Problem 26 - Chemistry: Principles and Reactions

Question

Calculate the molar masses (in grams per mole) of (a) gallium, Ga, a metal that literally melts in your hands. (b) plaster of paris, \(\mathrm{CaSO}_{4} \cdot \frac{1}{2} \mathrm{H}_{2} \mathrm{O}\), once used in making casts for broken bones. (c) benzoyl peroxide, \(\mathrm{C}_{14} \mathrm{H}_{10} \mathrm{O}_{4}\), the active ingredient in many lotions used to treat acne.

Step-by-Step Solution

Answer

Answer: The molar masses of the substances are as follows: 1. Gallium (Ga): 69.72 g/mol 2. Plaster of Paris (\(\mathrm{CaSO}_{4} \cdot \frac{1}{2} \mathrm{H}_{2}\mathrm{O}\)): 145.15 g/mol 3. Benzoyl Peroxide (\(\mathrm{C}_{14} \mathrm{H}_{10} \mathrm{O}_{4}\)): 242.24 g/mol

Step 1: 1. Gallium (Ga)

To find the molar mass of Gallium, look up the atomic mass for Ga on the periodic table which is 69.72 g/mol. So, the molar mass of Gallium is equal to its atomic mass: 69.72 g/mol.

Step 2: 2. Plaster of Paris (\(\mathrm{CaSO}_{4} \cdot \frac{1}{2} \mathrm{H}_{2}\mathrm{O}\))

To find the molar mass: 1. Identify the number of atoms of each element in the formula: 1 Ca, 1 S, 4 O, and 1 H2O (divided by 2 at the end). 2. Look up the atomic masses of each element on the periodic table: Ca = 40.08 g/mol, S = 32.06 g/mol, O = 16.00 g/mol, and H = 1.01 g/mol. 3. Multiply the atomic mass by the number of atoms of each element and sum the results: (1 × 40.08) + (1 × 32.06) + (4 × 16.00) + (0.5 × 18.02) 40.08 + 32.06 + 64.00 + 9.01 = 145.15 g/mol The molar mass of Plaster of Paris (\(\mathrm{CaSO}_{4} \cdot \frac{1}{2} \mathrm{H}_{2}\mathrm{O}\)) is 145.15 g/mol.

Step 3: 3. Benzoyl Peroxide (\(\mathrm{C}_{14} \mathrm{H}_{10} \mathrm{O}_{4}\))

To find the molar mass: 1. Identify the number of atoms of each element in the formula: 14 C, 10 H, and 4 O. 2. Look up the atomic masses of each element on the periodic table: C = 12.01 g/mol, H = 1.01 g/mol, and O = 16.00 g/mol. 3. Multiply the atomic mass by the number of atoms of each element and sum the results: (14 × 12.01) + (10 × 1.01) + (4 × 16.00) 168.14 + 10.10 + 64.00 = 242.24 g/mol The molar mass of Benzoyl Peroxide (\(\mathrm{C}_{14} \mathrm{H}_{10} \mathrm{O}_{4}\)) is 242.24 g/mol.

Key Concepts

Atomic Mass

Understanding atomic mass is fundamental when starting to explore chemistry. It is essentially the weight of a single atom of an element, usually expressed in atomic mass units (amu), where one amu is defined as one-twelfth of the mass of a carbon-12 atom.

Atomic masses are tabulated and found on the periodic table for each element. For example, in the textbook exercise, the atomic mass of gallium (Ga) is given as 69.72 amu. These values are averaged because natural elements typically exist as a blend of isotopes - atoms with the same number of protons but different numbers of neutrons.

The atomic mass is crucial because it provides the basis for calculating the molar mass of elements and compounds when performing chemical calculations.

Molar Mass of Elements

Molar mass of elements connects the microscopic world of atoms to the macroscopic world we can measure. It is the mass of one mole of atoms of an element and is usually expressed in grams per mole (g/mol).

By definition, a mole contains exactly 6.02214076×10²³ particles (Avogadro's number), whether they are atoms, molecules, ions, or other entities. The molar mass of an element is numerically equivalent to its atomic mass but expressed in g/mol instead of amu. For instance, the molar mass of gallium derived from the exercise is 69.72 g/mol, which means 1 mole of gallium atoms weighs 69.72 grams.

Molar Mass of Compounds

Moving forward from elements to compounds, the molar mass calculation becomes a sum of the molar masses of all the atoms in the molecular formula. An excellent example from the exercise is the molar mass of plaster of paris, \(\mathrm{CaSO}_{4} \cdot \frac{1}{2} \mathrm{H}_{2}\mathrm{O}\), which includes adding together the molar masses of calcium (Ca), sulfur (S), oxygen (O), and hydrogen (H) in the quantities they appear in the compound.

To find the overall molar mass, add the molar masses of the constituent elements, adjusted for their respective number of atoms. This practice is common in pharmaceutics for compounds like benzoyl peroxide (\

Stoichiometry

Stoichiometry is the term used to describe calculations that relate the quantities of substances in chemical reactions. It's predicated on the law of conservation of mass and the concept of the mole. Stoichiometry uses the balanced chemical equation as a basis to determine the amount of reactants needed and the amount of products formed.

Involving molar mass within stoichiometric calculations is a routine part of designing and analyzing chemical reactions. By understanding the molar mass of compounds, you can use stoichiometry to ascertain how much of a reactant is consumed or how much of a product is generated in a reaction. Practical applications span from manufacturing to medication dosing, showcasing the integral role stoichiometry plays in science and industry.