Question

A solution is prepared by dissolving \(20.2 \mathrm{~mL}\) of methanol \(\left(\mathrm{CH}_{3} \mathrm{OH}\right)\) in \(100.0 \mathrm{~mL}\) of water at \(25^{\circ} \mathrm{C} .\) The final volume of the solution is \(118 \mathrm{~mL}\). The densities of methanol and water at this temperature are \(0.782 \mathrm{~g} / \mathrm{mL}\) and \(1.00 \mathrm{~g} / \mathrm{mL}\), respectively. For this solution, calculate the concentration in each unit. a. molarity b. molality c. percent by mass d. mole fraction e. mole percent

Short answer

a. Molarity = 4.22 M. b. Molality = 4.52 m. c. Percent by mass = 15.47%. d. Mole fraction of methanol = 0.132. e. Mole percent = 13.2%.

Step-by-step solution

Calculate the mass of methanol and water

Using the density and volume of methanol and water, calculate their respective masses. For methanol: mass = density * volume = 0.782 g/mL * 20.2 mL. For water: mass = density * volume = 1.00 g/mL * 100.0 mL.

Convert mass of methanol to moles

Using the molar mass of methanol (32.04 g/mol), convert the mass of methanol to moles. Moles of methanol = mass of methanol / molar mass of methanol.

Calculate molarity (M)

Molarity is defined as moles of solute per liter of solution. Calculate the molarity by dividing the moles of methanol by the final volume of the solution in liters. Molarity = moles of methanol / volume of solution in liters.

Calculate molality (m)

Molality is defined as moles of solute per kilogram of solvent. Calculate the molality by dividing moles of methanol by the mass of water in kilograms. Molality = moles of methanol / mass of water in kg.

Calculate percent by mass (%)

Percent by mass is the mass of the solute divided by the total mass of the solution, multiplied by 100. Calculate using the equation: percent by mass = (mass of methanol / (mass of methanol + mass of water)) * 100.

Calculate mole fraction of methanol

Mole fraction is the moles of one component divided by total moles of all components. First, calculate the moles of water and then the mole fraction of methanol = moles of methanol / (moles of methanol + moles of water).

Calculate mole percent of methanol

Mole percent is the mole fraction multiplied by 100. Calculate the mole percent of methanol = (mole fraction of methanol) * 100.

Key concepts

Molarity

Molarity is a measure of the concentration of a solute in a solution, represented as moles of solute per liter of solution. It is often denoted as 'M'. In the context of our exercise, you calculate molarity by first determining the amount of solute in moles and the total volume of the solution in liters.
For methanol in water, you would find the moles of methanol using the density and volume provided, and then you would divide this by the final volume of the resulting solution after mixing, which has been converted to liters. Molarity is critical for reactions in solution because it directly relates to the number of moles of reactants that can interact.

Molality

Molality is another way to express the concentration of a solution, defined as the number of moles of solute per kilogram of solvent. It is represented by 'm'. Unlike molarity, molality does not change with temperature since it depends on the mass of the solvent rather than the volume of the solution.
To calculate molality, you use the mass of water (the solvent) in kilograms and the number of moles of methanol. This concentration measurement is particularly useful when dealing with temperature-dependent studies, as it is unaffected by expansion or contraction of the solution.

Percent by Mass

Percent by mass (or weight percent) is a concentration term expressing the ratio of the mass of the solute to the total mass of the solution, multiplied by 100 to give a percentage. It tells you how many grams of solute are present in every 100 grams of solution.
For the methanol-water solution, you would add the masses of methanol and water to find the total mass, then divide the mass of methanol by the total mass, and finally multiply by 100. This measurement is straightforward and does not involve the volume of the solution, which can be advantageous when dealing with mixtures of solids or when measuring out ingredients by mass.

Mole Fraction

Mole fraction is the ratio of the number of moles of a specific component to the total number of moles of all components in the mixture. It is unitless and thus independent of temperature or pressure. Mole fraction is often used in calculations involving gas laws and properties such as vapor pressure.
To calculate the mole fraction of methanol in our example, find the moles of both methanol and water, add them together to get the total moles, and then divide the moles of methanol by the total. This gives you the proportion of the solution that is methanol on a molecular level.

Mole Percent

Mole percent is similar to mole fraction but expressed as a percentage. It is the mole fraction of a component multiplied by 100. This makes it easier to understand in terms of percentages rather than a fraction between 0 and 1.
To obtain the mole percent of methanol in the solution, you simply multiply the mole fraction you have previously calculated by 100. It's a helpful concentration unit when you want to communicate the composition of a mixture in an intuitive, percentage-based format.