Problem 37 - Chemistry The Central Science In Si Units

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 mol has a mass of \(0.406 \mathrm{~g}\) ?

Step-by-Step Solution

Answer

(a) 0.3728 g; (b) 0.00573 mol Cl⁻; (c) Moles of caffeine: 1.28 × 10⁻³ mol; (d) Molar mass of cholesterol: 386.67 g/mol.

Step 1: Calculate Molar Mass of Ammonium Phosphate

To find the mass of ammonium phosphate, first calculate its molar mass. The formula for ammonium phosphate is \((\text{NH}_4)_3\text{PO}_4\). Calculate the molar mass as follows: \[\text{N: } 3 \times 14.01 = 42.03 \text{ g/mol}\]\[\text{H: } 12 \times 1.01 = 12.12 \text{ g/mol}\]\[\text{P: } 1 \times 30.97 = 30.97 \text{ g/mol}\]\[\text{O: } 4 \times 16.00 = 64.00 \text{ g/mol}\]Add them up: \[42.03 + 12.12 + 30.97 + 64.00 = 149.12 \text{ g/mol}\].

Step 2: Calculate Mass of Ammonium Phosphate in Grams

Once the molar mass is known, use the number of moles given to find the mass: \[\text{Mass} = \text{Moles} \times \text{Molar Mass}\]\[\text{Mass} = 2.50 \times 10^{-3} \text{ mol} \times 149.12 \text{ g/mol} = 0.3728 \text{ g}\].

Step 3: Find Molar Mass of Aluminum Chloride

The formula for aluminum chloride is \(\text{AlCl}_3\). Calculate the molar mass:\[\text{Al: } 26.98 \text{ g/mol}\]\[\text{Cl: } 3 \times 35.45 = 106.35 \text{ g/mol}\]Add them up: \[26.98 + 106.35 = 133.33 \text{ g/mol}\].

Step 4: Determine Moles of Aluminum Chloride

Using the mass and molar mass, calculate moles of aluminum chloride:\[\text{Moles} = \frac{\text{Mass}}{\text{Molar Mass}}\]\[\text{Moles} = \frac{0.2550 \text{ g}}{133.33 \text{ g/mol}} = 0.00191 \text{ mol}\].

Step 5: Calculate Moles of Chloride Ions

Since each formula unit of \(\text{AlCl}_3\) contains 3 chloride ions, calculate moles of chloride ions:\[\text{Moles of Cl}^- = 0.00191 \text{ mol} \times 3 = 0.00573 \text{ mol}\].

Step 6: Calculate Number of Moles of Caffeine

Using Avogadro's number \(6.022 \times 10^{23}\) molecules/mol, find moles of caffeine:\[\text{Moles} = \frac{7.70 \times 10^{20} \text{ molecules}}{6.022 \times 10^{23} \text{ molecules/mol}}\approx 1.28 \times 10^{-3} \text{ mol}\].

Step 7: Compute Molar Mass of Cholesterol

Molar mass is the mass divided by the moles. Here, calculate:\[\text{Molar Mass} = \frac{0.406 \text{ g}}{0.00105 \text{ mol}} = 386.67 \text{ g/mol}\].

Key Concepts

Molar Mass Determination

Understanding the molar mass is crucial for solving problems in chemistry as it serves as the bridge between the microscopic world of atoms and molecules and the macroscopic world we can measure. The molar mass tells us how much one mole of a substance weighs. It's calculated by summing the atomic masses of each atom in a molecule.
The periodic table is an invaluable tool here, as it provides the atomic mass of each element. For example, for ammonium phosphate, \((\text{NH}_4)_3\text{PO}_4\), we calculate its molar mass by adding:

• 3 nitrogen atoms (times 14.01 \text{ g/mol})
• 12 hydrogen atoms (times 1.01 \text{ g/mol})
• 1 phosphorus atom (30.97 \text{ g/mol})
• 4 oxygen atoms (times 16.00 \text{ g/mol})

Once we add these up, we obtain a total of 149.12 g/mol for the molar mass of ammonium phosphate. This number tells us the mass of one mole of ammonium phosphate in grams.

Mole Concept

The mole is a fundamental concept in chemistry that allows chemists to count particles like atoms, molecules, and ions by weighing them. Avogadro's number, approximately \(6.022 \times 10^{23}\), defines how many particles are in one mole.
Consider the caffeine molecules; to find moles from a given number of molecules, we divide by Avogadro's number: \[ \text{Moles} = \frac{7.70 \times 10^{20} \text{ molecules}}{6.022 \times 10^{23} \text{ molecules/mol}} \approx 1.28 \times 10^{-3} \text{ mol}\].
This calculation makes it possible to connect the macroscopic measurement to the number of molecules we're interested in, making converting between particles and moles straightforward. This is essential in predicting how much of a chemical is needed or will be produced in a reaction.

Stoichiometry

Stoichiometry is the area of chemistry that involves calculating the quantities of reactants and products in chemical reactions. It is based on the conservation of mass and the laws of definite and multiple proportions. To practice stoichiometry, one needs to be adept at using the mole concept and molar masses.
For example, we start by finding the moles of a given substance, such as aluminum chloride. From the mass of the substance, we divided by its molar mass to get the number of moles. This gives us a pathway to determine how these moles interact with other substances in a chemical equation.
If each formula unit of \(\text{AlCl}_3\) contains 3 chloride ions, then to find the moles of chloride ions, multiply the moles of aluminum chloride by three. This step ensures that the stoichiometric relationship dictated by the chemical formula is respected, allowing accurate calculation of all substances involved in a chemical reaction.