Sure, here is a detailed, step-by-step solution to find the molarity of the solution:
### Step 1: Identify the given data
1. Mass of ethanol ([tex]\(C_2H_6O\)[/tex]): 50.1 grams
2. Volume of solution: 2.04 liters
3. Molar mass of ethanol ([tex]\(C_2H_6O\)[/tex]): 46.07 grams/mole
### Step 2: Calculate the number of moles of ethanol
To determine the number of moles of ethanol, use the formula:
[tex]\[
\text{{Number of moles}} = \frac{{\text{{mass}}}}{{\text{{molar mass}}}}
\][/tex]
Plugging in the values:
[tex]\[
\text{{Number of moles}} = \frac{{50.1 \text{ grams}}}{{46.07 \text{ grams/mole}}} \approx 1.087 \text{ moles}
\][/tex]
### Step 3: Calculate the molarity of the solution
Molarity ([tex]\(M\)[/tex]) is defined as the number of moles of solute per liter of solution. The formula for molarity is:
[tex]\[
M = \frac{{\text{{number of moles}}}}{{\text{{volume of solution in liters}}}}
\][/tex]
Substituting in the values:
[tex]\[
M = \frac{{1.087 \text{ moles}}}{{2.04 \text{ liters}}} \approx 0.533 \enspace \text{moles/liter}
\][/tex]
### Step 4: Express the molarity to three significant figures
From the calculations above, the molarity of the ethanol solution can be expressed as:
[tex]\[
M = 0.533 \enspace \text{moles per liter}
\][/tex]
In summary, the molarity of the solution is [tex]\(0.533 \text{ M}\)[/tex], rounded to three significant figures.
