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In chemistry, the molar mass is a central concept used to bridge the gap between microscopic and macroscopic scales. It tells us the mass of one mole of a given chemical substance. A mole, defined as Avogadro's number (6.022 x 10^{23}) of molecules, serves as a convenient counting unit for chemists.
Molar mass is calculated by adding up the atomic masses of the elements present in a compound. For caffeine, \(\mathrm{C}_{8}\mathrm{H}_{10}\mathrm{~N}_{4}\mathrm{O}_{2}\), this means multiplying the number of each type of atom by the atomic mass of those atoms and summing them together:

• Carbon (C) has an atomic mass of 12.01 g/mol, and there are 8 carbon atoms.
• Hydrogen (H) has an atomic mass of 1.01 g/mol, and there are 10 hydrogen atoms.
• Nitrogen (N) has an atomic mass of 14.01 g/mol, and there are 4 nitrogen atoms.
• Oxygen (O) has an atomic mass of 16.00 g/mol, with 2 oxygen atoms.

Adding these, we get the molar mass of caffeine as 194.22 g/mol. Understanding molar mass is key in converting between moles and grams, allowing precise measurement in experiments.

Solution concentration is a way of describing how much solute is dissolved in a solvent. Several methods exist to quantify this, including molarity, molality, and mass percent. In this particular exercise, we are dealing with molality.
Molality is defined as the moles of solute divided by the kilograms of solvent, which differentiates it slightly from molarity. This is because molality is temperature-independent since it uses mass instead of volume.
For the given problem, the solution's molality is 0.0946 m, meaning there are 0.0946 moles of caffeine present per kilogram of chloroform. Given the solvent weight of 45.0 g, which is equal to 0.045 kg, we can multiply this to obtain moles of caffeine. This calculation provides us with a crucial step in determining how much caffeine is actually present in the sample, allowing conversion to grams using molar mass.

The ultimate goal here is to determine the mass of caffeine present in the solution. First, we identify the moles of caffeine using the relationship provided by molality:

• The molality (0.0946 m) is multiplied by the kilograms of chloroform (0.045 kg).
• This results in approximately 0.004257 moles of caffeine.

With this in hand, the calculation progresses to find the mass:

• The number of moles (0.004257) is multiplied by caffeine's molar mass (194.22 g/mol).
• This results in approximately 0.826 grams of caffeine.

This step-by-step approach not only determines the caffeine content but also enhances comprehension of how molar mass and solution concentration interplay in chemical calculations. Understanding these processes is essential for practical lab work and theoretical analysis.