500 gm of urea solution of mole fraction 0.2 is diluted to 1500 gm. Calculate the mole fractions of solute in the diluted solution ?

Answer

To solve the problem of calculating the mole fraction of urea in a diluted solution, we can follow these steps:

### Step 1: Determine the mass of urea in the initial solution
Given that the mole fraction of urea in the 500 g solution is 0.2, we can use the formula for mole fraction:

\[ \text{Mole fraction} = \frac{\text{moles of solute}}{\text{moles of solute} + \text{moles of solvent}} \]

Let \( x \) be the mass of urea in grams. The mass of water in the solution is \( 500 - x \) grams.

### Step 2: Calculate moles of urea and water
The molecular weight of urea (NH₂CONH₂) is approximately 60 g/mol, and the molecular weight of water (H₂O) is approximately 18 g/mol.

- Moles of urea:
\[ \text{Moles of urea} = \frac{x}{60} \]

- Moles of water:
\[ \text{Moles of water} = \frac{500 - x}{18} \]

### Step 3: Set up the equation using the mole fraction
Substituting into the mole fraction equation:
\[ 0.2 = \frac{\frac{x}{60}}{\frac{x}{60} + \frac{500 - x}{18}} \]

### Step 4: Solve for \( x \)
Cross-multiplying gives:
\[ 0.2 \left( \frac{x}{60} + \frac{500 - x}{18} \right) = \frac{x}{60} \]

Expanding and simplifying:
\[ 0.2 \cdot \frac{x}{60} + 0.2 \cdot \frac{500 - x}{18} = \frac{x}{60} \]

Multiply through by 60 to eliminate the denominator:
\[ 0.2x + \frac{6000 - 120x}{18} = x \]

Multiply through by 18 to eliminate the fraction:
\[ 3.6x + 6000 - 120x = 18x \]

Combine like terms:
\[ 6000 = 18x + 120x - 3.6x \]
\[ 6000 = 134.4x \]
\[ x = \frac{6000}{134.4} \approx 44.6 \text{ grams} \]

### Step 5: Calculate the mass of water
The mass of water is:
\[ 500 - x = 500 - 44.6 = 455.4 \text{ grams} \]

### Step 6: Dilute the solution
The solution is diluted to 1500 grams. Therefore, the mass of water added is:
\[ 1500 - 500 = 1000 \text{ grams} \]

### Step 7: Calculate the new mass of water
The new mass of water after dilution:
\[ 455.4 + 1000 = 1455.4 \text{ grams} \]

### Step 8: Calculate the new mole fraction of urea
Now, we can calculate the mole fraction of urea in the diluted solution:
- Moles of urea:
\[ \text{Moles of urea} = \frac{44.6}{60} \approx 0.7433 \text{ moles} \]

- Moles of water:
\[ \text{Moles of water} = \frac{1455.4}{18} \approx 80.8667 \text{ moles} \]

### Step 9: Calculate the mole fraction
The mole fraction of urea in the diluted solution is:
\[ \text{Mole fraction of urea} = \frac{0.7433}{0.7433 + 80.8667} \approx \frac{0.7433}{81.61} \approx 0.0091 \]

### Final Answer
The mole fraction of urea in the diluted solution is approximately **0.0091**.

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