A solution is prepared by adding 64 g of `CH_(3)OH` to 180 g of water. Calculate the mole fraction of `CH_(3)OH`. (Molar mass of `CH_(3)OH = 32 g`) .

Answer

To calculate the mole fraction of `CH₃OH` in the solution prepared by adding 64 g of `CH₃OH` to 180 g of water, we will follow these steps:

### Step 1: Calculate the moles of `CH₃OH`
The formula to calculate the number of moles is:
\[ \text{Moles} = \frac{\text{Mass (g)}}{\text{Molar Mass (g/mol)}} \]
Given:
- Mass of `CH₃OH` = 64 g
- Molar mass of `CH₃OH` = 32 g/mol

Calculating the moles of `CH₃OH`:
\[ \text{Moles of } CH₃OH = \frac{64 \, \text{g}}{32 \, \text{g/mol}} = 2 \, \text{moles} \]

### Step 2: Calculate the moles of water (`H₂O`)
The molar mass of water (`H₂O`) is 18 g/mol.

Given:
- Mass of water = 180 g

Calculating the moles of water:
\[ \text{Moles of } H₂O = \frac{180 \, \text{g}}{18 \, \text{g/mol}} = 10 \, \text{moles} \]

### Step 3: Calculate the total moles in the solution
The total moles in the solution is the sum of the moles of `CH₃OH` and the moles of water:
\[ \text{Total moles} = \text{Moles of } CH₃OH + \text{Moles of } H₂O = 2 \, \text{moles} + 10 \, \text{moles} = 12 \, \text{moles} \]

### Step 4: Calculate the mole fraction of `CH₃OH`
The mole fraction of a component in a solution is given by the formula:
\[ \text{Mole Fraction} = \frac{\text{Moles of the component}}{\text{Total moles of the solution}} \]

Calculating the mole fraction of `CH₃OH`:
\[ \text{Mole Fraction of } CH₃OH = \frac{2 \, \text{moles}}{12 \, \text{moles}} = \frac{1}{6} \approx 0.1667 \]

### Final Answer
The mole fraction of `CH₃OH` in the solution is approximately **0.1667**.

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