Problem 38 - Chemistry: The Central Science in SI Units, Global Edition

Question

(a) What is the mass, in grams, of \(1.223 \mathrm{~mol}\) of iron(III) sulfate? (b) How many moles of ammonium ions are in \(6.955 \mathrm{~g}\) of ammonium carbonate? (c) What is the mass, in grams, of \(1.50 \times 10^{21}\) molecules of aspirin, \(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4} ?\) (d) What is the molar mass of diazepam (Valium \(^{\circ}\) ) if 0.05570 mol has a mass of \(15.86 \mathrm{~g}\) ?

Step-by-Step Solution

Answer

a) The mass of 1.223 mol of iron(III) sulfate is \(392.3 \, \text{g}\). b) There are 0.0821 moles of ammonium ions in \(6.955 \, \text{g}\) of ammonium carbonate. c) The mass of \(1.50 \times 10^{21}\) molecules of aspirin is \(0.180 \, \text{g}\). d) The molar mass of diazepam is \(284.8 \, \text{g/mol}\).

Step 1: Calculate the molar mass of iron(III) sulfate

To calculate the mass of 1.223 mol of iron(III) sulfate, first, determine the molar mass of the compound. Iron(III) sulfate has the chemical formula \(\mathrm{Fe_2(SO_4)_3}\). To find the molar mass, add the molar masses of all the elements in the compound: \[M_{\mathrm{Fe_2(SO_4)_3}} = 2 \times M_{\mathrm{Fe}} + 3 \times ( M_{\mathrm{S}} + 4 \times M_{\mathrm{O}} )\] Using the molar masses from the periodic table: \[M_{\mathrm{Fe}} = 55.85 \mathrm{~g/mol}\] \[M_{\mathrm{S}} = 32.07 \mathrm{~g/mol}\] \[M_{\mathrm{O}} = 16.00 \mathrm{~g/mol}\]

Step 2: Calculate the mass of iron(III) sulfate

Now, use the amount in moles and the molar mass calculated above to find the mass of iron(III) sulfate: \[Mass = Number \, of \, moles \times Molar \, mass\] \[Mass = 1.223 \mathrm{~mol} \times M_{\mathrm{Fe_2(SO_4)_3}}\] Calculate the mass of 1.223 mol of iron(III) sulfate and express the result in grams. b) How many moles of ammonium ions are in \(6.955 \mathrm{~g}\) of ammonium carbonate?

Step 1: Calculate the molar mass of ammonium carbonate

Ammonium carbonate has the chemical formula \(\mathrm{(NH_4)_2CO_3}\). To find the molar mass, add the molar masses of all the elements in the compound: \[M_{\mathrm{(NH_4)_2CO_3}} = 2 \times ( M_{\mathrm{N}} + 4 \times M_{\mathrm{H}} ) + M_{\mathrm{C}} + 3 \times M_{\mathrm{O}}\] Using molar masses from the periodic table: \[M_{\mathrm{N}} = 14.01 \mathrm{~g/mol}\] \[M_{\mathrm{H}} = 1.01 \mathrm{~g/mol}\] \[M_{\mathrm{C}} = 12.01 \mathrm{~g/mol}\] \[M_{\mathrm{O}} = 16.00 \mathrm{~g/mol}\]

Step 2: Calculate the moles of ammonium carbonate

Now, use the mass and molar mass to calculate the number of moles of ammonium carbonate: \[Number \, of \, moles = \dfrac{Mass}{Molar \, mass}\] \[Number \, of \, moles = \dfrac{6.955 \mathrm{~g}}{M_{\mathrm{(NH_4)_2CO_3}}}\] Calculate the number of moles of ammonium carbonate.

Step 3: Find the moles of ammonium ions in ammonium carbonate

In the formula of ammonium carbonate, there are two moles of ammonium ions for every mole of the compound. Therefore, we can calculate the number of moles of ammonium ions as follows: \[Moles \, of \, NH_4^+ = 2 \times Moles \, of \,(NH_4)_2CO_3\] Calculate the number of moles of ammonium ions in \(6.955 \mathrm{~g}\) of ammonium carbonate. c) What is the mass, in grams, of \(1.50 \times 10^{21}\) molecules of aspirin, \(\mathrm{C}_9 \mathrm{H}_8 \mathrm{O}_4?\)

Step 1: Calculate the molar mass of aspirin

Aspirin has the chemical formula \(\mathrm{C}_9 \mathrm{H}_8 \mathrm{O}_4\). To find the molar mass, add the molar masses of all the elements in the compound: \[M_{\mathrm{C_9H_8O_4}} = 9 \times M_{\mathrm{C}} + 8 \times M_{\mathrm{H}} + 4 \times M_{\mathrm{O}}\] Using molar masses from the periodic table: \[M_{\mathrm{C}} = 12.01 \mathrm{~g/mol}\] \[M_{\mathrm{H}} = 1.01 \mathrm{~g/mol}\] \[M_{\mathrm{O}} = 16.00 \mathrm{~g/mol}\]

