Question

Silversmiths are warned to limit their exposure to silver in the air to \(1 \times 10^{-8} \mathrm{~g} \mathrm{Ag} / \mathrm{L}\) of air in a 40 -hour week. What is the allowed exposure in terms of atoms of \(\mathrm{Ag} / \mathrm{L} /\) week?

Short answer

Answer: The allowed exposure limit for silversmiths is 5.58 × 10^14 atoms of silver per liter per week.

Step-by-step solution

Convert grams to moles

To convert a mass of a substance to the number of moles, we need to know its molar mass. The molar mass of silver (Ag) is 107.87 g/mol. So, to convert the mass of silver in grams to moles, we can simply divide the mass by the molar mass: \(1 \times 10^{-8} \mathrm{~g~Ag} \cdot \dfrac{1 \mathrm{~mol~Ag}}{107.87 \mathrm{~g~Ag}}\)

Calculate the number of silver atoms

To find the number of silver atoms in a given number of moles, we use Avogadro's number, which is \(6.022 \times 10^{23}\) atoms/mol. Multiply the number of moles (from Step 1) by Avogadro's number to find the number of silver atoms: \((1 \times 10^{-8} \mathrm{~g~Ag} \cdot \dfrac{1 \mathrm{~mol~Ag}}{107.87 \mathrm{~g~Ag}}) \cdot (6.022 \times 10^{23} \mathrm{~atoms~/ ~mol})\)

Evaluate the expression

Simplify the expression above to find the number of silver atoms per liter per week allowed for silversmiths: \((1 \times 10^{-8} \mathrm{~g~Ag} \cdot \dfrac{1 \mathrm{~mol~Ag}}{107.87 \mathrm{~g~Ag}}) \cdot (6.022 \times 10^{23} \mathrm{~atoms~/ ~mol}) = 5.58 \times 10^{14} \mathrm{~atoms~Ag/\mathrm{L}/week}\) So, the allowed exposure limit for silversmiths is \(5.58 \times 10^{14}\) atoms of silver per liter per week.

Key concepts

Molar Mass

Understanding molar mass is essential for converting between mass and moles in chemical calculations. Molar mass is defined as the mass of one mole of a given substance and is expressed in grams per mole (g/mol).

It serves as a bridge between the macroscopic world (grams) and the microscopic world (atoms or molecules). Every element on the periodic table has a unique molar mass, based on the average mass of its isotopes. For instance, the molar mass of silver (Ag) is 107.87 g/mol, which implies that one mole of silver atoms weighs 107.87 grams.

To calculate the number of moles from a given mass, divide the mass by the element's molar mass. During this process, it's crucial to ensure unit consistency; the mass of the sample must be in grams to align with the molar mass unit. This step is often the starting point in solving many stoichiometry problems in chemistry.

Avogadro's Number

The mole concept hinges on a fundamental constant: Avogadro's number, which is approximately equal to \(6.022 \times 10^{23}\) entities per mole. These entities could be atoms, ions, molecules, or other particles, depending on the context of the chemical substance in question.

Named after the scientist Amedeo Avogadro, this number provides a way to count atoms or molecules in a way that relates to mass measurements. When you multiply the number of moles of a substance by Avogadro's number, you obtain the total number of atoms or molecules in that sample. This relationship is pivotal in chemical calculations, allowing us to convert between microscopic (number of atoms) and macroscopic (moles) quantities.

Practical Application

Using Avogadro's number, chemists can determine how many particles are in a given amount of substance, which is especially handy when dealing with quantities too small to weigh on any scale.

Mole Concept

The mole concept is one of the central pillars of chemistry. A mole is a unit of measurement used in chemistry to express amounts of a chemical substance. It provides a specific measure of the number of particles, whether atoms, molecules, or ions, in a sample.

The mole is important because it allows chemists to work with the subatomic world with macroscopic amounts of chemicals. One mole is equivalent to \(6.022 \times 10^{23}\) particles of the substance (Avogadro's number). The mole concept connects the mass of a substance to the number of particles it contains.

• To translate between mass and number of particles, you first convert mass to moles using the molar mass.
• Then, you convert moles to particles (or vice versa) using Avogadro's number.

Understanding the mole concept allows for precise formulation and manipulation of chemical reactions, calculation of concentrations, and could even apply to assessing safe levels of chemical exposure, as in the exercise involving the exposure limits of silver for silversmiths.