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 \(1.250 \times 10^{21} \times \frac{12}{6} = 2.5 \times 10^{21}\). (b) The number of glucose molecules is \(\frac{1.250 \times 10^{21}}{6} = 2.083 \times 10^{20}\). (c) The number of moles of glucose is \(\frac{2.083 \times 10^{20}}{6.022\times10^{23}} = 3.460 \times 10^{-4}\) moles. (d) The mass of the glucose sample is \(3.460 \times 10^{-4}\) moles \(\times\) 180.16 g/mol = 0.06234 g.

Step-by-step solution

Calculate the number of hydrogen atoms

In a glucose molecule, we have a ratio of 6:12:6 for the number of carbon, hydrogen, and oxygen atoms, respectively. To find the number of hydrogen atoms, we can use the given information about carbon atoms in the glucose sample. Given that there are \(1.250 \times 10^{21}\) carbon atoms, we can find the hydrogen atoms by multiplying the total number of carbon atoms by the ratio of hydrogen atoms to carbon atoms, which is \(\frac{12}{6}\). Number of hydrogen atoms = \(1.250 \times 10^{21} \times \frac{12}{6}\)

Calculate the number of glucose molecules

We can calculate the number of glucose molecules by dividing the number of carbon atoms (in the given sample) by the number of carbon atoms in a single glucose molecule. Number of glucose molecules = \(\frac{1.250 \times 10^{21}}{6}\)

Calculate the number of moles of glucose

To find the number of moles of glucose, we will divide the total number of glucose molecules by Avogadro's number (\(6.022\times10^{23}\) molecules/mol). Number of moles of glucose = \(\frac{\text{Number of glucose molecules}}{6.022\times10^{23}}\)

Calculate the mass of the glucose sample

To find the mass of the glucose sample, we will multiply the number of moles by the molar mass of glucose. The molar mass of glucose is 180.16 g/mol, as there are 6 carbon atoms (12.01 g/mol each), 12 hydrogen atoms (1.01 g/mol each), and 6 oxygen atoms (16.00 g/mol each). Mass of the glucose sample = Number of moles of glucose \(\times\) 180.16 g/mol Now let's perform the calculations.

Key concepts

Stoichiometry

Stoichiometry is a fundamental concept in chemistry that deals with the relationships between the amounts of reactants and products in chemical reactions. At its core, stoichiometry involves using balanced chemical equations to determine the ratio between different substances. These calculations enable chemists to predict the amount of a product formed from given reactants or how much reactant is needed to produce a desired amount of product.

In the context of our exercise, stoichiometry helps us relate the number of atoms in glucose. Using the given ratio of carbon, hydrogen, and oxygen atoms \(6:12:6\), we can find the number of hydrogen atoms if we know the quantity of carbon atoms. By deriving these numbers using stoichiometric relationships, chemists can further calculate the number of molecules, moles, and even the mass of a compound.

Glucose

Glucose is a simple sugar and an essential energy source for cells in the body. It is a monosaccharide with the molecular formula \(\text{C}_6\text{H}_{12}\text{O}_6\). Understanding the molecular makeup of glucose is critical because it forms the basis for more complex carbohydrates like starch and cellulose.

In our exercise, a sample of glucose containing a known number of carbon atoms allows us to determine other properties such as the number of hydrogen atoms, number of glucose molecules, and other vital calculations such as molecular mass. This emphasizes glucose's role in molecular calculations and its fundamental nature in chemistry.

Molecular Structure

The molecular structure of a compound describes the arrangement and number of atoms in its molecules. In glucose, each molecule consists of six carbon atoms, twelve hydrogen atoms, and six oxygen atoms. This structure is crucial because it determines how glucose engages in chemical reactions and interacts with other molecules.

By understanding molecular structures, chemists can predict how substances behave under different conditions. For example, knowing glucose's structure allows us to perform stoichiometric conversions, such as calculating the number of glucose molecules from a set number of carbon atoms. The arrangement and bonds between these atoms define crucial characteristics such as solubility, reactivity, and stability.

Mole Concept

The mole concept is a central idea in chemistry that provides a bridge between the atomic and macroscopic worlds. A mole is defined as the quantity of substance containing the same number of elementary entities (e.g., atoms, molecules) as there are in 12 grams of carbon-12, which is Avogadro's number, \(6.022 \times 10^{23}\). This concept allows chemists to count particles by weighing them, a fundamental technique in quantitative chemistry.

In the context of the exercise, determining the number of moles of glucose helps convert from the microscopic scale of atoms and molecules to quantities that are practical for laboratory measurements and reactions. It shows how knowing the number of molecules in a substance relates to its mass, facilitating the calculation of the mass of the glucose sample from the number of moles determined.