))`

Answer

To determine the correct relationship between the molar elevation constant (Kb) and the molar depression constant (Kf), we need to analyze the definitions and equations associated with each.

### Step-by-Step Solution:

1. **Understanding Kb and Kf**:
- Kb is the molar elevation constant, which is used in the context of boiling point elevation.
- Kf is the molar depression constant, which is used in the context of freezing point depression.

2. **Boiling Point Elevation**:
- The relationship for boiling point elevation is given by:
\[ \Delta T_b = K_b \cdot m \]
- Where:
- \(\Delta T_b\) = change in boiling point
- \(K_b\) = molar elevation constant
- \(m\) = molality of the solution

3. **Freezing Point Depression**:
- The relationship for freezing point depression is given by:
\[ \Delta T_f = K_f \cdot m \]
- Where:
- \(\Delta T_f\) = change in freezing point
- \(K_f\) = molar depression constant
- \(m\) = molality of the solution

4. **Calculating Molar Mass**:
- The molality can be expressed in terms of the mass of solute and solvent:
\[ m = \frac{W_2}{M_2} \cdot \frac{1000}{W_1} \]
- Where:
- \(W_2\) = grams of solute
- \(M_2\) = molecular mass of the solute
- \(W_1\) = grams of solvent

5. **Relating Kb and Kf to Latent Heat**:
- For Kb, it can be derived from the latent heat of vaporization:
\[ K_b = \frac{R T_b^2}{1000 L_v} \]
- For Kf, it can be derived from the latent heat of fusion:
\[ K_f = \frac{R T_f^2}{1000 L_f} \]

6. **Identifying Correct Relationships**:
- The relationships involving \(K_b\) and \(K_f\) must include the respective latent heats and molecular masses.
- The correct relationships are:
- For Kb: \(K_b = \frac{R T_b^2}{1000 L_v}\)
- For Kf: \(K_f = \frac{R T_f^2}{1000 L_f}\)

7. **Conclusion**:
- The correct relationships must include the latent heat and the molecular mass of the solvent. The relationships that do not include these factors are incorrect.