Problem 39 - Chemistry: The Central Science in SI Units, Global Edition

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) The molar mass of saccharin is approximately 183.18 g/mol. (b) In 2.00 mg of saccharin, there are about 1.09 × 10^{-5} moles. (c) In 2.00 mg of saccharin, there are approximately 6.56 × 10^{18} molecules. (d) In 2.00 mg of saccharin, there are approximately 4.59 × 10^{19} carbon atoms.

Step 1: Calculate the molar mass of saccharin

To calculate the molar mass of saccharin, we'll need to find the molar mass of each element in the molecule and add them together. The molecular formula of saccharin is \(\mathrm{C}_{7} \mathrm{H}_{5} \mathrm{NO}_{3} \mathrm{~S}\), so its molar mass will be: Molar mass of saccharin = (7 × molar mass of C) + (5 × molar mass of H) + (1 × molar mass of N) + (3 × molar mass of O) + (1 × molar mass of S) Using the molar masses of each element (C: 12.01 g/mol, H: 1.01 g/mol, N: 14.01 g/mol, O: 16.00 g/mol, S: 32.07 g/mol): Molar mass of saccharin ≈ (7 × 12.01) + (5 × 1.01) + (1 × 14.01) + (3 × 16.00) + (1 × 32.07) Molar mass of saccharin ≈ 183.18 g/mol

Step 2: Calculate the number of moles of saccharin

To find the number of moles of saccharin in 2.00 mg, we'll use the molar mass we calculated and convert the mass to grams: 2.00 mg = 0.00200 g Number of moles = (mass of substance) / (molar mass of substance) Number of moles of saccharin = 0.00200 g / 183.18 g/mol ≈ 1.09 × 10^{-5} moles

Step 3: Calculate the number of molecules of saccharin

To find the number of molecules in the sample, we'll use Avogadro's number (6.022 × 10^{23} molecules/mol): Number of molecules = (number of moles) × (Avogadro's number) Number of molecules of saccharin ≈ 1.09 × 10^{-5} moles × 6.022 × 10^{23} molecules/mol ≈ 6.56 × 10^{18} molecules

Step 4: Calculate the number of C atoms in 2.00 mg of saccharin

To find the number of carbon atoms in 2.00 mg of saccharin, first find the number of carbon atoms in one molecule of saccharin: C atoms in one molecule of saccharin = 7 (from the molecular formula \(\mathrm{C}_{7} \mathrm{H}_{5} \mathrm{NO}_{3} \mathrm{~S}\)) Next, multiply the number of carbon atoms in one molecule by the total number of molecules: C atoms in 2.00 mg of saccharin = (number of C atoms per molecule) × (number of molecules) C atoms in 2.00 mg of saccharin ≈ 7 × 6.56 × 10^{18} ≈ 4.59 × 10^{19} carbon atoms

Key Concepts

Saccharin Molecular Formula

Saccharin, a well-known artificial sweetener, has the molecular formula \(\mathrm{C}_{7} \mathrm{H}_{5} \mathrm{NO}_{3} \mathrm{~S}\). This formula tells us the types and numbers of atoms that make up each molecule of saccharin. Specifically:

• The **C** stands for carbon, and there are 7 carbon atoms.
• The **H** stands for hydrogen, with 5 atoms present.
• The **N** indicates nitrogen, with a single atom.
• The **O** means oxygen, and there are 3 oxygen atoms.
• The **S** is sulfur, with one sulfur atom.

This composition gives saccharin its unique properties and helps in calculating other chemical characteristics like molar mass. Understanding a molecule's formula is the foundation for studying its chemical behavior.

Avogadro's Number

Avogadro's Number is a fundamental constant in chemistry, denoted as \(6.022 \times 10^{23} \). It defines the number of atoms, molecules, or particles in one mole of a substance. This value is crucial when scaling chemical reactions from individual molecules to measurable quantities.When calculating the number of molecules in a sample, such as 2.00 mg of saccharin, Avogadro's Number helps to bridge the gap between the number of moles and the actual number of molecules. It turns the abstract concept of moles into tangible numbers. For example, once you have the number of moles, multiplying it by Avogadro's Number gives you the total count of molecules in your sample accurately.Hence, Avogadro's Number helps us understand and quantify the enormous number of entities involved in chemical processes, converting them into a form that can be observed and measured in experiments and industry.

Number of Moles Calculation

The number of moles in a substance connects the measurable mass of a sample to its molecular structure. This is done using the formula:\[\text{Number of moles} = \frac{\text{mass of substance}}{\text{molar mass of substance}}\]For saccharin, with a molar mass of approximately \(183.18 \text{ g/mol}\), calculating moles involves determining the mass in grams first. For example, 2.00 mg of saccharin is equivalent to \(0.00200 \text{ g}\). Using the formula, you calculate:\[\text{Number of moles of saccharin} = \frac{0.00200 \text{ g} }{183.18 \text{ g/mol} } \approx 1.09 \times 10^{-5} \text{ moles}\]Moles serve as a bridge between the mass of a material and the number of molecules or atoms it contains, allowing chemists to predict how substances will react and interact at the molecular level.

Atomic Structure of Compounds

The atomic structure of a compound is crucial to understanding its chemical properties and reactions. For a compound like saccharin, knowing its atomic structure involves recognizing the types and quantities of atoms present, as given by its molecular formula \(\mathrm{C}_{7} \mathrm{H}_{5} \mathrm{NO}_{3} \mathrm{~S}\).Each atom type in the structure contributes to the compound's overall mass, reactivity, and interactions. Chemical bonds formed between these atoms determine the shape and stability of the molecule.In saccharin:

• **Carbon** atoms form the backbone of the molecule, creating stable bonds with hydrogen and other atoms.
• **Hydrogen** completes the valence requirements of some carbon atoms, stabilizing the molecule.
• **Nitrogen**, **oxygen**, and **sulfur** provide functional groups that affect the compound's solubility and acidity.

Understanding atomic structures aids scientists and students in predicting how substances behave and react under various conditions, making it a fundamental aspect of chemistry studies.