Problem 37 - Chemistry: The Central Science

Question

(a) What is the mass, in grams, of \(2.50 \times 10^{-3} \mathrm{~mol}\) of ammonium phosphate? (b) How many moles of chloride ions are in \(0.2550 \mathrm{~g}\) of aluminum chloride? (c) What is the mass, in grams, of \(7.70 \times 10^{20}\) molecules of caffeine, \(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{~N}_{4} \mathrm{O}_{2}\) ? (d) What is the molar mass of cholesterol if \(0.00105 \mathrm{~mol}\) has a mass of \(0.406 \mathrm{~g}\) ?

Step-by-Step Solution

Answer

(a) \(Mass = 2.50 \times 10^{-3} mol \times 149 g/mol = 0.373 g\) (b) \(Moles_{Cl^-} = 3 \times \frac{0.2550}{133.5 g/mol} = 5.742 \times 10^{-3} mol\) (c) \(Mass_{Caffeine} = \frac{7.70 \times 10^{20}~molecules}{6.022 \times 10^{23}~molecules/mol} \times 194 g/mol = 2.50 \times 10^{-3} g\) (d) \(Molar~Mass_{Cholesterol} = \frac{0.406 g}{0.00105 mol} = 386.67 g/mol\)

Step 1: (a) Mass of Ammonium Phosphate

To determine the mass of ammonium phosphate, we need to multiply its molar mass by the number of moles. The formula for ammonium phosphate is (NH_4)_3PO_4. First, we need to find the molar mass of ammonium phosphate: \(Molar~Mass = 3 \times (M_{N} + 4 \times M_{H}) + M_{P} + 4 \times M_{O}\) Next, substitute the atomic masses (in grams/mol) of nitrogen (N), hydrogen (H), phosphorus (P), and oxygen (O): \(Molar~Mass = 3 \times (14.0 + 4 \times 1.0) + 31.0 + 4 \times 16.0 g/mol\) Calculate the molar mass, and then multiply it by the given amount in moles: \(Mass = 2.50 × 10^{-3} mol \times Molar~Mass g/mol\)

Step 2: (b) Moles of Chloride Ions

To determine the number of moles of chloride ions, we first need to calculate the moles present in \(0.2550 g\) of aluminum chloride (AlCl_3). The molar mass of aluminum chloride can be calculated as: \(Molar~Mass = M_{Al} + 3 \times M_{Cl}\) Substitute the atomic masses (in grams/mol) of aluminum (Al) and chlorine (Cl): \(Molar~Mass = 27.0 + 3 \times 35.5 g/mol\) Next, to find the moles of aluminum chloride, use the given mass and the molar mass: \(Moles_{AlCl_3} = \dfrac{0.2550}{Molar~Mass}\) Lastly, since there are 3 moles of chloride ions in each mole of aluminum chloride, multiply the moles of aluminum chloride by 3 to get the moles of chloride ions.

Step 3: (c) Mass of Caffeine

First, we need to find the molar mass of caffeine, which has the formula \(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{N}_{4} \mathrm{O}_{2}\). \(Molar~Mass = 8 \times M_{C} + 10 \times M_{H} + 4 \times M_{N} + 2 \times M_{O}\) Next, substitute the atomic masses (in grams/mol) of carbon (C), hydrogen (H), nitrogen (N), and oxygen (O): \(Molar~Mass = 8 \times 12.0 + 10 \times 1.0 + 4 \times 14.0 + 2 \times 16.0 g/mol\) Now, we have to convert the number of caffeine molecules given to moles using Avogadro's number (6.022 × 10^23 molecules/mol): \(Moles_{Caffeine} = \dfrac{7.70 \times 10^{20}~molecules}{6.022 \times 10^{23}~molecules/mol}\) Finally, multiply the moles of caffeine by its molar mass to determine the mass: \(Mass_{Caffeine} = Moles_{Caffeine} \times Molar~Mass\)

Step 4: (d) Molar Mass of Cholesterol

To determine the molar mass of cholesterol, we can divide the given mass by the given number of moles: \(Molar~Mass_{Cholesterol} = \dfrac{0.406 g}{0.00105 mol}\)

Key Concepts

Mole Concept

Understanding the mole concept is a fundamental part of mastering chemistry. The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as the amount of any substance that contains as many elementary entities (e.g., atoms, molecules, ions, electrons) as there are atoms in 12 grams of pure carbon-12. This number is known as Avogadro's number, approximately equal to \(6.02 \times 10^{23}\).

When we talk about \(1 \text{mole}\) of a substance, we mean \(6.02 \times 10^{23}\) of its elementary entities. This large number is incompressible for everyday counting but indispensable in chemistry for handling atoms and molecules, which are incredibly small. The importance of the mole is that it provides a bridge between the atomic and macroscopic worlds.

• For example, in exercise (c), we convert a specific number of caffeine molecules to moles to eventually find the mass in grams, which is a scaled-up quantity we can measure and comprehend.

Molecular Mass Determination

Determining the molecular mass (or molar mass) of a substance is a critical step in many chemistry calculations. Molecular mass is the sum of the atomic masses of all the atoms in a molecule. It's usually expressed in grams per mole (g/mol). The atomic masses of elements are found on the periodic table and are averaged weights accounting for the various isotopes and their abundances.

In practice, molecular mass determination involves adding up the atomic masses of each element in a compound, multiplied by the number of atoms of that element present in the molecule.

• For instance, in the solution for ammonium phosphate in part (a), we calculated the molecular mass by multiplying the atomic masses of nitrogen, hydrogen, phosphorus, and oxygen by the number of each atom in the molecule, and adding them all together.

Stoichiometry

Stoichiometry is the section of chemistry that involves the calculation of the reactants and products in chemical reactions. It allows chemists to predict the quantities of substances consumed and produced in a given reaction. Stoichiometric calculations are based on the conservation of mass and the concept of the mole.

The key to stoichiometry is the balanced chemical equation, which provides the ratio of moles of reactants and products. By using ratios from the balanced equation, you can calculate the mass, volume, or particles of a substance from the quantity of another substance in the reaction.

• In the aluminum chloride problem (b), we used stoichiometric principles to relate the mass of AlCl_3 to moles and then to moles of chloride ions, since we know the ratio of aluminum to chloride ions is 1:3 from the chemical formula.