Problem 37 - Chemistry: The Central Science in SI Units, Global Edition

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) The mass of \(2.50 \times 10^{-3} \mathrm{~mol}\) of ammonium phosphate is \(0.373 \mathrm{~g}\). (b) There are \(5.74 \times 10^{-3} \mathrm{~mol}\) of chloride ions in \(0.2550 \mathrm{~g}\) of aluminum chloride. (c) The mass of \(7.70 \times 10^{20}\) molecules of caffeine is \(0.248 \mathrm{~g}\). (d) The molar mass of cholesterol is \(386.67 \mathrm{~g/mol}\).

Step 1: Find the molar mass of ammonium phosphate

First, we need to determine the chemical formula of ammonium phosphate, which is (NH4)3PO4. Then, the molar mass can be calculated by finding the atomic masses of each element and adding them up: Molar mass = 3 × (1 × 14.01 + 4 × 1.01) + 1 × (15.99 + 1 × 15.99 + 4 × 1.008) = 149.09 g/mol.

Step 2: Calculate the mass of given moles

Next, we need to calculate the mass of given moles using the molar mass. We know that mass = moles × molar mass: Mass = (2.50 × 10⁻³ mol) × (149.09 g/mol) = 0.373 g. Question (b):

Step 1: Find the molar mass of aluminum chloride

First, we need to determine the chemical formula of aluminum chloride, which is AlCl3. Then, we can calculate the molar mass by finding the atomic masses of each element and adding them up: Molar mass = 1 × 26.98 + 3 × 35.45 = 133.33 g/mol.

Step 2: Calculate the moles of aluminum chloride

We can calculate the moles of aluminum chloride using the given mass and molar mass. We know that moles = mass/molar mass: Moles = 0.2550 g / 133.33 g/mol = 1.91 × 10⁻³ mol.

Step 3: Calculate the moles of chloride ions

For each molecule of aluminum chloride, there are three chloride ions. So, the moles of chloride ions can be found by multiplying the moles of aluminum chloride with 3: Moles of Cl⁻ = 1.91 × 10⁻³ mol × 3 = 5.74 × 10⁻³ mol. Question (c):

Step 1: Find the molar mass of caffeine

First, we need to determine the chemical formula of caffeine, which is C8H10N4O2. Then, we can calculate the molar mass by finding the atomic masses of each element and adding them up: Molar mass = 8 × 12.01 + 10 × 1.01 + 4 × 14.01 + 2 × 16.00 = 194.19 g/mol.

Step 2: Calculate moles of caffeine

We can calculate the moles of caffeine using Avogadro's number (6.022 × 10²³ molecules/mol) and the given number of molecules: Moles = (7.70 × 10²⁰ molecules) / (6.022 × 10²³ molecules/mol) = 1.28 × 10⁻³ mol.

Step 3: Calculate the mass of the given number of caffeine molecules

We can calculate the mass using the molar mass and the moles of caffeine. We know that mass = moles × molar mass: Mass = (1.28 × 10⁻³ mol) × (194.19 g/mol) = 0.248 g. Question (d):

Step 1: Convert mass and moles to molar mass

To find the molar mass, we can use the mass and moles of cholesterol. We know that molar mass = mass / moles: Molar mass = 0.406 g / 0.00105 mol = 386.67 g/mol.

Key Concepts

Moles and Mass Conversion

Understanding how to convert between moles and mass is crucial in chemistry. This conversion allows you to relate the mass of a substance to the number of moles, using the molar mass as a bridge. Essentially, the molar mass serves as a conversion factor between the number of moles and grams.

To perform this conversion, use the formula:

• Mass = Moles × Molar Mass

This equation tells us that once we know the number of moles and the molar mass of a substance, we can calculate the mass. Conversely, knowing the mass and the molar mass allows us to find the number of moles:

• Moles = Mass / Molar Mass

These calculations are pivotal in determining how much of a substance you have in a laboratory setting or any practical application. For instance, if you have 2.50 × 10⁻³ moles of a compound, and you know its molar mass, you can precisely determine its mass in grams.

Chemical Formulas

Chemical formulas are shorthand notations to represent atoms in a molecule or compound. They reveal the elements present and the number of each atom in a molecule.

For example, the chemical formula for caffeine is C extsubscript{8}H extsubscript{10}N extsubscript{4}O extsubscript{2}. This tells us that each caffeine molecule is formed by 8 carbon atoms, 10 hydrogen atoms, 4 nitrogen atoms, and 2 oxygen atoms.

Having the correct chemical formula is essential for calculating molar mass, which involves summing the atomic masses of all atoms in the formula. Therefore, accurate chemical formulas are imperative for precise molar mass calculations and experimentation in chemistry.

Atomic Masses

Atomic mass is the mass of an atom, typically measured in atomic mass units (amu). It's approximately equivalent to the number of protons and neutrons in an atom's nucleus.

To find atomic masses for molecules, you use the periodic table, which lists the average atomic masses for each element. For example, carbon has an atomic mass of about 12.01 amu, hydrogen has about 1.01 amu, and nitrogen is approximately 14.01 amu.

When calculating the molar mass of a compound like aluminum chloride (AlCl₃), you add up the atomic masses of aluminum and three chlorine atoms based on their occurrences in the compound. Each chlorine atom contributes about 35.45 amu, and aluminum about 26.98 amu. Summing these gives the molar mass, required for further calculations like mass conversion.

Avogadro's Number

Avogadro's number plays a fundamental role in chemistry. It is the key to converting between the number of particles and the amount of substance in moles. Avogadro's number is defined as 6.022 × 10²³

This number tells us the number of atoms, ions, or molecules in one mole of a substance. For example, 7.70 × 10²⁰ molecules of caffeine is a substantially smaller amount than a mole. We use Avogadro's number to find how many moles these molecules represent:

• Moles = Number of molecules / Avogadro's number

Through this conversion, you can proceed to other useful calculations in chemistry, like determining mass from the number of molecules. In this manner, Avogadro's number is a vital constant that links the microscopic world of atoms and molecules to the macroscopic world we can measure and observe.