Problem 56 - Chemistry The Molecular Science

Question

A 12 -oz \((355-\mathrm{mL})\) Pepsi contains \(38.9 \mathrm{mg}\) caffeine (molar mass \(=194.2 \mathrm{~g} / \mathrm{mol}\) ). Assume that the Pepsi, mainly water, has a density of \(1.01 \mathrm{~g} / \mathrm{mL}\). For such a Pepsi, calculate: (a) its caffeine concentration in ppm; (b) its molarity of caffeine; and (c) the molality of caffeine.

Step-by-Step Solution

Answer

Caffeine concentration is 108.49 ppm, molarity is 5.64 × 10^{-4} M, and molality is 5.58 × 10^{-4} m.

Step 1: Convert Volume to Mass

First, convert the volume of Pepsi from milliliters to grams using the density. The volume is given as \(355 \text{ mL}\). Using the density \(1.01 \text{ g/mL}\), we can find the mass of the Pepsi as follows:\[\text{Mass of Pepsi} = \text{Volume} \times \text{Density} = 355 \times 1.01 = 358.55 \text{ g}\]

Step 2: Calculate Caffeine Concentration in ppm

"Parts per million" (ppm) is calculated as \(\frac{\text{mass of solute}}{\text{mass of solution}} \times 10^6\).Here, the mass of caffeine is \(38.9 \text{ mg} = 0.0389 \text{ g}\) and the mass of the Pepsi is \(358.55 \text{ g}\).The caffeine concentration is\[\text{ppm} = \frac{0.0389}{358.55} \times 10^6 \approx 108.49 \text{ ppm}\]

Step 3: Convert Caffeine Mass to Moles

To find molarity, convert the mass of caffeine to moles. Use the molar mass of caffeine, \(194.2 \text{ g/mol}\):\[\text{Moles of caffeine} = \frac{0.0389 \text{ g}}{194.2 \text{ g/mol}} \approx 2.002 \times 10^{-4} \text{ mol}\]

Step 4: Calculate Molarity of Caffeine

Molarity is defined as moles of solute per liter of solution. First, convert \(355 \text{ mL}\) to liters:\[\text{Volume in liters} = 355 \times 10^{-3} = 0.355 \text{ L}\]Then calculate the molarity:\[\text{Molarity} = \frac{2.002 \times 10^{-4} \text{ mol}}{0.355 \text{ L}} \approx 5.64 \times 10^{-4} \text{ M}\]

Step 5: Convert Pepsi Mass to Kilograms

To find molality, first convert the mass of the solution (Pepsi) to kilograms:\[\text{Mass in kg} = \frac{358.55 \text{ g}}{1000} = 0.35855 \text{ kg}\]

Step 6: Calculate Molality of Caffeine

Molality is defined as moles of solute per kilogram of solvent. Assume solvent and solution as same considering aqueous solution here.\[\text{Molality} = \frac{2.002 \times 10^{-4} \text{ mol}}{0.35855 \text{ kg}} \approx 5.58 \times 10^{-4} \text{ m}\]

Key Concepts

ppm calculation

When we talk about ppm, which stands for "parts per million," we are essentially dealing with a unit of measurement used to describe the concentration of a solute in a solution. Think of ppm like measuring one tiny speck of dust in a million specks of flour—it's a way to visualize extremely small amounts.

To calculate caffeine concentration in ppm, first ensure you have the mass of both the caffeine (our solute) and the Pepsi (our solution). The formula for ppm is:

• ppm = \( \frac{\text{mass of solute}}{\text{mass of solution}} \times 10^6 \)

In our example, the mass of caffeine is 0.0389 grams, and the mass of the Pepsi is 358.55 grams. Plug these values into the formula:

• ppm = \( \frac{0.0389}{358.55} \times 10^6 \approx 108.49 \text{ ppm} \)

This denotes that out of a million parts of the solution, 108.49 parts are caffeine. It's a clear way to express tiny concentrations, convenient for comparing trace substances in solutions.

molarity calculation

Molarity is all about understanding how many moles of a solute are present in one litre of solution. In simpler terms, it's like having a recipe card for a specific soup—how much of one ingredient is needed in our large pot of soup.

To calculate the molarity of caffeine in Pepsi, we need:

• The moles of caffeine: We find this by dividing the mass of caffeine (0.0389 grams) by the molar mass of caffeine (194.2 g/mol):\[\text{Moles of caffeine} = \frac{0.0389}{194.2} \approx 2.002 \times 10^{-4} \text{ mol}\]
• The volume of the solution in liters: Convert 355 mL to liters which gives \(0.355 \text{ L}\).

With both pieces of information, employ the formula for molarity:

• Molarity = \( \frac{\text{moles of solute}}{\text{liters of solution}} \)
• Molarity = \( \frac{2.002 \times 10^{-4}}{0.355} \approx 5.64 \times 10^{-4} \text{ M} \)

This result tells us that in every litre of this Pepsi, there are about \(5.64 \times 10^{-4}\) moles of caffeine. Molarity is very useful in chemistry for reactions and concentrations.

molality calculation

Molality provides a slightly different perspective—it's about how many moles of a solute exist per kilogram of solvent, rather than per litre of solution. Think of molality as a chef wanting to know precisely how much of an ingredient goes into each pound of dough, regardless of volume changes.

To determine the molality of caffeine, remember that:

• Moles of caffeine are already calculated as \(2.002 \times 10^{-4} \) mol.
• Convert the mass of the solution, 358.55 grams, to kilograms, which is \(0.35855 \text{ kg}\).

Utilize the formula for molality:

• Molality = \( \frac{\text{moles of solute}}{\text{kilograms of solvent}} \)
• Molality = \( \frac{2.002 \times 10^{-4}}{0.35855} \approx 5.58 \times 10^{-4} \text{ m} \)

Understanding molality helps when you need to know concentrations under varying conditions, such as temperature changes, as it doesn't depend on the volume but rather the weight of the solvent.