An aqueous solution containing 4.9 g of solute dissolved in 500 mL of the soluiton shows an osmotic pressure of 2.1 atmosphere at `27^(@)C` . What is the nature of the solute ( associated or dissociated , if the molar mass of solute is 57a.m.u) ?

Explanation

To solve the problem, we need to determine the nature of the solute based on the given data about its osmotic pressure, mass, and molar mass. Here’s a step-by-step solution:

### Step 1: Convert the given temperature to Kelvin
The temperature is given as 27°C. To convert it to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273.15 \]
So,
\[ T = 27 + 273.15 = 300.15 \, K \approx 300 \, K \]

**Hint:** Always convert Celsius to Kelvin when dealing with gas laws and osmotic pressure calculations.

### Step 2: Calculate the concentration (C) of the solution
The osmotic pressure (\(\Pi\)) is related to the concentration (C) of the solution by the formula:
\[ \Pi = C \cdot R \cdot T \]
Where:
- \(\Pi\) = osmotic pressure = 2.1 atm
- \(R\) = universal gas constant = 0.0821 L·atm/(K·mol)
- \(T\) = temperature in Kelvin = 300 K

Rearranging the formula to find concentration (C):
\[ C = \frac{\Pi}{R \cdot T} \]
Substituting the values:
\[ C = \frac{2.1}{0.0821 \cdot 300} \]
Calculating:
\[ C = \frac{2.1}{24.63} \approx 0.0853 \, \text{mol/L} \]

**Hint:** Remember that concentration is expressed in moles per liter (mol/L).

### Step 3: Calculate the number of moles of solute
We know the concentration and the volume of the solution. The number of moles (n) can be calculated using:
\[ n = C \cdot V \]
Where:
- \(V\) = volume in liters = 0.5 L

Substituting the values:
\[ n = 0.0853 \cdot 0.5 = 0.04265 \, \text{mol} \]

**Hint:** Always convert volume from mL to L when calculating moles.

### Step 4: Calculate the molar mass (M) of the solute
The molar mass can be calculated using the formula:
\[ M = \frac{m}{n} \]
Where:
- \(m\) = mass of solute = 4.9 g
- \(n\) = number of moles = 0.04265 mol

Substituting the values:
\[ M = \frac{4.9}{0.04265} \approx 114.8 \, \text{g/mol} \]

**Hint:** Molar mass is calculated by dividing the mass of the solute by the number of moles.

### Step 5: Compare the calculated molar mass with the given molar mass
The given molar mass of the solute is 57 g/mol. We calculated the molar mass to be approximately 114.8 g/mol. Since the calculated molar mass (114.8 g/mol) is significantly higher than the given molar mass (57 g/mol), this indicates that the solute is associating in solution.

### Conclusion
Since the calculated molar mass is greater than the given molar mass, it suggests that the solute is associated in solution.

**Final Answer:** The nature of the solute is associated.