Problem 36 - Chemistry: The Central Science

Question

(a) What is the mass, in grams, of \(0.0714\) mol of iron(III) sulfate? (b) How many moles of ammonium ions are in \(8.776 \mathrm{~g}\) of ammonium carbonate? (c) What is the mass, in grams, of \(6.52 \times 10^{21}\) molecules of aspirin, \(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4} ?\) (d) What is the molar mass of diazepam (Valium \(^{\otimes}\) ) if \(0.05570\) mol weighs \(15.86 \mathrm{~g}\) ?

Step-by-Step Solution

Answer

(a) The mass of 0.0714 mol of iron(III) sulfate is 28.56 g. (b) There are 0.1826 moles of ammonium ions in 8.776 g of ammonium carbonate. (c) The mass of \(6.52 \times 10^{21}\) molecules of aspirin is 1.95 g. (d) The molar mass of diazepam is 284.74 g/mol.

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

To find the mass of \(0.0714\) mol of iron(III) sulfate, we first need to calculate the molar mass of iron(III) sulfate (Fe2(SO4)3). The molar mass is the sum of the atomic masses of all the atoms in the compound. Molar mass of Fe2(SO4)3 = 2(Fe) + 3(S) + 12(O)

Step 2: a.1 Find atomic masses

Look up the atomic masses from the Periodic Table: - Fe (Iron): 55.85 g/mol - S (Sulfur): 32.07 g/mol - O (Oxygen): 16.00 g/mol

Step 3: a.2 Calculate molar mass of iron(III) sulfate

Using the atomic masses, we calculate the molar mass: Molar mass of Fe2(SO4)3 = 2(55.85) + 3(32.07) + 12(16.00) = 111.7 + 96.21 + 192 = 399.91 g/mol

Step 4: a.3 Calculate the mass of 0.0714 mol of iron(III) sulfate

Now, use the molar mass and the given number of moles to find the mass: Mass = Moles × Molar mass Mass = 0.0714 mol × 399.91 g/mol = 28.56 g So, the mass of 0.0714 mol of iron(III) sulfate is 28.56 g.

Step 5: (b) Determine the molar mass of ammonium carbonate

To find how many moles of ammonium ions are in 8.776 g of ammonium carbonate, we first need to calculate the molar mass of ammonium carbonate (NH4)2CO3. Molar mass of (NH4)2CO3 = 2(NH4) + 1(C) + 3(O)

Step 6: b.1 Find atomic masses

Look up the atomic masses from the Periodic Table: - N (Nitrogen): 14.01 g/mol - H (Hydrogen): 1.01 g/mol - C (Carbon): 12.01 g/mol - O (Oxygen): 16.00 g/mol

Step 7: b.2 Calculate molar mass of ammonium carbonate

Using the atomic masses, we calculate the molar mass: Molar mass of (NH4)2CO3 = 2(14.01 + 4(1.01)) + 12.01 + 3(16.00) = 96.12 g/mol

Step 8: b.3 Calculate moles of ammonium carbonate

Now, use the molar mass and the given mass to find the moles: Moles = Mass / Molar mass Moles = 8.776 g / 96.12 g/mol = 0.0913 mol

Step 9: b.4 Calculate moles of ammonium ions

There are 2 moles of ammonium ions (NH4+) for every mole of ammonium carbonate, so the moles of ammonium ions are: Moles of NH4+ = 2 × 0.0913 mol = 0.1826 mol So, there are 0.1826 moles of ammonium ions in 8.776 g of ammonium carbonate.

Step 10: (c) Calculate the mass of 6.52 x 10^21 molecules of aspirin

In this part, we are given the number of molecules for a specific substance (aspirin) and need to find its mass. To do this, we will first find the molar mass of aspirin and then use Avogadro's number to determine the mass. Aspirin has a chemical formula of C9H8O4. Molar mass of C9H8O4 = 9(C) + 8(H) + 4(O)

Step 11: c.1 Find atomic masses

Using the atomic masses from the Periodic Table, we have: - C (Carbon): 12.01 g/mol - H (Hydrogen): 1.01 g/mol - O (Oxygen): 16.00 g/mol

Step 12: c.2 Calculate molar mass of aspirin

Now, we calculate the molar mass: Molar mass of C9H8O4 = 9(12.01) + 8(1.01) + 4(16.00) = 108.09 + 8.08 + 64 = 180.17 g/mol

Step 13: c.3 Calculate moles of aspirin

Next, calculate moles of aspirin using the given number of molecules and Avogadro's number: Moles = (Number of molecules) / (Avogadro's number) Moles = (6.52 × 10^21) / (6.022 × 10^23) = 0.01083 mol

Step 14: c.4 Calculate the mass of aspirin

Now, find the mass using the molar mass and moles: Mass = Moles × Molar mass Mass = 0.01083 mol × 180.17 g/mol = 1.95 g So, the mass of 6.52 × 10^21 molecules of aspirin is 1.95 g.

