Problem 39 - Chemistry The Central Science In Si Units

Question

The molecular formula of saccharin, an artificial sweetener, is \(\mathrm{C}_{7} \mathrm{H}_{5} \mathrm{NO}_{3} \mathrm{~S} .(\mathbf{a})\) What is the molar mass of saccharin? (b) How many moles of sachharin are in \(2.00 \mathrm{mg}\) of this substance?(c) How many molecules are in \(2.00 \mathrm{mg}\) of this substance? (d) How many C atoms are present in \(2.00 \mathrm{mg}\) of saccharin?

Step-by-Step Solution

Answer

(a) 183.18 g/mol, (b) 1.092 x 10^-5 moles, (c) 6.57 x 10^18 molecules, (d) 4.60 x 10^19 C atoms.

Step 1: Calculate Molar Mass of Saccharin

To calculate the molar mass of saccharin, add the atomic masses of all atoms in its molecular formula. The formula is \( \mathrm{C}_{7} \mathrm{H}_{5} \mathrm{NO}_{3} \mathrm{S} \). Using the periodic table: \( \mathrm{C} \) (Carbon) = 12.01 g/mol, \( \mathrm{H} \) (Hydrogen) = 1.008 g/mol, \( \mathrm{N} \) (Nitrogen) = 14.01 g/mol, \( \mathrm{O} \) (Oxygen) = 16.00 g/mol, \( \mathrm{S} \) (Sulfur) = 32.07 g/mol. Calculate as follows: \( (7 \times 12.01) + (5 \times 1.008) + (1 \times 14.01) + (3 \times 16) + (1 \times 32.07) = 183.18 \) g/mol.

Step 2: Convert Milligrams to Grams

Convert 2.00 mg of saccharin to grams for easier calculations in moles. Recall that 1 mg = 0.001 g. So, \( 2.00 \) mg = \( 2.00 \times 0.001 \) g = \( 0.002 \) g.

Step 3: Calculate Moles of Saccharin

Use the molar mass to find moles of saccharin in 0.002 g. Formula: \( \text{moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \). Calculate: \( \frac{0.002 \text{ g}}{183.18 \text{ g/mol}} = 1.092 \times 10^{-5} \text{ moles} \).

Step 4: Calculate Number of Molecules

To find the number of molecules, use Avogadro's number: \(6.022 \times 10^{23} \text{ molecules/mol}\). Multiply moles by Avogadro's number: \(1.092 \times 10^{-5} \text{ moles} \times 6.022 \times 10^{23} \text{ molecules/mol} \approx 6.57 \times 10^{18} \text{ molecules} \).

Step 5: Calculate Number of Carbon Atoms

Each molecule of saccharin has 7 carbon atoms. Multiply the number of molecules by 7 to find carbon atoms. Calculation: \(6.57 \times 10^{18} \text{ molecules} \times 7 \text{ C atoms/molecule} = 4.60 \times 10^{19} \text{ C atoms} \).

Key Concepts

Molecular Formula

The molecular formula represents the actual number of atoms of each element contained in a single molecule of a compound. This formula is essential because it gives specific insight into the composition of molecules, which is crucial in chemistry. For saccharin, the molecular formula is \(\mathrm{C}_{7}\mathrm{H}_{5}\mathrm{NO}_{3}\mathrm{S}\). This indicates that in each molecule of saccharin:

• There are 7 carbon atoms.
• There are 5 hydrogen atoms.
• There is 1 nitrogen atom.
• There are 3 oxygen atoms.
• There is 1 sulfur atom.

Each of these elements contributes to the overall mass and properties of saccharin. Knowing the molecular formula allows chemists to calculate other critical quantities, such as molar mass and the number of atoms or molecules in a given sample. By understanding this foundation, transitioning to complex calculations, like converting mass to moles or identifying the number of specific atoms, becomes more manageable.

Avogadro's Number

Avogadro's number is a cornerstone in the world of chemistry, providing a link between the macroscopic and microscopic worlds. It is defined as \(6.022 \times 10^{23}\), and represents the number of atoms, molecules, or particles in one mole of a substance. This number allows chemists to count particles by weighing them, making laboratory work much more practical. Avogadro's number is crucial when transitioning from moles to actual molecules. For instance, if you know the amount of a substance in moles, multiplying it by Avogadro's number will give you the number of molecules present. This relationship is pivotal, especially in experiments where precise quantities and reactions are essential. For students, understanding Avogadro’s number lets you compute how a seemingly tiny mass can translate into an enormous quantity of particles. It’s this bridge from the scale of lab measurements to atomic precision that highlights Avogadro's number as a fundamental concept in chemistry.

Moles Calculation

Moles are a standard unit of measurement in chemistry used to express amounts of a chemical substance. Calculating moles involves knowing the mass of your sample and the molar mass of the compound in question. The basic formula for calculating moles is:\[\text{moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}}\]This equation allows chemists to convert the mass of a substance into the number of moles, making it easier to work with substances quantitatively. For example, to find the number of moles in 2.00 mg of saccharin:

• First, convert the mass from milligrams to grams (since the molar mass is typically in g/mol), which gives 0.002 g.
• Next, use the molecular formula \(\mathrm{C}_{7}\mathrm{H}_{5}\mathrm{NO}_{3}\mathrm{S}\) to find the molar mass: \(183.18 \text{ g/mol}\).
• Finally, apply the formula: \(\frac{0.002 \text{ g}}{183.18 \text{ g/mol}} \approx 1.092 \times 10^{-5} \text{ moles}\).

By mastering mole calculations, you not only handle basic conversions but also set the groundwork for more elaborate chemistry topics, such as stoichiometry and chemical reactions. Knowing how to calculate moles is thus foundational for anyone studying chemistry or related sciences.