Problem 40 - Chemistry: The Central Science in SI Units, Global Edition

Question

The molecular formula of salicylic acid, a compound commonly found in facial cleanser, is \(\mathrm{C}_{7} \mathrm{H}_{6} \mathrm{O}_{3}\). (a) What is the molar mass of salicylic acid? (b) How many moles of salicylic acid are present in \(0.5 \mathrm{mg}\) of this substance? (c) How many molecules of salicylic acid are in \(0.5 \mathrm{mg}\) of this substance? (d) How many oxygen atoms are present in \(0.5 \mathrm{mg}\) of salicylic acid?

Step-by-Step Solution

Answer

(a) The molar mass of salicylic acid is 138.13 g/mol. (b) There are 3.62 x 10^(-6) moles of salicylic acid in 0.5 mg of the substance. (c) There are 2.18 x 10^(18) molecules of salicylic acid in 0.5 mg of the substance. (d) There are 6.53 x 10^(18) oxygen atoms in 0.5 mg of salicylic acid.

Step 1: Calculate the molar mass of salicylic acid

To calculate the molar mass of salicylic acid (C₇H₆O₃), we need to find the sum of the molar mass of each of its elements (C, H, and O) as they appear in the molecular formula: Molar mass of salicylic acid = (7 x molar mass of Carbon) + (6 x molar mass of Hydrogen) + (3 x molar mass of Oxygen) Using the standard atomic weights: Carbon: 12.01 g/mol Hydrogen: 1.01 g/mol Oxygen: 16 g/mol Molar mass of salicylic acid = (7 x 12.01) + (6 x 1.01) + (3 x 16) = 84.07 + 6.06 + 48 = 138.13 g/mol

Step 2: (b) Calculate the number of moles of salicylic acid in 0.5 mg

To find the number of moles of salicylic acid in 0.5 mg, we will use the following formula: Number of moles = mass of substance / molar mass First, convert 0.5 mg to grams: 0.5 mg = 0.0005 g Now, divide the mass of substance by its molar mass: Number of moles = 0.0005 g / 138.13 g/mol = 3.62 x 10^(-6) moles

Step 3: (c) Calculate the number of molecules of salicylic acid in 0.5 mg

To find the number of molecules of salicylic acid in 0.5 mg, we will use Avogadro's number, which is approximately 6.022 x 10^(23) particles (atoms, ions, or molecules) per mole. Number of molecules = number of moles x Avogadro's number Number of molecules = 3.62 x 10^(-6) moles x 6.022 x 10^(23) molecules/mole = 2.18 x 10^(18) molecules

Step 4: (d) Calculate the number of oxygen atoms in 0.5 mg of salicylic acid

To find the number of oxygen atoms in 0.5 mg of salicylic acid, we will first determine the number of moles of oxygen present in our given amount of salicylic acid. Since there are 3 oxygen atoms in one molecule of salicylic acid, the number of moles of oxygen will be three times the number of moles of salicylic acid. Number of moles of oxygen = 3 x 3.62 x 10^(-6) moles = 10.86 x 10^(-6) moles Now, we will multiply the number of moles of oxygen by Avogadro's number to obtain the total number of oxygen atoms: Number of oxygen atoms = 10.86 x 10^(-6) moles x 6.022 x 10^(23) atoms/mole = 6.53 x 10^(18) oxygen atoms

Key Concepts

Molecular Formula

The molecular formula of a compound is like its unique identifier which reveals the types and numbers of atoms present in a single molecule of that compound. In the case of salicylic acid, the molecular formula is \( \mathrm{C}_{7} \mathrm{H}_{6} \mathrm{O}_{3} \). This tells us that each molecule contains:

• 7 carbon atoms
• 6 hydrogen atoms
• 3 oxygen atoms

Each element in this formula contributes a specific mass based on its atomic weight, which is essential for calculating the molar mass of the compound.
Understanding the molecular formula is crucial for doing any calculations that involve the compound's molar mass or determining the number of atoms in a given sample. For instance, if we know how many molecules of salicylic acid we have, we can easily calculate how many atoms of carbon, hydrogen, and oxygen are there by looking at its molecular formula.

Avogadro's Number

Avogadro's number is a fundamental concept in chemistry that connects the macroscopic and microscopic worlds. It is approximately \( 6.022 \times 10^{23} \), which represents the number of particles, such as atoms, molecules, or ions, contained in one mole of a substance.
This large number allows chemists to count incredibly tiny and numerous entities by weighing out practical amounts of a substance in moles. For example, when we calculated the number of molecules present in \( 0.5 \) mg of salicylic acid, we used Avogadro's number to convert moles into individual molecules. Knowing the number of moles (\( 3.62 \times 10^{-6} \) moles in this specific case), multiplying by Avogadro's number gives us the total number of molecules present.
Avogadro's number is critical for understanding chemical reactions because it helps determine how many molecules or atoms participate in a reaction, enabling precise stoichiometric calculations.

Oxygen Atoms

Oxygen atoms are an essential component in many chemical compounds, including salicylic acid (\( \mathrm{C}_{7} \mathrm{H}_{6} \mathrm{O}_{3} \)). Each molecule of salicylic acid contains 3 oxygen atoms. As we calculated earlier, in \( 0.5 \) mg of salicylic acid, there are \( 2.18 \times 10^{18} \) molecules. To find the total number of oxygen atoms, we multiply the number of molecules by the number of oxygen atoms per molecule.
Therefore, we have:

• Number of moles of oxygen: \( 3 \times 3.62 \times 10^{-6} \) moles
• Total number of oxygen atoms: \( 6.53 \times 10^{18} \) oxygen atoms

Understanding the presence and quantity of specific atoms like oxygen in compounds is crucial for chemical analysis and reactions, influencing how these compounds interact in biological systems and industrial applications.