Step 15: (d) Determine the molar mass of diazepam

In this part, we are asked to find the molar mass of diazepam given its moles (0.05570 mol) and mass (15.86 g). We can use the following equation: Molar mass = Mass / Moles

Step 16: d.1 Calculate the molar mass of diazepam

Substitute the given values into the equation: Molar mass = 15.86 g / 0.05570 mol = 284.74 g/mol So, the molar mass of diazepam is 284.74 g/mol.

Key Concepts

Iron(III) Sulfate

Iron(III) sulfate is a chemical compound with the formula \( \text{Fe}_2(\text{SO}_4)_3 \). It's composed of iron (Fe), sulfur (S), and oxygen (O). Calculating the molar mass involves adding up the atomic masses of all the atoms, which are found on the Periodic Table: two iron atoms, three sulfur atoms, and twelve oxygen atoms.
Here's how it breaks down:

• Iron (Fe): 55.85 g/mol
• Sulfur (S): 32.07 g/mol
• Oxygen (O): 16.00 g/mol

When you combine these values, you calculate the molar mass: \( 2 \times 55.85 + 3 \times 32.07 + 12 \times 16.00 = 399.91 \text{ g/mol} \).
This molar mass is essential for converting moles into grams, which is crucial in various chemical calculations.

Ammonium Ions

Ammonium ions (\( \text{NH}_4^+ \)) are found in ammonium carbonate, \((\text{NH}_4)_2\text{CO}_3\). When working with compounds like this, you determine the moles of a specific ion, such as ammonium ions, by understanding how many of those ions are present per molecule of the compound.
In the case of ammonium carbonate, each formula unit contains two ammonium ions. Therefore, if you know the moles of ammonium carbonate, you simply multiply by two to find the moles of ammonium ions:

• If there are \(0.0913 \text{ mol}\) of ammonium carbonate, then there would be \(2 \times 0.0913 = 0.1826 \text{ mol}\) of ammonium ions.

Understanding these relationships is critical for quantifying ions in various chemical reactions.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry, specifically \(6.022 \times 10^{23}\). It represents the number of atoms, ions, or molecules in one mole of a substance.
This number is key to converting between the number of particles and the amount of substance in moles. If you have a specific number of molecules, you divide by Avogadro's number to find the moles:

• For example, with \(6.52 \times 10^{21}\) molecules of aspirin, you divide by \(6.022 \times 10^{23}\) to find the moles.

This conversion is essential for any chemical calculation involving particles, whether it's reacting them or calculating their mass.

Diazepam

Diazepam, commonly known by the brand name Valium, is a medication with a known formula weight which you can calculate based on its molar mass. In chemical terms, the molar mass allows you to convert between mass and moles.
If you know the weight of a sample and its molarity, you can calculate its molar mass:

• For instance, given \(0.05570 \text{ mol}\) of diazepam weighing \(15.86 \text{ g}\), the molar mass is \(15.86 / 0.05570 = 284.74 \text{ g/mol}\).

Understanding how to calculate the molar mass is vital for dosing decisions in pharmaceuticals and other applications.

Moles to Mass Conversion

Converting moles to mass is a common task in chemistry. It involves using the molar mass of a substance, which is the mass of one mole of that substance.
Here's how you do it:

• First, determine the number of moles you have.
• Second, multiply the moles by the molar mass (\( \text{g/mol} \)).

For example, with \(0.0714 \text{ mol}\) of iron(III) sulfate, you use its molar mass \(399.91 \text{ g/mol}\) to find the mass: \(0.0714 \times 399.91 = 28.56 \text{ g}\).
This conversion is integral to preparing solutions and determining the quantities needed for chemical reactions.