To solve the problem step by step, we will calculate the molarity and mole fraction of ethanol after diluting it with water.
### Step 1: Calculate the mass of ethanol
Given:
- Volume of ethanol = 100 mL
- Density of ethanol = 0.8 g/cc
To find the mass of ethanol, we use the formula:
\[
\text{Mass} = \text{Volume} \times \text{Density}
\]
Converting 100 mL to liters (1 mL = 1 cc):
\[
\text{Mass of ethanol} = 100 \, \text{mL} \times 0.8 \, \text{g/mL} = 80 \, \text{g}
\]
### Step 2: Calculate the number of moles of ethanol
The molecular mass of ethanol (C₂H₅OH) is calculated as follows:
- Carbon (C): 2 × 12 g/mol = 24 g/mol
- Hydrogen (H): 6 × 1 g/mol = 6 g/mol
- Oxygen (O): 1 × 16 g/mol = 16 g/mol
\[
\text{Molecular mass of ethanol} = 24 + 6 + 16 = 46 \, \text{g/mol}
\]
Now, we can calculate the number of moles of ethanol:
\[
\text{Moles of ethanol} = \frac{\text{Mass}}{\text{Molecular mass}} = \frac{80 \, \text{g}}{46 \, \text{g/mol}} \approx 1.74 \, \text{mol}
\]
### Step 3: Calculate the mass of water
The total volume after dilution is 1 L (1000 mL). The volume of water used is:
\[
\text{Volume of water} = 1000 \, \text{mL} - 100 \, \text{mL} = 900 \, \text{mL}
\]
Since the density of water is 1 g/cc, the mass of water is:
\[
\text{Mass of water} = 900 \, \text{mL} \times 1 \, \text{g/mL} = 900 \, \text{g}
\]
### Step 4: Calculate the number of moles of water
The molecular mass of water (H₂O) is:
- Hydrogen (H): 2 × 1 g/mol = 2 g/mol
- Oxygen (O): 1 × 16 g/mol = 16 g/mol
\[
\text{Molecular mass of water} = 2 + 16 = 18 \, \text{g/mol}
\]
Now, we can calculate the number of moles of water:
\[
\text{Moles of water} = \frac{\text{Mass}}{\text{Molecular mass}} = \frac{900 \, \text{g}}{18 \, \text{g/mol}} = 50 \, \text{mol}
\]
### Step 5: Calculate the total number of moles
\[
\text{Total moles} = \text{Moles of ethanol} + \text{Moles of water} = 1.74 + 50 = 51.74 \, \text{mol}
\]
### Step 6: Calculate the molarity of the solution
Molarity (M) is defined as the number of moles of solute per liter of solution:
\[
\text{Molarity} = \frac{\text{Moles of ethanol}}{\text{Volume of solution in L}} = \frac{1.74 \, \text{mol}}{1 \, \text{L}} = 1.74 \, \text{M}
\]
### Step 7: Calculate the mole fraction of ethanol
The mole fraction (X) of ethanol is given by:
\[
X_{\text{ethanol}} = \frac{\text{Moles of ethanol}}{\text{Total moles}} = \frac{1.74}{51.74} \approx 0.0337 \approx 0.03
\]
### Final Answers:
(a) Molarity of ethanol = 1.74 M
(b) Mole fraction of ethanol = 0.03
