Question

A solution of ethanol ( \({C}_{2} {H}_{5} {OH}\) ) in water is prepared by dissolving 75.0 \({mL}\) of ethanol (density \(=0.79 {g} / {cm}^{3} )\) in enough

Short answer

The molality of ethanol in the solution is approximately 3.024 mol/kg.

Step-by-step solution

Calculate the mass of ethanol

Given the volume of ethanol (75.0 \({mL}\)) and the density of ethanol (0.79 \({g}/{cm}^{3}\)), we can calculate the mass of ethanol. Since 1 \({mL}\) is equal to 1 \({cm}^{3}\), we can use density as follows: \( Mass\_of\_ethanol = Density \times Volume \) \( Mass\_of\_ethanol = 0.79 \frac{g}{cm^{3}} \times 75.0 cm^{3}\) Upon calculating, we find: \( Mass\_of\_ethanol = 59.25 g \)

Calculate the moles of ethanol

Now that we have calculated the mass of ethanol, we need to convert it into moles using the molar mass of ethanol, which is 12.01 (C) + 6*1.01 (H) + 16.00 (O) = 46.07 g/mol. \( Moles\_of\_ethanol = \frac{Mass\_of\_ethanol}{Molar\_mass} \) \( Moles\_of\_ethanol = \frac{59.25 g}{46.07 \frac{g}{mol}} \) Upon calculating, we find: \( Moles\_of\_ethanol = 1.286 mol \)

Calculate the mass of the solvent (water)

Knowing the volume of the solution (500.0 \({mL}\)) and the volume of ethanol (75.0 \({mL}\)), we can find the volume of water: \( Volume\_of\_water = Volume\_of\_solution - Volume\_of\_ethanol \) \( Volume\_of\_water = 500.0 mL - 75.0 mL \) \( Volume\_of\_water = 425 mL \) Next, we use the density of water (1.00 \({g}/{cm}^{3}\)) to calculate the mass of water: \( Mass\_of\_water = Density \times Volume\_of\_water \) \( Mass\_of\_water = 1.00 \frac{g}{cm^{3}} \times 425 cm^{3}\) Upon calculating, we find: \( Mass\_of\_water = 425 g \)

Calculate the molality of the solution

Molality is defined as the number of moles of solute (ethanol) per kilogram of solvent (water). We can calculate molality using the moles of ethanol and mass of water from the previous steps: \( Molality = \frac{Moles\_of\_ethanol}{Mass\_of\_water (in\, kg)} \) \( Molality = \frac{1.286 mol}{0.425 kg} \) Upon calculating, we find: \( Molality = 3.024 mol/kg \) Hence, the molality of ethanol in the solution is approximately 3.024 mol/kg.

Key concepts

Mass Calculation

Mass calculation is fundamental when preparing solutions because it allows us to quantify the amount of a substance present. For ethanol, you begin with its volume, which is a fairly common way to start since liquids are easily measured in terms of volume.
Given the volume, you can then use the substance's density to calculate its mass. The formula you'll use is straightforward:

• Mass = Density × Volume

This formula tells us that by multiplying a liquid's density (in g/cm³) with its volume (in cm³), the result is the mass in grams. For example, ethanol with a density of 0.79 g/cm³ and a volume of 75.0 mL results in a mass of 59.25 grams. Remember, 1 mL is equal to 1 cm³, so you don't need to convert units here. Calculating mass helps understand how much substance you're actually handling beyond just its volume measure.

Density

Density is how we relate the mass of a substance to its volume. It's a key factor in converting between mass and volume—a common requirement for solving chemisty problems related to solutions. In a sense, it tells us how heavy or light a substance is, per unit of volume.

• Formula for density: Density = Mass/Volume

If the density of ethanol is 0.79 g/cm³, it means that each cm³ of ethanol has a mass of 0.79 grams. Densities for other substances, like water, are different; water's density at room temperature is approximately 1.00 g/cm³. Understanding density helps to calculate mass when only the volume is known and vice versa. It provides vital context about a substance’s mass relative to its space occupation.

Molar Mass

Molar mass links the mass of a substance to the amount of particles (like atoms or molecules) it contains. This is represented in units of grams per mole (g/mol) and is unique to each chemical compound.

• Formula: Sum of atomic masses of all atoms in the molecule

For ethanol (C₂H₅OH), the calculation steps are:

• Total molar mass = (2 × 12.01 g/mol for C) + (6 × 1.01 g/mol for H) + (16.00 g/mol for O)
• This sums up to 46.07 g/mol.

Knowing the molar mass is crucial for converting a given mass of the substance to moles, helping determine how many moles are present, using:

• Moles = Mass / Molar Mass

Such conversions are essential for preparing solutions by molarity or molality, which rely on moles of solute.

Solution Preparation

Preparing a solution correctly involves understanding each component's amount and behavior. It often starts with calculating the solute’s mass, converting to moles when needed, and mixing these with the solvent while respecting specific ratios or concentrations like molality. Molality, unlike molarity, depends only on the solvent's mass. The formula is defined as:

• Molality = Moles of solute / Mass of solvent (in kg)

In the ethanol solution example, after calculating the moles of ethanol (1.286 mol) and the mass of the water solvent (425 g or 0.425 kg), molality comes to 3.024 mol/kg. Molality is favored in some applications because it does not change with temperature, providing a consistent measure of concentration.