15g urea and 20g NaOH dissolved in water. Total mass in solution is 250g. Mole fraction of NaOH in the mixture.

Options

A. 0.039
B. 0.62
C. 0.5
D. 0.4

Answer

To find the mole fraction of NaOH in the solution, we will follow these steps:

### Step 1: Calculate the number of moles of NaOH
The formula to calculate the number of moles is:
\[ \text{Number of moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \]
The molar mass of NaOH (Sodium Hydroxide) is calculated as follows:
- Sodium (Na) = 23 g/mol
- Oxygen (O) = 16 g/mol
- Hydrogen (H) = 1 g/mol

Thus, the molar mass of NaOH = 23 + 16 + 1 = 40 g/mol.

Given mass of NaOH = 20 g,
\[ \text{Number of moles of NaOH} = \frac{20 \text{ g}}{40 \text{ g/mol}} = 0.5 \text{ moles} \]

### Step 2: Calculate the number of moles of Urea
The molar mass of Urea (NH₂CONH₂) is calculated as follows:
- Nitrogen (N) = 14 g/mol (2 Nitrogens)
- Carbon (C) = 12 g/mol
- Oxygen (O) = 16 g/mol
- Hydrogen (H) = 1 g/mol (4 Hydrogens)

Thus, the molar mass of Urea = (14 * 2) + 12 + 16 + (1 * 4) = 28 + 12 + 16 + 4 = 60 g/mol.

Given mass of Urea = 15 g,
\[ \text{Number of moles of Urea} = \frac{15 \text{ g}}{60 \text{ g/mol}} = 0.25 \text{ moles} \]

### Step 3: Calculate the mass of water in the solution
The total mass of the solution is given as 250 g. The mass of water can be calculated as:
\[ \text{Mass of water} = \text{Total mass} - (\text{Mass of Urea} + \text{Mass of NaOH}) \]
\[ \text{Mass of water} = 250 \text{ g} - (15 \text{ g} + 20 \text{ g}) = 250 \text{ g} - 35 \text{ g} = 215 \text{ g} \]

### Step 4: Calculate the number of moles of water
The molar mass of water (H₂O) is:
- Hydrogen (H) = 1 g/mol (2 Hydrogens)
- Oxygen (O) = 16 g/mol

Thus, the molar mass of water = 2 + 16 = 18 g/mol.

Now, calculate the number of moles of water:
\[ \text{Number of moles of water} = \frac{215 \text{ g}}{18 \text{ g/mol}} \approx 11.94 \text{ moles} \]

### Step 5: Calculate the total number of moles in the solution
\[ \text{Total moles} = \text{Moles of NaOH} + \text{Moles of Urea} + \text{Moles of Water} \]
\[ \text{Total moles} = 0.5 + 0.25 + 11.94 = 12.69 \text{ moles} \]

### Step 6: Calculate the mole fraction of NaOH
The mole fraction of NaOH is given by:
\[ \text{Mole fraction of NaOH} = \frac{\text{Moles of NaOH}}{\text{Total moles}} \]
\[ \text{Mole fraction of NaOH} = \frac{0.5}{12.69} \approx 0.0394 \]

### Final Answer
The mole fraction of NaOH in the mixture is approximately **0.039**.

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