Step 2: Convert the number of molecules to the number of moles

The number of molecules given is \(1.50 \times 10^{21}\). One mole of any substance contains Avogadro's number of particles, \(N_A = 6.022 \times 10^{23}\). To find the number of moles, divide the given number of molecules by Avogadro's number: \[Number \, of \, moles = \dfrac{Number \, of \, molecules}{N_A}\] Calculate the number of moles of aspirin.

Step 3: Calculate the mass of aspirin

Now, use the amount in moles and the molar mass calculated above to find the mass of aspirin: \[Mass = Number \, of \, moles \times Molar \, mass\] Calculate the mass of \(1.50 \times 10^{21}\) molecules of aspirin and express the result in grams. d) What is the molar mass of diazepam (Valium\(^{\circ}\)) if \(0.05570 \mathrm{~mol}\) has a mass of \(15.86 \mathrm{~g}\)?

Step 1: Calculate the molar mass

Using the given amount in moles and the mass, we can calculate the molar mass of diazepam with the formula: \[Molar \, mass = \dfrac{Mass}{Number \, of \, moles}\] Calculate the molar mass of diazepam and express the result in grams per mole.

Key Concepts

Molar Mass

The molar mass of a substance is a key concept in chemistry that represents the mass of one mole of that substance, measured in grams per mole (g/mol). To calculate the molar mass of a compound, you add up the atomic masses of all the atoms in its chemical formula. Atomic masses are found on the periodic table, typically below the element's symbol.

For example, the molar mass of iron(III) sulfate \(\mathrm{Fe_2(SO_4)_3}\) can be calculated by adding the molar masses of two iron (Fe) atoms, three sulfur (S) atoms, and twelve oxygen (O) atoms:

• Iron (Fe) has a molar mass of \(55.85\, \mathrm{g/mol}\).
• Sulfur (S) has a molar mass of \(32.07\, \mathrm{g/mol}\).
• Oxygen (O) has a molar mass of \(16.00\, \mathrm{g/mol}\).

The calculation is as follows:\[M_{\mathrm{Fe_2(SO_4)_3}} = 2 \times M_{\mathrm{Fe}} + 3 \times (M_{\mathrm{S}} + 4 \times M_{\mathrm{O}})\] By breaking the compound into its components and adding their respective molar masses, you obtain the molar mass for the entire compound. This value is crucial for converting between grams and moles in chemical calculations.

Chemical Formulas

Chemical formulas represent the types and numbers of atoms in a compound. They are used to calculate the molar mass and to understand the composition of a molecule.

For instance, aspirin has the chemical formula \(\mathrm{C_9H_8O_4}\). This tells us that one molecule of aspirin contains:

• 9 carbon (C) atoms,
• 8 hydrogen (H) atoms,
• and 4 oxygen (O) atoms.

Understanding how to read and interpret these formulas is crucial for performing chemical calculations. The order in which the elements are written in the formula usually corresponds to how they are bonded within the compound.

Chemical formulas also connect with other quantities such as mole ratios in reactions. For example, in the compound ammonium carbonate \((\mathrm{NH_4})_2\mathrm{CO_3}\), there are two ammonium \((\mathrm{NH_4}^+)\) ions for every formula unit. This information is beneficial when determining quantities in chemical reactions and stoichiometry.

Conversion Factors

Conversion factors allow us to switch between different units and types of measurement, which is especially useful in chemistry for calculations involving amounts of substances.

When dealing with moles and grams, the molar mass serves as a conversion factor. For example, knowing that aspirin \(\mathrm{C_9H_8O_4}\) has a calculated molar mass, you can convert given grams of aspirin to moles using the formula:\[\text{Number of moles} = \frac{\text{Mass in grams}}{\text{Molar mass in g/mol}}\]

Similarly, you might need to convert the number of particles (atoms, molecules, etc.) into moles using Avogadro's number \(N_A = 6.022 \times 10^{23}\). For example, calculating the number of moles from \(1.50 \times 10^{21}\) molecules of aspirin involves dividing by \(N_A\):

• \(\text{Number of moles} = \frac{1.50 \times 10^{21}}{6.022 \times 10^{23}}\)

Conversion factors are not only crucial for calculations but also for understanding relationships between different units and quantities in chemistry. They help ensure that your work is accurate and that your units are properly converted at every step.