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 \(\mathrm{C}_{2} \mathrm{H}\), and \(\mathrm{O}\) in cellulose? (b) What is the molar mass of cellulose?

Short answer

Answer: The mass percentages of C, H, and O in cellulose are 44.44%, 6.22%, and 49.34%, respectively. The molar mass of cellulose is 1,621,400 g/mol.

Step-by-step solution

Calculate the molar mass of one C6H10O5 unit.

Using the periodic table, we find the molar masses of C, H, and O. C: 12.01 g/mol H: 1.008 g/mol O: 16.00 g/mol Now, we can find the molar mass of one C6H10O5 unit: \(M = 6 \times 12.01 + 10 \times 1.008 + 5 \times 16 = 72.06 + 10.08 + 80 = 162.14 \mathrm{g/mol}\)

Calculate the mass percentages of C, H, and O in C6H10O5.

Divide the molar mass of each element by the molar mass of the C6H10O5 unit, and then multiply by 100 to find the percentages: Percent C: \(\frac{72.06 \mathrm{g/mol}}{162.14 \mathrm{g/mol}} \times 100 = 44.44 \% C\) Percent H: \(\frac{10.08 \mathrm{g/mol}}{162.14 \mathrm{g/mol}} \times 100 = 6.22 \% H\) Percent O: \(\frac{80 \mathrm{g/mol}}{162.14 \mathrm{g/mol}} \times 100 = 49.34 \% O\) (a) The mass percentages of C, H, and O in cellulose are 44.44%, 6.22%, and 49.34%, respectively.

Calculate the molar mass of cellulose.

To find the molar mass of cellulose, multiply the molar mass of one C6H10O5 unit by the number of units (10,000): \(M_\text{cellulose} = 162.14\ \mathrm{g/mol} \times 10,000 = 1,621,400\ \mathrm{g/mol}\) (b) The molar mass of cellulose is 1,621,400 g/mol.

Key concepts

Molar Mass Calculation

Understanding molar mass is crucial for grasping the basics of chemistry, especially when dealing with compounds like cellulose. The molar mass of a substance is the weight of 6.022 x 10^23 (Avogadro's number) of its particles, usually atoms or molecules, and is expressed in grams per mole (g/mol).

When calculating the molar mass of a compound like cellulose, which is a polymer consisting of many repeating units, we start by determining the molar mass of a single repeating unit. Each compound is made up of elements, and each element has an atomic mass that is found on the periodic table. For a single unit of C6H10O5, found within cellulose, the atomic masses for carbon (C), hydrogen (H), and oxygen (O) are respectively 12.01 g/mol, 1.008 g/mol, and 16.00 g/mol.

You simply multiply these atomic masses by the number of atoms of each element in one molecule of the substance and then add them together. This process reveals that a single repeating unit of cellulose has a molar mass of 162.14 g/mol. If you were to calculate the molar mass of cellulose itself, which is made of about 10,000 of these units, the sum would be significantly larger, illustrating the macromolecular nature of cellulose.

Mass Percent Composition

The mass percent composition of an element within a compound tells us what portion of the compound's total mass comes from that specific element. It's a way of breaking down the relative contribution of different atoms to the overall mass. This is particularly useful when you want to understand the makeup of large, complex molecules like cellulose.

To find the mass percent composition, divide the combined atomic mass of all atoms of the element in a formula by the molar mass of the entire compound and multiply by 100%. For instance, cellulose's formula unit C6H10O5 contains carbon, hydrogen, and oxygen. By dividing the total mass contributed by each type of atom by the molar mass of a single repeating unit and multiplying the result by 100, we can determine that cellulose consists of 44.44% carbon, 6.22% hydrogen, and 49.34% oxygen by mass. These percentages are vital for applications in fields like material science and biochemistry where the composition of materials determines their properties and suitability for specific uses.

Molecular Formula

A molecular formula is a representation of the numbers and types of atoms in a molecule. It provides a way of specifying the precise composition of a compound, which in turn relates to its properties and behaviours. Cellulose, for instance, has the basic repeating unit expressed as C6H10O5, indicating six carbon atoms, ten hydrogen atoms, and five oxygen atoms in each unit.

However, as a polymer, cellulose consists of a very large number of these repeating units linked together - around 10,000 in typical cellulose molecules found in nature. The molecular formula for the entire cellulose molecule would therefore include approximately 60,000 carbon atoms, 100,000 hydrogen atoms, and 50,000 oxygen atoms, giving insight into the sheer size and complexity of this natural polymer. Understanding the molecular formula can help students and researchers understand the reaction mechanisms and interactions of cellulose at a molecular level, which is essential for its various applications in industries such as textiles, paper, and biodegradable materials.