Problem 57 - General Chemistry: Principles and Modern Applications

Question

Determine (a) the number of moles of \(\mathrm{Zn}\) in a \(415.0 \mathrm{g}\) sample of zinc metal (b) the number of \(\mathrm{Cr}\) atoms in \(147.4 \mathrm{kg}\) chromium (c) the mass of a one-trillion-atom \(\left(1.0 \times 10^{12}\right)\) sample of metallic gold (d) the mass of one fluorine atom

Step-by-Step Solution

Answer

For (a), the number of moles in \(415.0\) g of Zn is given in Step 1. For (b), the number of Cr atoms in \(147.4 kg\) of Cr is given in Step 2. For (c), the mass of a \(1.0 * 10^{12}\)-atom sample of Au is provided in Step 3. Finally, for (d), the mass of one F atom is computed in Step 4.

Step 1: Conversion of Zn grams into moles

The number of Zn moles is determined using the formula: moles = mass / molar mass. The molar mass of Zn is 65.38 g/mol. Therefore, Moles of Zn = \(415.0/65.38\)

Step 2: Conversion of Cr kilograms into atoms

Firstly, convert kg to g: 1kg = 1000g. So Cr mass = \(147.4 * 1000g\). The number of moles can then be determined using the formula: moles = mass / molar mass. The molar mass of Cr is 52.00 g/mol. Therefore, Moles of Cr = \(147.4 * 1000 / 52.00\). Lastly, convert moles to atoms using Avogadro's number (\(6.02 * 10^{23}\)): Number of Cr atoms = Moles of Cr * Avogadro's number.

Step 3: Calculation of Au sample mass

The number of moles can be determined using the formula: moles = number of atoms / Avogadro's number. Therefore, Moles of Au = \(1.0 * 10^{12} / 6.02 * 10^{23}\). Finally, mass can be determined using the formula: mass = moles * molar mass. The molar mass of Au is 197.0 g/mol. Total mass of Au = Moles of Au * molar mass.

Step 4: Computation of F atom mass

The mass of a single atom is determined using the formula: mass = molar mass / Avogadro's number. The molar mass of F is 18.998 g/mol. Therefore, the mass of one F atom = molar mass / Avogadro's number.

Key Concepts

Moles

Moles are a fundamental concept in chemistry, representing a specific quantity of substance. When we talk about moles, we are referring to a huge number of atoms or molecules bundled into one unit: the mole. Just like a dozen equals 12, a mole equals Avogadro's number of particles, often atoms or molecules. This measurement allows chemists to traverse the microscopic world of atoms with macroscopic amounts of material.

To calculate the number of moles from a known mass, the formula is:

• Moles = Mass / Molar Mass

You simply divide the mass of the element by its molar mass (atomic weight expressed in g/mol). This calculation provides you with how many moles are present in that sample of material. For example, if you have a 415.0 g sample of zinc, you would use the molar mass of zinc (65.38 g/mol) to find the number of moles by dividing the mass by the molar mass.

Avogadro's Number

Avogadro's Number is an incredibly large constant used in chemistry to provide clarity when dealing with very small entities like atoms and molecules. Its value is approximately \(6.022 imes 10^{23}\). This is the number of atoms or molecules present in one mole of any substance.

Avogadro's Number is crucial for converting between the number of moles and the number of individual atoms or molecules. For instance, if you have the number of moles of chromium, multiplying it by Avogadro's Number gives the total number of chromium atoms you possess.

• Number of Atoms = Moles × Avogadro's Number

In step by step solutions, once you've calculated moles using mass and molar mass, you can find the number of atoms by simply multiplying the moles by Avogadro's constant.

Molar Mass

Molar Mass is the bridge between the microscopic and macroscopic worlds in chemistry. It's the mass of one mole of a given substance and is expressed in grams per mole (g/mol). The molar mass is critical for converting between grams and moles of a substance.

Every element has a unique molar mass, which is numerically equivalent to its atomic weight from the periodic table. For conversions, if you know the mass of a sample and the molar mass, you can calculate the number of moles it contains. In a case like step 1, using the molar mass of zinc (65.38 g/mol) allows you to find out how many moles are in a 415.0 g sample.

• Moles = Mass / Molar Mass
• Mass = Moles × Molar Mass

Knowing the molar mass, you can perform stoichiometry calculations, helping to determine how much reactant is needed or how much product is formed in a chemical reaction.

Conversion between units

Conversion between units is an essential skill when dealing with chemical calculations. Accurate conversions allow you to switch between different units of measurement easily - such as grams, moles, and atoms - without losing track of the actual amounts involved.

Let's say you need to find the number of moles from a mass in kilograms. First, you convert the mass to grams by multiplying by 1000 since 1 kilogram equals 1000 grams. Afterward, from grams, you can move to moles using the molar mass of the substance, and eventually to atoms using Avogadro's Number.

• From kilograms to grams: Multiply by 1000
• From grams to moles: Use Molar Mass
• From moles to atoms: Use Avogadro's Number

In the original exercise, these conversion skills are used effectively to change mass units and calculate moles to finally determine the mass or number of atoms in a sample of an element.