Problem 41 - Chemistry The Central Science In Si Units

Question

A sample of glucose, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6},\) contains \(1.250 \times 10^{21}\) carbon atoms. (a) How many atoms of hydrogen does it contain? (b) How many molecules of glucose does it contain? (c) How many moles of glucose does it contain? (d) What is the mass of this sample in grams?

Step-by-Step Solution

Answer

(a) 2.500 × 10²¹ hydrogen atoms; (b) 2.083 × 10²⁰ glucose molecules; (c) 3.46 × 10⁻⁴ moles; (d) 0.0623 grams.

Step 1: Determine the number of hydrogen atoms

In one molecule of glucose, which is \(\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6\), there are 6 carbon atoms and correspondingly 12 hydrogen atoms. Thus, for every carbon atom, there are \(12/6 = 2\) hydrogen atoms. Therefore, the sample contains \(2 \times 1.250 \times 10^{21} = 2.500 \times 10^{21}\) hydrogen atoms.

Step 2: Find the number of glucose molecules

Each glucose molecule contains 6 carbon atoms. Thus, the number of glucose molecules is the total number of carbon atoms divided by the number of carbon atoms per molecule: \[\text{Number of glucose molecules} = \frac{1.250 \times 10^{21}}{6}\approx 2.083 \times 10^{20}.\]

Step 3: Calculate the number of moles of glucose

The number of moles can be calculated by dividing the number of molecules by Avogadro's number \(6.022 \times 10^{23}\) molecules/mole: \[\text{Moles of glucose} = \frac{2.083 \times 10^{20}}{6.022 \times 10^{23}} \approx 3.46 \times 10^{-4}\text{ moles}.\]

Step 4: Determine the mass of the glucose sample in grams

The molar mass of glucose is calculated as follows: - Carbon: 6 atoms \(\times\) 12.01 g/mol = 72.06 g/mol- Hydrogen: 12 atoms \(\times\) 1.008 g/mol = 12.096 g/mol- Oxygen: 6 atoms \(\times\) 16.00 g/mol = 96.00 g/molTotal Molar Mass = 180.156 g/molThe mass is then: \[\text{Mass} = 3.46 \times 10^{-4} \text{ moles} \times 180.156 \text{ g/mol} \approx 0.0623 \text{ grams}.\]

Key Concepts

Glucose Molecular Formula

The molecular formula for glucose is \( \mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6 \). This comprehensive formula tells us about the number and type of atoms present in a single molecule of glucose. It consists of:

• 6 carbon atoms \((\mathrm{C}_6)\)
• 12 hydrogen atoms \((\mathrm{H}_{12})\)
• 6 oxygen atoms \((\mathrm{O}_6)\)

This combination of carbon, hydrogen, and oxygen structures glucose, which is a carbohydrate crucial for energy storage in living organisms. In chemical equations, knowing the molecular formula helps determine the amounts of each element involved. For instance, if you know how many carbon atoms you have in a glucose sample, you can easily calculate the hydrogen and oxygen atoms using the molecular ratio from the formula.

Avogadro's Number

Avogadro's Number, \( 6.022 \times 10^{23} \), is a fundamental constant in chemistry that represents the number of atoms, ions, or molecules in one mole of a substance. This incredibly large number is named after the scientist Amedeo Avogadro and is essential for converting between the number of particles and the amount of substance in moles.
Why do we need Avogadro's Number? Consider if you have a measured number of glucose molecules and you want to understand how many moles it represents. Use this relationship:

• Number of Moles = \( \frac{\text{Number of Particles}}{\text{Avogadro's Number}} \)

For instance, if a sample contains \( 2.083 \times 10^{20} \) glucose molecules, you divide this by \( 6.022 \times 10^{23} \) to find approximately \( 3.46 \times 10^{-4} \) moles. This constant bridges the molecular scale with the macroscopic scale, making it possible to relate bulk quantities of substances to the counts of atoms and molecules.

Molar Mass Calculation

Molar mass is the mass of one mole of a given substance. It's expressed in grams per mole (g/mol) and calculated by adding up the atomic masses of all atoms in a molecular formula. Let's consider glucose, \( \mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6 \), for example.
To find the molar mass of glucose:

• Determine the atomic masses: Carbon (\(\mathrm{C}\)) = 12.01 g/mol, Hydrogen (\(\mathrm{H}\)) = 1.008 g/mol, and Oxygen (\(\mathrm{O}\)) = 16.00 g/mol.
• Multiply each by the number of atoms:
  • Carbon: 6 atoms \( \times 12.01 = 72.06 \text{ g/mol} \)
  • Hydrogen: 12 atoms \( \times 1.008 = 12.096 \text{ g/mol} \)
  • Oxygen: 6 atoms \( \times 16.00 = 96.00 \text{ g/mol} \)
• Add up all the contributions: Total molar mass = \(72.06 + 12.096 + 96.00 = 180.156 \text{ g/mol} \)

This calculated molar mass is crucial, especially when you need to convert between moles and grams in chemical calculations. For example, knowing that a sample has \( 3.46 \times 10^{-4} \) moles of glucose, you can multiply by the molar mass to find its mass in grams, which would be approximately 0.0623 grams.