Problem 67 - Chemistry The Central Science In Si Units

Question

(a) How many grams of ethanol, \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH},\) should you dissolve in water to make \(1.00 \mathrm{~L}\) of vodka (which is an aqueous solution that is \(6.86 \mathrm{M}\) ethanol)? (b) Using the density of ethanol \((0.789 \mathrm{~g} / \mathrm{mL})\), calculate the volume of ethanol you need to make \(1.00 \mathrm{~L}\) of vodka.

Step-by-Step Solution

Answer

316.13 g of ethanol and 400.67 mL are needed.

Step 1: Calculate Moles of Ethanol

To find the grams, we first need to find how many moles of ethanol are in the solution. The molarity (M) is defined as moles of solute per liter of solution. Given: \[ M = 6.86 \] and the volume of solution is 1.00 L, we calculate the moles of ethanol: \[ \text{moles of ethanol} = 6.86 \times 1.00 = 6.86 \text{ moles} \]

Step 2: Calculate Grams of Ethanol

The molar mass of ethanol, \( \mathrm{CH}_3 \mathrm{CH}_2 \mathrm{OH} \), is calculated as follows: Carbon: \( 12.01 \times 2 = 24.02 \) g/mol Hydrogen: \( 1.01 \times 6 = 6.06 \) g/mol Oxygen: \( 16.00 \) g/mol Total molar mass: \( 46.08 \) g/mol To find the grams of ethanol: \[ 6.86 \text{ moles} \times 46.08 \text{ g/mol} = 316.13 \text{ grams} \]

Step 3: Calculate Volume Using Density

Given the density \(0.789 \text{ g/mL}\): Convert the mass of ethanol into mL using the formula: \[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \] \[ \text{Volume} = \frac{316.13 \text{ g}}{0.789 \text{ g/mL}} = 400.67 \text{ mL} \]

Key Concepts

Molality and Molarity

When discussing molarity and molality, it's important to note that although the terms sound similar, they differ fundamentally. Molarity (M) measures the concentration of a solute in a solution, specifically as moles of solute per liter of solution. For example, when we say a vodka solution is 6.86 M in ethanol, it means there are 6.86 moles of ethanol dissolved in every liter of vodka.

Molality, on the other hand, takes into account the weight of the solvent. It measures how many moles of solute are present per kilogram of solvent, not the entire solution. This makes molality particularly useful for situations where the temperature changes because it does not vary with temperature like molarity can, as it is based on mass, not volume.

A quick summary to differentiate:

• Molarity: Moles of solute per liter of solution (temperature-dependent).
• Molality: Moles of solute per kilogram of solvent (temperature-independent).

Density of Substances

Density is a crucial property that connects the mass and volume of a substance. It is defined as mass per unit volume, often indicated in units like grams per milliliter (g/mL). For ethanol, the density is given as 0.789 g/mL.

Understanding density is important for converting between mass and volume. In our vodka example, to find out how much ethanol (by volume) is needed, we use the density to convert the mass we calculated (316.13 grams of ethanol) into a volume in milliliters. This conversion is possible because density provides a direct relationship between these two properties.

Keep in mind:

• Density Formula: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \)
• Volume Calculation: Given density \( d = 0.789 \, \text{g/mL} \), we use \( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \)

Molar Mass Calculation

Molar mass is the weight of one mole of a substance and is used to relate grams of a substance to moles. To calculate the molar mass of ethanol (\( \text{CH}_3\text{CH}_2\text{OH} \)), you sum the atomic masses of each element based on its occurrence in the molecular formula.

The calculation for ethanol:

• Carbon: Each carbon atom is approx. 12.01 g/mol, so \( 12.01 \times 2 = 24.02 \) g/mol for two carbon atoms.
• Hydrogen: Each hydrogen atom is approx. 1.01 g/mol, so \( 1.01 \times 6 = 6.06 \) g/mol for six hydrogen atoms.
• Oxygen: 16.00 g/mol for one oxygen atom.
• Total Molar Mass: Adding these gives \( 24.02 + 6.06 + 16.00 = 46.08 \) g/mol.

Knowing the molar mass is essential for converting between moles and grams, particularly when preparing solutions with specific molarities, as seen in our ethanol example.

Solution Concentration

Solution concentration describes how much solute is present in a given quantity of solvent or solution. It can be expressed in several ways, such as molarity, molality, or even weight/volume percentage, depending on the context and requirement.

For the vodka example, we use molarity as our measure of concentration, indicating how many moles of ethanol are present per liter of vodka solution. This highlights the importance of being precise about measuring volumes and masses when preparing solutions to ensure consistency and accuracy.

In practical terms:

• Understanding Concentration: Helps in determining whether a solution will behave as intended, be it in a laboratory setting or culinary application, ensuring correct chemical reactions or desired tastes and strengths.
• Consistency in Measurements: Ensures reproducible results across different batches or preparations.

Understanding and measuring solution concentration is a foundational concept in chemistry that aids in various fields like pharmaceuticals, environmental science, and food production.