Problem 32 - Chemistry The Central Science In Si Units

Question

Without doing any detailed calculations (but using a periodic table to give atomic weights), rank the following samples in order of increasing numbers of atoms: \(0.2 \mathrm{~mol} \mathrm{PCl}_{5}\), molecules, \(80 \mathrm{~g} \mathrm{Fe}_{2} \mathrm{O}_{3}, 3.0 \times 10^{23} \mathrm{CO}\) molecules.

Step-by-Step Solution

Answer

Rank: \(0.2 \text{ mol} PCl_5\) < \(0.4982\, \text{mol CO}\) < \(0.5016\, \text{mol} \ Fe_2O_3\).

Step 1: Determine Moles from Molecules

First, we need to determine the number of moles for the sample given in molecules for simpler comparison. The number of molecules given is \(3.0 \times 10^{23}\) for CO. We use Avogadro's number, which is \(6.022 \times 10^{23}\) molecules/mol, to find moles. Divide 3.0 x 10^{23} molecules by Avogadro's number \(6.022 \times 10^{23}\) molecules/mol to get approximately \(0.4982\) mol of CO.

Step 2: Calculate Moles from Mass

Next, calculate the number of moles from the mass given for \(Fe_2O_3\). The molar mass of \(Fe_2O_3\) can be calculated as follows: iron has an atomic weight of about 55.85 amu and oxygen 16.00 amu. Therefore, \(Fe_2O_3\) weighs \(2 \times 55.85 + 3 \times 16.00 = 159.7\, \text{g/mol}\). Given 80 g of \(Fe_2O_3\), the moles present are \(\frac{80}{159.7} \approx 0.5016\) mol.

Step 3: Compare and Rank Moles

We now have the number of moles for each sample, which can represent the number of formula units (molecules or atoms depending on the context since each molecule has an Avogadro's number worth of units in a mol). The samples are as follows: \(0.2\, \text{mol of} \ PCl_5\), \(0.4982\, ext{mol of CO}\), and \(0.5016\, ext{mol of} \ Fe_2O_3\).Therefore, the order of increasing numbers of atoms based on the calculated moles is: \(0.2 \text{ mol} PCl_5\) < \(0.4982\, \text{mol CO}\) < \(0.5016\, \text{mol} \ Fe_2O_3\).

Key Concepts

Avogadro's Number

Avogadro's Number is a key figure in chemistry, especially when dealing with quantities of atoms or molecules in macroscopic amounts. Named after the scientist Amedeo Avogadro, this number, approximately equal to \(6.022 \times 10^{23}\), represents the quantity of constituent particles, usually atoms or molecules, contained in one mole of a substance.

Understanding Avogadro's Number is crucial because it bridges the gap between the microscopic world of atoms and the macroscopic measurements we make in the laboratory. For example, when we say we have one mole of a substance, it means we have \(6.022 \times 10^{23}\) units of that substance, whether they are atoms, molecules, ions, or electrons.

To find the number of moles from a given number of molecules, divide the number of molecules by Avogadro's Number. This concept applies universally, making it easier to switch between counting individual molecules or atoms and measuring their collective mass.

Molar Mass

Molar Mass refers to the mass of one mole of a substance, usually measured in grams per mole (g/mol). It combines the atomic masses of all the atoms specified in a chemical formula. To find the molar mass of a compound, you need to use the atomic masses from the periodic table.

Molar mass is critical because it allows conversion between the mass of a substance and the number of moles, facilitating calculations in chemistry. For instance, in the context of the exercise, to find the number of moles of \(Fe_2O_3\), you first calculate its molar mass:

• Iron (Fe) atomic mass: \(55.85\) amu
• Oxygen (O) atomic mass: \(16.00\) amu
• So, \(Fe_2O_3\) molar mass is: \(2 \times 55.85 + 3 \times 16.00 = 159.7\, \text{g/mol}\)

This conversion allows chemists to determine the composition of compounds in terms of moles, thus aiding in further stoichiometric calculations.

Stoichiometry

Stoichiometry is the aspect of chemistry that deals with the relationships between the quantities of reactants and products in chemical reactions. It involves using balanced chemical equations to determine the ratios of products and reactants.

In practical terms, stoichiometry helps calculate how much of a substance is needed or will be produced in a chemical reaction. Understanding stoichiometry means understanding how to interpret and use balanced equations, using coefficients to relate moles of different substances.

For example, the reaction involving \(Fe_2O_3\) could be used to illustrate stoichiometry because once the chemical equation is balanced, you can use the mole ratio from the equation to convert between moles of \(Fe_2O_3\) and other substances in the reaction. This aids in efficient use of reactants and predicting the yield of products, vital for both academic exercises and industrial applications.