Problem 42 - Chemistry Matter and Change

Question

Ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\) , a domestically produced fuel source, is often blended with gasoline. A sample of ethanol has a mass of 45.6 \(\mathrm{g} .\) a. How many carbon atoms does the sample contain? b. How many hydrogen atoms are present? c. How many oxygen atoms are present?

Step-by-Step Solution

Answer

The given sample of ethanol contains \(1.193 \times 10^{24}\) carbon atoms, \(3.580 \times 10^{24}\) hydrogen atoms, and \(5.963 \times 10^{23}\) oxygen atoms.

Step 1: Calculate the molar mass of ethanol

To do this, we need to find the atomic mass of each atom and add them up. The atomic masses are: C: 12.01 g/mol, H: 1.008 g/mol, and O: 16.00 g/mol. Ethanol (C2H5OH) contains 2 carbon (C) atoms, 6 hydrogen (H) atoms, and 1 oxygen (O) atom. So, the molar mass of ethanol = 2(12.01) + 6(1.008) + 16.00 = 24.02 + 6.048 + 16.00 = 46.068 g/mol.

Step 2: Determine the moles of ethanol in the given sample

We are given the mass of ethanol as 45.6 g. To find the moles of ethanol, divide the mass by the molar mass. Moles of ethanol = \(\frac{45.6 \,\text{g}}{46.068\, \text{g/mol}} = 0.9898 \, \text{mol}\)

Step 3: Calculate the number of carbon, hydrogen, and oxygen atoms (a, b, and c)

To find the number of atoms, multiply the moles by Avogadro's number (6.022 x 10^23). a. For carbon atoms: Number of carbon atoms = Number of moles of ethanol x 2 x Avogadro's number = (0.9898 mol) x 2 x 6.022 x 10^23 = 1.193 x 10^24 carbon atoms b. For hydrogen atoms: Number of hydrogen atoms = Number of moles of ethanol x 6 x Avogadro's number = (0.9898 mol) x 6 x 6.022 x 10^23 = 3.580 x 10^24 hydrogen atoms c. For oxygen atoms: Number of oxygen atoms = Number of moles of ethanol x 1 x Avogadro's number = (0.9898 mol) x 1 x 6.022 x 10^23 = 5.963 x 10^23 oxygen atoms So, the given sample of ethanol contains 1.193 x 10^24 carbon atoms, 3.580 x 10^24 hydrogen atoms, and 5.963 x 10^23 oxygen atoms.

Key Concepts

Molar Mass of Ethanol

Understanding the molar mass of a substance is critical for converting between mass and number of moles. For ethanol \( \mathrm{C}_{2}\mathrm{H}_{5}\mathrm{OH} \), the molar mass is the sum of the atomic masses of all the atoms in one molecule of ethanol.

To calculate it, we need the atomic masses for carbon (C), hydrogen (H), and oxygen (O), which are approximately 12.01 g/mol, 1.008 g/mol, and 16.00 g/mol, respectively. Ethanol has 2 carbon atoms, 6 hydrogen atoms, and 1 oxygen atom. Thus, the molar mass is calculated as

\[\text{Molar Mass of Ethanol} = 2(12.01 \, \text{g/mol}) + 6(1.008 \, \text{g/mol}) + 16.00 \, \text{g/mol} = 46.068 \, \text{g/mol}\.\]
When you know the mass of ethanol, you can find the number of moles by dividing the mass by its molar mass. This step is crucial for all subsequent calculations involving the amount of substance.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry that represents the number of particles in one mole of a substance. The fixed numerical value is approximately \(6.022 \times 10^{23}\).

This value allows chemists to convert between moles and the actual number of atoms or molecules. For example, if we have one mole of ethanol, according to Avogadro's principle, it contains \(6.022 \times 10^{23}\) ethanol molecules. To find the total number of atoms in a mass of ethanol, one would multiply the number of moles of ethanol by Avogadro's number and the number of each type of atom within an ethanol molecule.

For instance, for carbon atoms in ethanol:\[\text{Number of Carbon Atoms} = (\text{Number of Moles of Ethanol}) \times 2 \times (\text{Avogadro's Number})\.\]
This process applies similarly to calculate the quantity of hydrogen and oxygen atoms as part of ethanol's composition.

Empirical and Molecular Formulas

The empirical formula of a compound gives the simplest whole-number ratio of atoms of each element in the compound, while the molecular formula represents the actual number of atoms of each element in a molecule of the compound.

For ethanol \(\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{OH}\), \(\mathrm{C}_{2}\mathrm{H}_{6}\mathrm{O}\) is its molecular formula, indicating two carbon atoms, six hydrogen atoms, and one oxygen atom. However, the empirical formula for ethanol would be simply \(\mathrm{CH}_{3}\mathrm{O}\), showing the relative ratios of the atoms.

Determining empirical and molecular formulas is a leap towards understanding the composition and reactions of substances. It's important to know that while the empirical formula offers valuable information on the proportion of elements, only the molecular formula unveils the absolute number of atoms present in a single molecule, which can be quantified into moles using Avogadro's number for practical applications in scientific calculations.