The expression relating molality (m) and mole fraction `(x_(2))` of solute in a solution is :

Options

A. `x_(2)=(mM_(1))/(1+mM_(1))`
B. `x_(2)=(mM_(1))/(1-mM_(1))`
C. `x_(2)=(1+mM_(1))/(mM_(1))`
D. `x_(2)=(1-mM_(1))/(mM_(1))`

Answer

To derive the expression relating molality (m) and mole fraction (x₂) of the solute in a solution, we can follow these steps:

### Step 1: Define the terms
- Let \( n_2 \) be the number of moles of the solute.
- Let \( n_1 \) be the number of moles of the solvent.
- The mole fraction of the solute \( x_2 \) can be expressed as:
\[ x_2 = \frac{n_2}{n_1 + n_2} \]
- The mole fraction of the solvent \( x_1 \) is:
\[ x_1 = \frac{n_1}{n_1 + n_2} \]

### Step 2: Relate moles to mass
- The number of moles can be expressed in terms of mass and molar mass:
\[ n_2 = \frac{m_2}{M_2} \quad \text{(for solute)} \]
\[ n_1 = \frac{m_1}{M_1} \quad \text{(for solvent)} \]
where \( m_2 \) is the mass of the solute, \( M_2 \) is the molar mass of the solute, \( m_1 \) is the mass of the solvent, and \( M_1 \) is the molar mass of the solvent.

### Step 3: Express molality
- Molality \( m \) is defined as the number of moles of solute per kilogram of solvent:
\[ m = \frac{n_2}{m_1 \text{ (in kg)}} \]
Substituting for \( n_2 \):
\[ m = \frac{m_2/M_2}{m_1/1000} \]
Rearranging gives:
\[ m = \frac{1000 \cdot m_2}{M_2 \cdot m_1} \]

### Step 4: Substitute for \( n_1 \) and \( n_2 \)
- Substitute \( n_2 \) and \( n_1 \) in terms of their respective masses into the mole fraction equation:
\[ x_2 = \frac{n_2}{n_1 + n_2} = \frac{m_2/M_2}{(m_1/M_1) + (m_2/M_2)} \]

### Step 5: Rearranging to find the relationship
- To express \( x_2 \) in terms of \( m \):
\[ x_2 = \frac{m_2}{m_2 + \frac{M_1}{1000} \cdot m} \]
By substituting \( m_1 \) with \( 1000 \) grams (1 kg) for simplicity, we can simplify the expression.

### Step 6: Final expression
- After simplification, we can derive that:
\[ x_2 = \frac{m \cdot M_1}{1000 + m \cdot M_1} \]

This expression relates the molality \( m \) of the solution to the mole fraction \( x_2 \) of the solute.