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 chloroform \(\left(\mathrm{CHCl}_{3}\right)\) as a solvent has a concentration of \(0.0500 \mathrm{~m}\), calculate (a) the percent caffeine by mass, (b) the mole fraction of caffeine.

Short answer

(a) The percent of caffeine by mass in the given solution is approximately \(0.960\% \). (b) The mole fraction of caffeine in the given solution is approximately \(0.005940\).

Step-by-step solution

Calculate the molar mass of caffeine and chloroform

Determine the molar mass of caffeine (\(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{N}_{4} \mathrm{O}_{2}\)) and chloroform (\(\mathrm{CHCl}_{3}\)). Molar mass of caffeine = (8 × 12.01) + (10 × 1.01) + (4 × 14.01) + (2 × 16.00) = 194.19 g/mol Molar mass of chloroform = (1 × 12.01) + (1 × 1.01) + (3 × 35.45) = 119.38 g/mol

Calculate the mass of solute and solvent

The concentration of the solution is given as 0.0500 m, which means there are 0.0500 moles of caffeine in 1 kg of chloroform. Mass of solute (caffeine) = moles × molecular weight = 0.0500 mol × 194.19 g/mol = 9.7095 g Mass of solvent (chloroform) = 1000 g (because 1 kg is equal to 1000 g)

Calculate the percent of caffeine by mass

Divide the mass of solute (caffeine) by the total mass of the solution (mass of solute + mass of solvent), and multiply by 100. Percent caffeine by mass = \[\frac{\mathrm{mass~of~solute}}{\mathrm{mass~of~solute} + \mathrm{mass~of~solvent}} × 100\% \] = \[\frac{9.7095 g}{9.7095 g + 1000 g} × 100\% \approx 0.960 \% \] (a) The percent of caffeine by mass in the given solution is approximately 0.960 %.

Calculate the mole fraction of caffeine

To find the mole fraction of caffeine, first calculate moles of caffeine and moles of chloroform in 1 kg of the solvent. Moles of solute (caffeine) = 0.0500 mol Moles of solvent (chloroform) = \[\frac{\mathrm{mass~of~solvent}}{\mathrm{molar~mass~of~solvent}} = \frac{1000 g}{119.38 g/mol} = 8.3786 mol \] Mole fraction of caffeine = \[\frac{\mathrm{moles~of~caffeine}}{\mathrm{moles~of~caffeine} + \mathrm{moles~of~chloroform}}\] = \[\frac{0.0500 mol}{0.0500 mol + 8.3786 mol} = 0.005940 \] (b) The mole fraction of caffeine in the given solution is approximately 0.005940.

Key concepts

Molar Mass Calculation

Understanding molar mass is fundamental when working with chemical solutions. It represents the mass of one mole of a substance, typically expressed in grams per mole (g/mol). To calculate the molar mass of a compound like caffeine, which has the formula \( \mathrm{C}_{8} \mathrm{H}_{10} \mathrm{N}_{4} \mathrm{O}_{2} \), you simply add up the atomic masses of all atoms in the molecule. This is done by multiplying the number of each type of atom by its respective atomic weight (found on the periodic table) and summing these values.

For caffeine, this looks like:
- Carbon (C): 8 atoms × 12.01 g/mol
- Hydrogen (H): 10 atoms × 1.01 g/mol
- Nitrogen (N): 4 atoms × 14.01 g/mol
- Oxygen (O): 2 atoms × 16.00 g/mol
The total gives you caffeine’s molar mass: \( 194.19 \ g/mol \). Likewise, for chloroform \( \mathrm{CHCl}_{3} \), the molar mass is calculated using the atomic weights of carbon, hydrogen, and chlorine. The ability to calculate molar mass is pivotal as it serves as a bridge between the mass of a substance and the number of moles, which is a core concept in stoichiometry and concentration calculations.

Percent Composition by Mass

The percent composition by mass gives us a way to express the concentration of a particular component in a mixture. It’s particularly useful in solutions where we need to know the proportion of solute versus solvent by mass. To find it, we divide the mass of the solute by the total mass of the solution and then multiply by 100% to convert the fraction to a percentage.

For example, if we dissolve caffeine in chloroform, the percent composition by mass of caffeine tells us how much of the solution’s total mass is due to caffeine. Once the molar mass and the mass of solute and solvent are known, the calculation is straightforward. A key point to stress is accuracy in mass measurements and unit conversions. A small mistake here can dramatically change the result. Additionally, ensure students understand that the solution's total mass is the sum of the mass of the solute and the solvent, not just the mass of the solute.

Mole Fraction

The mole fraction is a dimensionless number representing the ratio of the number of moles of a component to the total number of moles in the mixture. This concept is crucial in the study of mixtures and solutions where we are interested in the relative quantity of substances rather than their absolute mass.

Calculating the mole fraction involves two steps: first determining the number of moles of each component, and then dividing the number of moles of the component of interest by the total number of moles in the mixture. For the caffeine in chloroform example, we calculate both the moles of caffeine and moles of chloroform, then apply the formula for mole fraction. As fractions, mole fractions will always range from 0 to 1 and they are often useful in calculating partial pressures in gas mixtures or in other scenarios where concentrations on a molar basis are needed. Such calculations are common in the fields of chemistry and chemical engineering, particularly when dealing with solutions of varying concentrations.