Problem 68 - Introductory Chemistry Atoms First

Question

Consider a sample of \(92.5 \mathrm{~g}\) of \(\mathrm{O}_{2}\) molecules. (a) How many moles of \(\mathrm{O}_{2}\) molecules are present? (b) How many moles of \(\mathrm{O}\) atoms are present?

Step-by-Step Solution

Answer

(a) There are \(2.890625\) moles of \(\mathrm{O}_{2}\) molecules present. (b) There are \(5.78125\) moles of \(\mathrm{O}\) atoms present.

Step 1: Determine the molar mass of O2

The molar mass of O2 can be found by multiplying the molar mass of a single O atom by 2 (since there are two O atoms in an O2 molecule). The molar mass of O is approximately 16 g/mol. Therefore, the molar mass of O2 is: \[ M_{O_{2}} = 2 \times M_{O} = 2 \times 16 \mathrm{\ g/mol} = 32 \mathrm{\ g/mol} \]

Step 2: Calculate the number of moles of O2 molecules

Use the given sample mass and the molar mass of O2 to determine the number of moles of O2 molecules. The formula to do this is: \[ \text{moles} = \frac{\text{mass of the sample}}{\text{molar mass}} \] In this case: \[ \text{moles of O2} = \frac{92.5 \mathrm{\ g}}{32 \mathrm{\ g/mol}} \] Calculate the number of moles: \[ \text{moles of O2} = 2.890625 \mathrm{\ mol} \]

Step 3: Calculate the number of moles of O atoms

Since there are twice as many O atoms as there are O2 molecules, we can find the moles of O atoms by multiplying the moles of O2 molecules by 2: \[ \text{moles of O atoms} = 2 \times \text{moles of O2} \] Calculate the number of moles of O atoms: \[ \text{moles of O atoms} = 2 \times 2.890625 \mathrm{\ mol} = 5.78125 \mathrm{\ mol} \] The final answers are: (a) There are 2.890625 moles of O2 molecules present. (b) There are 5.78125 moles of O atoms present.

Key Concepts

Molar Mass Calculation

Understanding molar mass is crucial when it comes to dealing with measurements in chemistry. Molar mass is defined as the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It corresponds to the mass of 6.022 x 10²³ (Avogadro's number) particles of that substance. For elements, the molar mass is the mass of one mole of atoms and is numerically equivalent to the atomic weight given in the periodic table.

In our exercise, the calculation of the molar mass of an oxygen molecule (\r\(O_2\)) is performed by adding the molar masses of two oxygen atoms. Since one oxygen atom has a molar mass of approximately 16 g/mol, for the molecule \r\(O_2\), the molar mass is double this value: \r\[M_{O_{2}} = 2 \times 16 \mathrm{\ g/mol} = 32 \mathrm{\ g/mol}\]. Knowing the molar mass is the first step before we can convert between grams and moles, which is a foundational skill in chemical stoichiometry.

Mole Concept in Chemistry

The mole concept is a bridge between the microscopic world of atoms and molecules with the macroscopic world we encounter in the laboratory. A mole is simply a certain number of particles (usually atoms or molecules) and is one of the seven SI base units. Avogadro's number (named after the scientist Amedeo Avogadro) is the number of particles in one mole, which is approximately 6.022 x 10²³ particles.

When dealing with compounds, moles refer to this fixed number of molecules. In our example with oxygen, we've determined that 92.5 g of \r\(O_2\) correspond to \r\(2.890625 \mathrm{\ mol}\) of oxygen molecules, using its molar mass. This helps us quantify the substance in terms of a number that connects to the amount of molecules present, rather than dealing with an incomprehensible number of individual atoms.

Stoichiometry

Stoichiometry is the area of chemistry that deals with the quantitative relationships between the reactants and products in a chemical reaction, based on the principles of the mole concept and conservation of mass. It comes from the Greek words \r\(\text{\'stoicheion\'}\) (element) and \r\(\text{\'metron\'}\) (measure).

In practice, stoichiometry allows us to predict how much product will form from a given amount of reactants, or vice versa. In the given exercise, we calculated moles of \r\(O_2\) molecules and then converted that to moles of oxygen atoms. This step is a form of stoichiometric calculation because it involves the ratio of the number of molecules of \r\(O_2\) to the number of atoms of oxygen, which is 2:1, according to the chemical formula. Understanding the stoichiometric relationships in chemical compounds is essential for tasks including creating reaction mechanisms, determining theoretical yields of a reaction, and analyzing reactant-product conversion efficiency.