Which of the following is the expression of Raoult’s law? (`p^(o)` = vapour pressure of pure solvent, `p^(s)` = vapour pressure of the solution)

Options

A. `(p^(o) - p^(s))/(p^(o)) = (n)/(n+N)`
B. `(p^(s) - p^(o))/(p^(o)) = (N)/(N+n)`
C. `(p^(o) - p^(s))/(p^(o)) = (N)/(N-n)`
D. `(p^(s) - p^(o))/(p^(s)) = (N-n)/(N)`

Answer

To determine the expression of Raoult's law, we can follow these steps:

### Step 1: Understand Raoult's Law
Raoult's law states that the vapor pressure of a solution (Ps) is directly related to the vapor pressure of the pure solvent (P0) and the mole fraction of the solvent in the solution.

### Step 2: Write the Relative Lowering of Vapor Pressure
The relative lowering of vapor pressure can be expressed as:
\[ \text{Relative Lowering} = P_0 - P_s \]
This represents the difference between the vapor pressure of the pure solvent and the vapor pressure of the solution.

### Step 3: Relate Relative Lowering to Mole Fraction of Solute
According to Raoult's law, the relative lowering of vapor pressure is proportional to the mole fraction of the solute (X_s):
\[ P_0 - P_s \propto X_s \]
Where \( X_s \) is the mole fraction of the solute.

### Step 4: Introduce a Constant of Proportionality
To remove the proportionality, we can introduce a constant:
\[ P_0 - P_s = k \cdot X_s \]
Where \( k \) is a constant.

### Step 5: Express Mole Fraction of Solute
The mole fraction of the solute (X_s) can be expressed as:
\[ X_s = \frac{n_s}{n_s + N} \]
Where \( n_s \) is the number of moles of solute and \( N \) is the number of moles of solvent.

### Step 6: Substitute Mole Fraction into the Equation
Substituting the mole fraction into the equation gives:
\[ P_0 - P_s = k \cdot \frac{n_s}{n_s + N} \]

### Step 7: Rearranging the Equation
To find the expression of Raoult's law, we can rearrange this to:
\[ \frac{P_0 - P_s}{P_0} = \frac{n_s}{n_s + N} \]
This shows the relationship between the vapor pressures and the mole fraction of the solute.

### Conclusion
The correct expression of Raoult's law can be summarized as:
\[ \frac{P_0 - P_s}{P_0} = X_s \]