Problem 39 - Chemistry: The Central Science

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

The glucose sample contains \(2.500 \times 10^{21}\) hydrogen atoms, \(2.083 \times 10^{20}\) glucose molecules, \(3.461 \times 10^{-4}\) moles of glucose, and has a mass of \(6.240 \times 10^{-2}\) grams.

Step 1: (Step 1: Find the number of hydrogen atoms in the glucose sample)

To find the number of hydrogen atoms in the glucose sample, first, notice that for every one glucose molecule, there are 12 hydrogen atoms. Since we know there are \(1.250 \times 10^{21}\) carbon atoms in the glucose sample and in one glucose molecule there is 6 carbon atoms, we can calculate the number of hydrogen atoms as follows: Number of hydrogen atoms = (Number of carbon atoms / 6) * 12 Now, plug in the given value of carbon atoms and calculate the number of hydrogen atoms: Number of hydrogen atoms = (\((1.250 \times 10^{21}) / 6) * 12 = 2.500 \times 10^{21}\) .

Step 2: (Step 2: Find the number of glucose molecules in the glucose sample)

Since every glucose molecule has 6 carbon atoms, we can find the number of glucose molecules by dividing the total number of carbon atoms by 6: Number of glucose molecules = (Number of carbon atoms) / 6 Now, plug in the given value of carbon atoms and calculate the number of glucose molecules: Number of glucose molecules = \((1.250 \times 10^{21})/6 = 2.083 \times 10^{20}\).

Step 3: (Step 3: Find the number of moles of glucose in the glucose sample)

To find the number of moles of glucose, we will use Avogadro's number (\(6.022 \times 10^{23}\)) which is the number of particles (atoms/molecules) in one mole of a substance. We will divide the number of glucose molecules by Avogadro's number: Number of moles of glucose = (Number of glucose molecules) / (Avogadro's number) Plug in the calculated number of glucose molecules and Avogadro's number: Number of moles of glucose = \((2.083 \times 10^{20})/(6.022 \times 10^{23}) = 3.461 \times 10^{-4}\) moles.

Step 4: (Step 4: Find the mass of the glucose sample in grams)

To find the mass of the glucose sample, we will use the formula: Mass = (Number of moles) * (Molar mass of glucose) The molar mass of glucose is given by the sum of the molar mass of each element multiplied by the respective number of atoms in the glucose formula (\(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\)). We have: Molar mass of glucose = (6 * 12.01) + (12 * 1.01) + (6 * 16.00) = 72.06 + 12.12 + 96.00 = 180.18 g/mol Now, plug in the calculated number of moles of glucose and the molar mass of glucose: Mass = \((3.461 \times 10^{-4}) \times 180.18 = 6.240 \times 10^{-2}\) grams.

Key Concepts

Moles Calculation

Understanding how to calculate moles is a fundamental aspect of chemistry and helps to quantify substances in chemical reactions. Moles are a unit of measurement that denote the amount of a chemical substance. Calculating moles allows chemists to relate macroscopic quantities to the molecular level.

The concept of moles relates heavily to Avogadro's number, as one mole of any substance contains exactly Avogadro's number of particles, which is approximately \(6.022 \times 10^{23}\) particles. In the case of the glucose problem, the number of glucose molecules was divided by Avogadro's number to determine the number of moles:

• Glucose molecules = \(2.083 \times 10^{20}\)
• Avogadro's number = \(6.022 \times 10^{23}\)

The division provides us with \(3.461 \times 10^{-4}\) moles of glucose.

Molar Mass Determination

Molar mass is the mass per mole of a substance and is measured in grams per mole (g/mol). It acts as a bridge between the atomic and macroscopic worlds, allowing us to calculate the mass of grams we have when we know the number of moles.

To find the molar mass of a compound like glucose (\(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\)), add together the molar masses of all atoms in a molecule. Each atom's molar mass can be found on the periodic table. For glucose:

• Carbon: \(6 \times 12.01 = 72.06\) g/mol
• Hydrogen: \(12 \times 1.01 = 12.12\) g/mol
• Oxygen: \(6 \times 16.00 = 96.00\) g/mol

Adding these together gives the molar mass of glucose as \(180.18\) g/mol. Knowing this value is crucial to finding the total mass of glucose when the number of moles is known.

Avogadro's Number

Avogadro's number \((6.022 \times 10^{23})\) is an essential figure in chemistry that represents the number of atoms, ions, or molecules in one mole of a substance. Named after the scientist Amedeo Avogadro, it creates a direct link between the mass-based macroscopic scale and the molecular scale.

In the context of the glucose calculation, Avogadro's number allows the translation of individual glucose molecules into moles. Understanding this constant is vital for any operations involving conversions between number of particles and amount in moles. For instance, when you know how many molecules you have, simply divide by Avogadro's number to get the amount in moles.

Molecule Counting

Counting molecules can be quite an involved process, especially when large numbers are involved. However, by breaking it down into manageable parts, the task becomes more straightforward. For the glucose problem, the number of carbon atoms was used as a key entry point.

A molecule of glucose has 6 carbon atoms, and if you know the total number of carbon atoms, you can easily find the number of entire glucose molecules. This is done by dividing the total carbon atoms by 6. For example:

• Total carbon atoms = \(1.250 \times 10^{21}\)
• Molecules of glucose = \(\frac{1.250 \times 10^{21}}{6} = 2.083 \times 10^{20}\)

Such calculations are essential when dealing with reactions in which the number of molecules affects the reaction's stoichiometry or its yield.