Problem 23 - Chemistry Matter and Change

Question

Challenge What volume of ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{OH}\right)\) is in 100.0 \(\mathrm{mL}\) of 0.15 \(\mathrm{M}\) solution? The density of ethanol is 0.7893 \(\mathrm{g} / \mathrm{mL} .\)

Step-by-Step Solution

Answer

0.8757 mL of ethanol is in the solution.

Step 1: Calculate the moles of ethanol

To determine the moles of ethanol in the solution, use the formula: \[ \text{moles of ethanol} = Molarity \times Volume \]. Given the molarity \(0.15\,\text{Moles/L}\) and volume \(0.1\,\text{L}\) (converted from \(100.0\,\text{mL}\)), the calculation is \(0.15 \times 0.1 = 0.015\,\text{moles of ethanol}\).

Step 2: Determine the mass of ethanol

Next, use the molar mass of ethanol to find the mass. Ethanol's molar mass is approximately \(46.08\,\text{g/mol}\). Multiply the moles by the molar mass: \(0.015 \times 46.08 = 0.6912\,\text{g of ethanol}\).

Step 3: Calculate the volume of ethanol

To find the volume, use the density formula: \[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \]. With a mass of \(0.6912\,\text{g}\) and density \(0.7893\,\text{g/mL}\), the volume is \(\frac{0.6912}{0.7893} = 0.8757\,\text{mL}\).

Key Concepts

Molarity

Molarity is an important concept when dealing with solutions. It helps us understand how concentrated a solution is. Molarity, often represented by the letter \( M \), is defined as the number of moles of solute (the substance being dissolved) per liter of solution.

The formula for molarity is:

• \( M = \frac{n}{V} \)

where \( n \) is the number of moles of solute, and \( V \) is the volume of the solution in liters.

In the exercise, we have a 0.15 molar (\( M \)) solution of ethanol. This means there are 0.15 moles of ethanol in every liter of solution. Converting the initial volume from 100.0 mL to liters (0.1 L), we can calculate the moles of ethanol using:

• \( ext{moles of ethanol} = 0.15 \times 0.1 = 0.015 \text{ moles} \)

Understanding molarity is crucial because it allows us to calculate how much of a substance is present in a given volume of something else, like a liquid.

Density

Density describes how much mass is packed into a given volume. For liquids, density is often expressed in grams per milliliter (g/mL). It helps us convert between mass and volume, which is essential for solving problems like the one in the exercise.

The formula for density is:

• \( ext{Density} = \frac{ ext{Mass}}{ ext{Volume}} \)

This can be rearranged to find volume when mass and density are known:

• \( ext{Volume} = \frac{ ext{Mass}}{ ext{Density}} \)

In the exercise, we are given the density of ethanol as 0.7893 g/mL. This means that every milliliter of ethanol has a mass of 0.7893 grams. Knowing the density allowed us to derive the volume of ethanol from its mass.

Molar Mass

Molar mass is a property of chemical compounds that tells us the mass of one mole of molecules. It's a critical step when transitioning from moles to grams, as both are common units used in chemical calculations.

For ethanol (\( \text{C}_2\text{H}_5\text{OH} \)), the molar mass is calculated by adding the atomic masses of all the atoms in a molecule:

• Carbon (C): \( 2 \times 12.01 = 24.02 \text{ g/mol} \)
• Hydrogen (H): \( 6 \times 1.01 = 6.06 \text{ g/mol} \)
• Oxygen (O): \( 1 \times 16.00 = 16.00 \text{ g/mol} \)
• Total: \( 46.08 \text{ g/mol} \)

Using the molar mass, we can translate the 0.015 moles of ethanol from the solution into grams:

• \( 0.015 \times 46.08 = 0.6912 \text{ g} \)

Understanding molar mass helps us convert between the number of moles and the mass in grams, aiding in comprehensive calculations involving chemical substances.

Volume Calculation

Calculating volume is often the concluding step in solving problems related to solutions. By using known values of mass and density, we can determine the volume of a substance.

Formula for calculating volume is simple:

• \( ext{Volume} = \frac{ ext{Mass}}{ ext{Density}} \)

In our exercise, we needed to find the volume of ethanol after determining its mass to be 0.6912 grams. The given density of ethanol is \( 0.7893 \text{ g/mL} \). Substituting these values into the formula, we find:

• \( ext{Volume} = \frac{0.6912}{0.7893} = 0.8757 \text{ mL} \)

This calculation demonstrates how density and mass interplay to determine the volume. Understanding and applying the volume calculation is critical in practical chemistry and many real-life applications, from mixing solutions to industrial chemical processes.