Question

Cellulose consists of about \(10,000 \mathrm{C}_{6} \mathrm{H}_{10} \mathrm{O}_{5}\) units linked together. (a) What are the mass percents of \(\mathrm{C}, \mathrm{H},\) and \(\mathrm{O}\) in cellulose? (b) What is the molar mass of cellulose?

Short answer

Answer: The mass percents of carbon (C), hydrogen (H), and oxygen (O) in cellulose are 44.44%, 6.23%, and 49.33%, respectively. The molar mass of cellulose is 1,621,600 g/mol.

Step-by-step solution

Determine the molecular mass of C6H10O5

To find the molar mass of one C6H10O5 unit, we need to multiply the molar mass of each element by its number of atoms and add them together. The molar mass of carbon (C) is 12.01 g/mol, hydrogen (H) is 1.01 g/mol, and oxygen (O) is 16.00 g/mol. Molecular mass of C6H10O5 = (6 * 12.01) + (10 * 1.01) + (5 * 16.00) = 72.06 + 10.10 + 80.00 = 162.16 g/mol

Calculate the mass percent of each element in C6H10O5

To find the mass percent of each element in cellulose, divide the mass of each element in one C6H10O5 unit by the total mass of one C6H10O5 unit and multiply by 100. Mass percent of C = (Mass of C / Molecular mass of C6H10O5) * 100 = (72.06 / 162.16) * 100 = 44.44% Mass percent of H = (Mass of H / Molecular mass of C6H10O5) * 100 = (10.10 / 162.16) * 100 = 6.23% Mass percent of O = (Mass of O / Molecular mass of C6H10O5) * 100 = (80.00 / 162.16) * 100 = 49.33%

Calculate the molar mass of cellulose

Now that we know the mass percent of each element in cellulose, we can find the molar mass of cellulose, which is the product of the molar mass of one C6H10O5 unit and the number of C6H10O5 units in cellulose (10,000). Molar mass of cellulose = Molecular mass of C6H10O5 * Number of units = 162.16 g/mol * 10,000 = 1,621,600 g/mol So, the mass percents of C, H, and O in cellulose are 44.44%, 6.23%, and 49.33%, respectively, and the molar mass of cellulose is 1,621,600 g/mol.

Key concepts

Molar Mass Calculation

Understanding the molar mass of substances is crucial for gaining insight into their chemical composition and for performing various calculations in chemistry. The molar mass represents the weight of one mole of a substance, typically measured in grams per mole (g/mol). One mole is Avogadro's number, approximately equal to 6.022 x 10²³ particles—atoms, molecules, ions, or electrons.

To calculate the molar mass of a compound like cellulose or even a simpler molecule, you need to sum up the molar masses of the individual atoms within a single unit of the substance. For instance, glucose (\text(C_6H_12O_6\text)) is a simple sugar with a molar mass that is calculated by adding the molar masses of 6 carbon atoms, 12 hydrogen atoms, and 6 oxygen atoms.

Here is a simplified representation using the molecular formula of glucose as an example: \[ MolarMass_{C_6H_12O_6} = (6 \times MolarMass_C) + (12 \times MolarMass_H) + (6 \times MolarMass_O) \]The same process is used for cellulose, which is a polymer consisting of repeating glucose units.

Mass Percent Composition

Mass percent composition indicates the proportion of each element in a compound. It's an important concept, particularly when dealing with chemical formulas and reactions. To calculate the mass percent of an element, one divides the total mass of that element in the molecule by the total molecular mass of the compound and then multiplies by 100 to convert it into a percentage.

For practical purposes, let's consider water (\text(H_2O\text)). Water consists of 2 hydrogen atoms and 1 oxygen atom. The mass percent composition of hydrogen in water would be calculated using its molar mass (about 1.01 g/mol for H) and the molar mass of water (about 18.02 g/mol), resulting in a mass percent for hydrogen: \[ MassPercent_H = \left( \frac{2 \times 1.01}{18.02} \right) \times 100%\]And similarly for oxygen. Understanding mass percent composition is vital, especially when determining the formula of a compound from experimental data or when preparing solutions in chemistry.

Molecular Mass of Compounds

Molecular mass, often used interchangeably with molecular weight, refers to the sum of the molar masses of the atoms in a molecule. It's a key aspect in stoichiometry, which is the quantitative relationship between reactants and products in a chemical reaction. Knowing the molecular mass helps determine how much of a compound reacts or is produced in a reaction.

For a compound like cellulose, which is a large molecule made up of many repeated glucose units, the molecular mass is essentially the mass of one of these units multiplied by the number of units in the molecule. Calculating molecular mass requires precise knowledge of the molecular formula and the molar mass of each element present. Additionally, it's important to note that for large polymers like cellulose, the molecular mass can be very large and the terms 'molecular mass' and 'molar mass' might be used in specific contexts when discussing the mass of one repeating unit versus the entire macromolecule.