Question

Calculate the number of carbon atoms in \(1.00 \mathrm{~g}\) of blood sugar, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\).

Short answer

There are approximately \(2.00 \times 10^{22}\) carbon atoms in \(1.00\, \text{g}\) of glucose.

Step-by-step solution

Determine Molar Mass of Glucose

Identify the molecular formula for glucose, which is \( \text{C}_6\text{H}_{12}\text{O}_6 \). Calculate its molar mass by adding the atomic masses of all atoms in the formula: \( 6 \times \text{(Carbon)} + 12 \times \text{(Hydrogen)} + 6 \times \text{(Oxygen)} \). The atomic masses are approximately \(12.01\, \text{amu} \) for carbon, \(1.01\, \text{amu} \) for hydrogen, and \(16.00\, \text{amu} \) for oxygen. Hence, the molar mass \[ \text{Molar Mass of Glucose} = 6 \times 12.01 + 12 \times 1.01 + 6 \times 16.00 = 180.18 \, \text{g/mol}. \]

Calculate Moles of Glucose

Use the formula for moles: \( \text{moles} = \frac{\text{mass}}{\text{molar mass}} \). Insert the given mass of glucose \(1.00\, \text{g} \) and molar mass \(180.18\, \text{g/mol} \) to find the moles of glucose: \[ \text{Moles of Glucose} = \frac{1.00}{180.18} \approx 0.00555 \, \text{mol}. \]

Determine Number of Glucose Molecules

Utilize Avogadro's number \(6.022 \times 10^{23} \) to find the number of glucose molecules in the moles calculated. Multiply moles of glucose by Avogadro's number: \[ \text{Number of Glucose Molecules} = 0.00555 \times 6.022 \times 10^{23} \approx 3.34 \times 10^{21} \, \text{molecules}. \]

Calculate Number of Carbon Atoms

Since each molecule of glucose contains 6 carbon atoms, multiply the number of glucose molecules by 6 to get the number of carbon atoms. \[ \text{Number of Carbon Atoms} = 3.34 \times 10^{21} \times 6 \approx 2.00 \times 10^{22} \, \text{atoms}. \]

Key concepts

Avogadro's Number

Avogadro's Number is a fundamental constant in chemistry that connects the macroscopic and microscopic worlds. This number, approximately \(6.022 \times 10^{23}\), represents the number of atoms, ions, or molecules in one mole of a substance. It helps chemists easily convert between the mass of a substance and the number of particles it contains.
When dealing with chemical computations, such as calculating the number of molecules or atoms in a given sample, Avogadro's Number is a crucial tool. It allows us to determine how many individual entities (like atoms or molecules) are present by simply multiplying the amount in moles by this constant.
For example, if we have \(0.00555\) moles of glucose, we can find the total number of glucose molecules by multiplying by Avogadro's Number: \(0.00555 \times 6.022 \times 10^{23} \approx 3.34 \times 10^{21}\) molecules. This gives us a detailed understanding of the quantity at a molecular level, making Avogadro's Number an indispensable part of the mole concept.

Molar Mass Calculation

In chemistry, the molar mass is the mass of one mole of a substance, often expressed in grams per mole (g/mol). Calculating the molar mass is essential for stoichiometry and various chemical calculations. It allows us to convert between the mass of a substance and the amount in moles, bridging macroscopic measurements and molecular quantities.
To calculate the molar mass of a compound, you add up the atomic masses of all atoms in its molecular formula. With glucose (\( \text{C}_6\text{H}_{12}\text{O}_6 \)), the calculation involves the addition of atomic masses of 6 carbons, 12 hydrogens, and 6 oxygens:

• Carbon: \(6 \times 12.01\, \text{g/mol} \)
• Hydrogen: \(12 \times 1.01\, \text{g/mol} \)
• Oxygen: \(6 \times 16.00\, \text{g/mol} \)

Adding these gives us a molar mass of \(180.18\, \text{g/mol} \) for glucose.
Understanding how to determine the molar mass is crucial as it is a stepping stone to more complex stoichiometric calculations and is used in virtually every aspect of chemistry dealing with reactions and compounds.

Stoichiometry

Stoichiometry is the study of the quantitative relationships between reactants and products in a chemical reaction. It is grounded in the principle that matter is neither created nor destroyed during a chemical reaction. This makes stoichiometry an essential tool for predicting the amounts of products formed in reactions or the reactants needed to achieve a desired product quantity.
In the context of our exercise with glucose, stoichiometry starts by using the molar mass to determine the number of moles present in a given mass. Here, knowing the molar mass of glucose as \(180.18\, \text{g/mol} \) allows us to convert \(1.00\, \text{g} \) of glucose into moles: \(\text{moles of glucose} = \frac{1.00}{180.18} \approx 0.00555\, \text{mol} \).
Once we have the moles, stoichiometry and Avogadro's Number help us calculate the number of molecules or atoms. For glucose, by knowing there are \(3.34 \times 10^{21}\) molecules of glucose, we can further determine the number of specific atoms, like carbon. Each molecule of glucose has 6 carbon atoms, so \(3.34 \times 10^{21} \times 6\). This results in around \(2.00 \times 10^{22}\) carbon atoms.
Stoichiometry is, therefore, the bridge that allows us to understand the connection between mass and amounts at the molecular level, making it an essential component of chemical problem-solving.