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?

Short answer

(a) The number of hydrogen atoms in the glucose sample is \(2.500 \times 10^{21}\). (b) The number of glucose molecules in the sample is \(\approx 2.08 \times 10^{20}\). (c) The number of moles of glucose is \(\approx 3.46 \times 10^{-3}\). (d) The mass of the glucose sample is \(\approx 0.623 \ \text{grams}\).

Step-by-step solution

Determine the number of hydrogen atoms

Since we know the number of carbon atoms in the glucose sample, we can determine the corresponding number of hydrogen atoms using the ratio of hydrogen to carbon atoms. The chemical formula for glucose is \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\) which indicates that there are two hydrogen atoms for each carbon atom. Thus, the number of hydrogen atoms can be calculated by: Number of hydrogen atoms = Number of carbon atoms × (Hydrogen atoms / Carbon atoms) Number of hydrogen atoms = \((1.250 \times 10^{21} \ \text{carbon atoms})(\frac{12 \ \text{hydrogen atoms}}{6 \ \text{carbon atoms}})\)

Calculate the number of hydrogen atoms

Now we can calculate the number of hydrogen atoms in the glucose sample: Number of hydrogen atoms = \((1.250 \times 10^{21})(2)\) Number of hydrogen atoms = \(2.500 \times 10^{21}\) hydrogen atoms

Calculate the number of glucose molecules

Since for each glucose molecule the number of carbon atoms is 6, we can determine the number of glucose molecules by dividing the total number of carbon atoms by 6: Number of glucose molecules = Number of carbon atoms / 6 Number of glucose molecules = \((1.250 \times 10^{21})/6\)

Calculate the number of moles of glucose

To find the number of moles of glucose, we need to use Avogadro's number (\(6.022 \times 10^{23}\) particles/mol). The equation for calculating the number of moles is: Moles = Number of particles / Avogadro's number Moles of glucose = Number of glucose molecules / \(6.022 \times 10^{23}\) mol\(^{-1}\) Moles of glucose = \((1.250 \times 10^{21})/(6 \times 6.022 \times 10^{23})\)

Calculate the mass of the glucose sample in grams

To find the mass of the glucose sample, we will use the molar mass of glucose (\(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\)) which is \(180.16 \ \text{g/mol}\). The mass can be determined by multiplying the number of moles by the molar mass: Mass = Moles × Molar mass Mass = Moles of glucose × \(180.16 \ \text{g/mol}\) Mass = [\((1.250 \times 10^{21})/(6 \times 6.022 \times 10^{23})\)] × \(180.16 \ \text{g/mol}\) By calculating these expressions, we will find the answers for parts (a), (b), (c), and (d): (a) The number of hydrogen atoms is \(2.500 \times 10^{21}\). (b) The number of glucose molecules is \(\approx 2.08 \times 10^{20}\). (c) The number of moles of glucose is \(\approx 3.46 \times 10^{-3}\). (d) The mass of the glucose sample is \(\approx 0.623 \ \text{grams}\).

Key concepts

Molar Mass

The concept of molar mass is essential in chemistry as it links the mass of a substance to the amount of material it contains. Molar mass is expressed in units of grams per mole (g/mol) and represents the mass of one mole of a given substance. For glucose, which has the chemical formula \( \mathrm{C}_{6}\mathrm{H}_{12}\mathrm{O}_{6} \), the molar mass is calculated by adding up the atomic masses of all the atoms in a glucose molecule.
Let's break it down:

• Carbon (C): There are 6 carbon atoms in glucose, each with an atomic mass of approximately 12.01 g/mol, so \( 6 \times 12.01 = 72.06 \) g/mol.
• Hydrogen (H): There are 12 hydrogen atoms, each with an atomic mass of about 1.01 g/mol, so \( 12 \times 1.01 = 12.12 \) g/mol.
• Oxygen (O): There are 6 oxygen atoms, each weighing around 16.00 g/mol, giving us \( 6 \times 16.00 = 96.00 \) g/mol.

Adding these together, the molar mass of glucose is \( 72.06 + 12.12 + 96.00 = 180.18 \) g/mol. This value is used to convert between grams and moles of a substance, aiding us in determining amounts needed or produced in a chemical reaction.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry, represented as \( 6.022 \times 10^{23} \). It defines the number of atoms, molecules, or particles in one mole of a substance. Essentially, it's the bridge that connects the atomic or molecular scale to the macroscopic scale that we can measure and observe.
This number is especially important when we want to convert between the number of particles (like atoms or molecules) and moles, which are more practically handled when measuring amounts of substances in the lab.
In the context of the exercise, using Avogadro's number allows us to calculate the number of moles of glucose from the number of glucose molecules. For example, if we have \( 2.08 \times 10^{20} \) molecules of glucose, we can calculate the moles of glucose by dividing the number of molecules by Avogadro's number. This results in a very small number: \( \approx 3.46 \times 10^{-3} \) moles of glucose.

Moles of Glucose

The concept of moles is central to understanding and relating the quantities of substances in chemistry. A mole is a unit that measures the amount of a chemical substance. It is similar to saying 'a dozen' for 12, but rather, a mole signifies \( 6.022 \times 10^{23} \) entities, thanks to Avogadro's number.
In the given problem, determining the moles of glucose involves a few straightforward calculations. We have a certain number of glucose molecules, and by dividing this number by Avogadro's number, we can find how many moles of glucose we have. The solution to the exercise presented the calculation:
\[ \text{Moles of glucose} = \frac{1.250 \times 10^{21}}{6 \times 6.022 \times 10^{23}} \approx 3.46 \times 10^{-3} \text{ moles} \]
Understanding the concept of moles helps when you're scaling reactions in a lab or simply calculating how much of a compound is present in a sample. It's a practical way to count particles when dealing with quantities large enough to weigh or measure with standard laboratory equipment.