Problem 58 - Chemistry The Central Science In Si Units

Question

Caffeine \(\left(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{~N}_{4} \mathrm{O}_{2}\right)\) is a stimulant found in coffee and tea. If a solution of caffeine in the solvent chloroform \(\left(\mathrm{CHCl}_{3}\right)\) has a concentration of \(0.0500 \mathrm{~m},\) calculate \((\mathbf{a})\) the percentage of caffeine by mass, (b) the mole fraction of caffeine in the solution.

Step-by-Step Solution

Answer

(a) 0.962% by mass caffeine, (b) mole fraction of caffeine is 0.00595.

Step 1: Write down the Given Information

First, note the given data: the molecular formula for caffeine is \( \mathrm{C}_{8} \mathrm{H}_{10} \mathrm{N}_{4} \mathrm{O}_{2} \), and the molality of the caffeine solution is \( 0.0500 \mathrm{~m} \). The solvent is chloroform \( \mathrm{CHCl}_{3} \).

Step 2: Calculate the Molar Mass of Caffeine

Calculate the molar mass of caffeine by adding up the atomic masses of all the atoms in its formula: \( (8 \times 12.01) + (10 \times 1.01) + (4 \times 14.01) + (2 \times 16.00) = 194.22 \text{ g/mol} \).

Step 3: Calculate the Molar Mass of Chloroform

Determine the molar mass of chloroform: \( \mathrm{CHCl}_3 \) consists of 1 carbon atom, 1 hydrogen atom, and 3 chlorine atoms. Thus, \( 12.01 + 1.01 + (3 \times 35.45) = 119.37 \text{ g/mol} \).

Step 4: Define Molality and Calculate the Mass of Solvent

Molality \( m \) is the number of moles of solute (caffeine) per kilogram of solvent (chloroform). For a concentration of \( 0.0500 \text{ m} \), this means \( 0.0500 \text{ moles of caffeine per 1 kg of chloroform} \). Since \( 1 \text{ kg of chloroform} = 1000 \text{ g} \), use this information moving forwards.

Step 5: Calculate the Mass of Caffeine

Calculate the mass of caffeine from its moles: \( 0.0500 \text{ moles} \times 194.22 \text{ g/mol} = 9.711 \text{ g} \).

Step 6: Calculate the Total Mass of Solution

The total mass of the solution is the sum of the masses of the solute (caffeine) and the solvent (chloroform). So, \( 9.711 \text{ g (caffeine)} + 1000 \text{ g (chloroform)} = 1009.711 \text{ g} \).

Step 7: Calculate the Percentage by Mass of Caffeine

The percentage by mass is given by \( \text{Percent by mass} = \left(\frac{\text{mass of solute (caffeine)}}{\text{total mass of solution}}\right) \times 100 \). Substitute the values: \( \left(\frac{9.711}{1009.711}\right) \times 100 = 0.962\% \).

Step 8: Calculate the Mole Fraction of Caffeine

The mole fraction \( \chi \) of caffeine is calculated by dividing the moles of caffeine by the total moles in the solution. The moles of chloroform are \( \frac{1000}{119.37} = 8.377 \text{ moles} \). The mole fraction of caffeine is \( \frac{0.0500}{0.0500 + 8.377} \approx 0.00595 \).

Key Concepts

Mole Fraction

The mole fraction is a way to express the concentration of a component in a solution without involving its mass or volume. It is the ratio of the number of moles of a particular substance to the total number of moles of all substances present in the solution.

To calculate the mole fraction, begin by determining the number of moles of each component. For example, in a solution where caffeine is dissolved in chloroform, first calculate the moles of caffeine and chloroform using their respective molar masses.

• Caffeine (C_{8}H_{10}N_{4}O_{2}) has a molar mass of 194.22 g/mol.
• Chloroform (CHCl_{3}) has a molar mass of 119.37 g/mol.

Using the given molality, you can find that 0.0500 moles of caffeine are in the solution. For chloroform, calculate the moles using its mass in the solution. For instance, 1000 g of chloroform is approximately 8.377 moles.
Now, the mole fraction of caffeine is obtained by dividing the moles of caffeine by the total moles of all components:\[ \chi_{\text{caffeine}} = \frac{\text{moles of caffeine}}{\text{moles of caffeine} + \text{moles of chloroform}} = \frac{0.0500}{0.0500 + 8.377} \approx 0.00595 \]This value reflects the small proportion of caffeine in the solution given the large amount of chloroform.

Molality

Molality is a measure of solute concentration defined as the number of moles of solute per kilogram of solvent. Unlike molarity, molality does not change with temperature because it depends solely on mass, not volume.

To calculate molality (m), divide the moles of solute by the mass of the solvent in kilograms. For instance, if you dissolve caffeine into chloroform, first find the moles of caffeine from its mass and molar mass.

• Given: Molality is 0.0500 m, which means there are 0.0500 moles of caffeine per kilogram of chloroform.

This direct relationship between moles and mass makes molality a useful concentration measure, especially in thermodynamic equations where temperature may influence solution volume but not its mass.

Mass Percentage

The mass percentage is a way to express the concentration of a component in a solution by comparing the mass of the solute to the total mass of the solution, expressed as a percentage. It indicates how many parts of solute are present in 100 parts of the solution.
To calculate mass percentage, use the formula:\[ \text{mass percent} = \left(\frac{\text{mass of solute}}{\text{total mass of solution}}\right) \times 100 \]Given the mass of caffeine as 9.711 g, and the total mass of the solution (caffeine plus chloroform) as 1009.711 g, the mass percentage of caffeine is:
\[ \text{mass percent of caffeine} = \left(\frac{9.711}{1009.711}\right) \times 100 = 0.962 \% \]
This means that in every 100 grams of the solution, about 0.962 grams is caffeine, offering a clear view of the solution's composition. Such information is particularly handy in chemical formulations where precise solute quantities are crucial.