Question

Calculate the number of \(\mathrm{C}, \mathrm{H},\) and \(\mathrm{O}\) atoms in \(1.50 \mathrm{~g}\) of glucose \(\left(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\right),\) a sugar.

Short answer

There are approximately \(3.01 \times 10^{22}\) C atoms, \(6.01 \times 10^{22}\) H atoms, and \(3.01 \times 10^{22}\) O atoms in 1.50 g of glucose.

Step-by-step solution

Calculate the molar mass of glucose

The molar mass of glucose (\(\mathrm{C}_{6}\mathrm{H}_{12}\mathrm{O}_{6}\)) is calculated by summing the atomic masses of all atoms in the molecule. Carbon's atomic mass is approximately \(12.01\,\text{g/mol}\), hydrogen's is \(1.01\,\text{g/mol}\), and oxygen's is \(16.00\,\text{g/mol}\). Therefore, the molar mass of glucose is calculated as:\[6(12.01) + 12(1.01) + 6(16.00) = 180.18\,\text{g/mol}.\]

Calculate the number of moles of glucose

Using the molar mass, calculate the number of moles of glucose in \(1.50\,\text{g}\) using the formula: \[\text{moles} = \frac{\text{mass}}{\text{molar mass}}.\] Hence, the number of moles is \(\frac{1.50}{180.18} \approx 0.00832\,\text{mol}.\)

Determine the number of molecules of glucose

To find the number of molecules, multiply the number of moles by Avogadro's number \(6.022 \times 10^{23}\ \text{mol}^{-1}\):\[0.00832\,\text{mol} \times 6.022 \times 10^{23}\ \text{mol}^{-1} \approx 5.01 \times 10^{21}\ \text{molecules}.\]

Calculate the number of C, H, and O atoms

Each glucose molecule \(\mathrm{C}_{6}\mathrm{H}_{12}\mathrm{O}_{6}\) contains 6 C atoms, 12 H atoms, and 6 O atoms. Multiply the number of molecules by the number of each type of atom per molecule. For C atoms: \[5.01 \times 10^{21} \times 6 \approx 3.01 \times 10^{22}\ \text{C atoms}.\] For H atoms: \[5.01 \times 10^{21} \times 12 \approx 6.01 \times 10^{22}\ \text{H atoms}.\] For O atoms: \[5.01 \times 10^{21} \times 6 \approx 3.01 \times 10^{22}\ \text{O atoms}.\]

Key concepts

Molar Mass Calculation

Understanding the molar mass calculation is crucial to converting between grams and moles of a substance. The process involves adding up the atomic masses of all the atoms in a molecule. For glucose, represented by the chemical formula \(\mathrm{C}_{6}\mathrm{H}_{12}\mathrm{O}_{6}\), it's a simple matter of summing up the atomic masses:

• Each carbon (C) atom has an atomic mass of approximately \(12.01\,\text{g/mol}\).
• Each hydrogen (H) atom weighs about \(1.01\,\text{g/mol}\).
• Each oxygen (O) atom comes in at \(16.00\,\text{g/mol}\).

For glucose, we multiply the atomic mass of each type of atom by the number of these atoms present in the molecule. Thus, the total molar mass of glucose can be calculated as: \[6(12.01) + 12(1.01) + 6(16.00) = 180.18\,\text{g/mol}.\]This value indicates how much one mole of glucose weighs.

Avogadro's Number

Avogadro's number is a fundamental constant used to calculate the number of particles, such as atoms or molecules, in a mole. Specifically, it is \(6.022 \times 10^{23} \text{mol}^{-1}\). This huge number stems from the fact that atoms and molecules are incredibly small.
To find how many entities there are in a given number of moles, you simply multiply the number of moles by Avogadro's number. When we have a substance measured in moles, we can multiply by Avogadro's number to find the number of particles making up that substance. For instance, with \(0.00832 \, \text{mol}\) of glucose, the number of glucose molecules can be calculated as: \[0.00832 \, \text{mol} \times 6.022 \times 10^{23} \, \text{mol}^{-1} = 5.01 \times 10^{21} \, \text{molecules}.\]This approach helps in translating abstract moles into tangible numbers.

Atomic Mass

Atomic mass is a critical concept in chemistry, especially when dealing with calculations related to compounds like glucose. It is the measure of an atom's mass, generally expressed in atomic mass units (amu) or grams per mole. Each element's atomic mass can be found on the periodic table and represents the average mass of atoms in a sample of the element, taking isotopic composition into account.
For glucose, the necessity of knowing atomic masses is explicit as it lays the foundation for accurately determining the molar mass.

• Carbon (C) has an atomic mass of \(12.01\,\text{amu}\).
• Hydrogen (H) comes in at \(1.01\,\text{amu}\).
• Oxygen (O) is about \(16.00\,\text{amu}\).

Understanding these values allows scientists to compute the precise molar mass of compounds based on their chemical formula.

Mole Concept

The mole concept is an essential framework in chemistry that allows chemists to work effectively with chemical quantities. It connects the microscopic world of atoms and molecules to the macroscopic world we can measure. A mole, often likened to a chemist's dozen, represents a specific number: the number of particles found in \(12\,\text{g}\) of carbon-12, which is Avogadro's number \(6.022 \times 10^{23}\).
The mole allows for a direct relationship between a substance’s mass in grams and the number of atoms or molecules it contains. For instance, when we found that we had \(0.00832\,\text{mol}\) of glucose in \(1.50\,\text{g}\), we essentially translated mass into number of moles. With this, we can easily determine the number of atoms of each element in the sample by recognizing that each mole of glucose contains a specific number of carbon, hydrogen, and oxygen atoms: 6, 12, and 6 respectively. Thus, this enables calculations like the conversion of moles to atoms and molecules.