Problem 37 - Chemistry The Central Science

Question

(a) What is the mass, in grams, of \(2.50 \times 10^{-3}\) mol of ammonium phosphate? (b) How many moles of chloride ions are in 0.2550 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 \(2.50 \times 10^{-3} mol\) of ammonium phosphate is 0.373 g. (b) There are 0.00573 moles of chloride ions in 0.2550 g of aluminum chloride. (c) The mass of \(7.70 \times 10^{20}\) molecules of caffeine is 0.248 g. (d) The molar mass of cholesterol is 386.67 g/mol.

Step 1: (Problem (a): Mass of ammonium phosphate)

First, we need to get the molar mass of ammonium phosphate, which is (NH4)3PO4. According to the periodic table, the molar masses of nitrogen, hydrogen, phosphorus, and oxygen are 14 g/mol, 1 g/mol, 31 g/mol, and 16 g/mol, respectively. The molar mass of ammonium phosphate can be calculated by: Molar mass of (NH4)3PO4 = 3(N+4H) + P + 4O = 3(14 + 4(1)) + 31 + 4(16) = 3(14 + 4) + 31 + 64 = 3(18) + 31 + 64 = 54 + 31 + 64 = 149 g/mol Now, we have moles of ammonium phosphate as \(2.50 \times 10^{-3} mol\). We can calculate the mass using the molar mass: Mass (in g) = Moles × Molar mass = \(2.50 \times 10^{-3} mol \times 149 g/mol\) = 0.373 g So, the mass of \(2.50 \times 10^{-3} mol\) of ammonium phosphate is 0.373 g.

Step 2: (Problem (b): Moles of chloride ions in aluminum chloride)

First, we need the molar mass of aluminum chloride (AlCl3). The molar masses for aluminum and chlorine are 27 g/mol and 35.5 g/mol, respectively. Molar mass of AlCl3 = Al + 3Cl = 27 + 3(35.5) = 27 + 106.5 = 133.5 g/mol Now, we are given the mass of aluminum chloride: 0.2550 g. We can find the moles of aluminum chloride: Moles of AlCl3 = mass / molar mass = 0.2550 g / 133.5 g/mol = 0.00191 mol For each molecule of aluminum chloride (AlCl3), there are 3 molecules of chloride ions (Cl-). Therefore, moles of chloride ions (Cl-) are: Moles of Cl- = 3 × Moles of AlCl3 = 3 × 0.00191 mol = 0.00573 mol So, there are 0.00573 moles of chloride ions in 0.2550 g of aluminum chloride.

Step 3: (Problem (c): Mass of caffeine molecules)

The molecular formula for caffeine is C8H10N4O2. We need to find the molar mass of caffeine: Molar mass of C8H10N4O2 = 8C + 10H + 4N + 2O = 8(12) + 10(1) + 4(14) + 2(16) = 96 + 10 + 56 + 32 = 194 g/mol Now we are given that there are \(7.70 \times 10^{20}\) molecules of caffeine. To find the mass of these molecules, first, find the moles of caffeine using Avogadro's number: Moles = (number of molecules) / (Avogadro's number) = \(\frac{7.70 \times 10^{20}}{6.02 \times 10^{23} mol^{-1}}\) = \(1.28 \times 10^{-3} mol\) Now, we can calculate the mass: Mass (in g) = Moles × Molar mass = \(1.28 \times 10^{-3} mol \times 194 g/mol\) = 0.248 g So, the mass of \(7.70 \times 10^{20}\) molecules of caffeine is 0.248 g.

Step 4: (Problem (d): Molar mass of cholesterol)

In this problem, we are given the mass (0.406 g) and the moles (0.00105 mol) of cholesterol. We can use the mass and the moles to determine the molar mass: Molar mass = mass / moles = 0.406 g / 0.00105 mol = 386.67 g/mol So, the molar mass of cholesterol is 386.67 g/mol.

Key Concepts

Molar Mass Calculation

Understanding molar mass calculation is essential for mastering stoichiometry problems. To calculate the molar mass of a compound, add up the molar mass of each element in the compound multiplied by the number of atoms of that element present in the molecule. For example, to find the molar mass of ammonium phosphate, (NH₄)₃PO₄, you would calculate:

• Molar mass of N: 14 g/mol
• Molar mass of H: 1 g/mol
• Molar mass of P: 31 g/mol
• Molar mass of O: 16 g/mol

Then, you can set up the calculation as shown in the solution, taking care to properly account for each atom present in the molecule. When done correctly, the molar mass of ammonium phosphate is 149 g/mol. It's important to remember that molar mass units are grams per mole (g/mol), enabling further conversions between moles and mass.

Mole-to-Mass Conversion

Converting moles to mass is a two-step process that starts with calculating the molar mass of the substance. Once you have the molar mass, use the relationship Mass in grams = Moles \times Molar mass in g/mol. For instance, to convert moles of ammonium phosphate to mass, you multiply the number of moles by its molar mass. As observed in the solution for part (a) of the exercise, 0.373 grams is the mass corresponding to\(2.50 \times 10^{-3}\) moles of ammonium phosphate. This calculation is critical for quantifying the amounts of substances in chemical reactions.

Avogadro's Number Application

Avogadro's number (\(6.02 \times 10^{23}\) molecules/mol) is the bridge between the microscopic world of atoms and the macroscopic world we measure in the lab. When you have a certain number of molecules and need to find the mass, you first convert molecules to moles using Avogadro's number, then multiply by the molar mass. For example, with caffeine, knowing that one mole of caffeine has \(6.02 \times 10^{23}\) molecules, calculate the number of moles for\(7.70 \times 10^{20}\) molecules. The solution shows that after finding the number of moles, a quick multiplication by the molar mass of caffeine gives you the mass in grams, allowing you to handle tangible quantities of the substance.

Molecular Mass Determination

Molecular mass determination involves finding the molar mass of a compound based on known quantities of mass and moles. In the case of cholesterol, the solution to the exercise demonstrates this by dividing the mass by the moles. Specifically, the molar mass of cholesterol is calculated as 386.67 g/mol by using the given mass (0.406 g) and the moles (0.00105 mol). This value indicates how much one mole of cholesterol weighs, and it's often used to deduce the molecular formula of unknown compounds or to prepare specific molar concentrations for reactions.