Problem 190 - Chemistry Matter and Change

Question

How many grams of \(\mathrm{CoCl}_{2} \cdot 6 \mathrm{H}_{2} \mathrm{O}\) must you measure out in a container to have exactly Avogadro's number of particles?

Step-by-Step Solution

Answer

To have exactly Avogadro's number of particles of \(CoCl_{2} \cdot 6H_{2}O\), you need to measure out 237.95 grams in the container.

Step 1: Calculate the molecular weight of \(CoCl_{2} \cdot 6H_{2}O\)

This can be calculated by adding the atomic weights of each atom in the molecule, multiplied by the number that atom appears: - Cobalt (Co): 58.93 g/mol - Chlorine (Cl): 35.45 g/mol, there are 2 Chlorine in this molecule - Hydrogen (H): 1.01 g/mol, there are 12 Hydrogen atoms in this molecule because of the water - Oxygen (O): 16.00 g/mol, there are 6 Oxygen atoms in this molecule because of the water Combine these to find the molecular weight: \(1 \cdot 58.93 \, g/mol + 2 \cdot 35.45 \, g/mol + 12 \cdot 1.01 \, g/mol + 6 \cdot 16.00 \, g/mol\)

Step 2: Calculate the total Molecular weight

Adding up the individual molecular weights gives: \(58.93 \, g/mol + 70.90 \, g/mol + 12.12 \, g/mol + 96.00 \, g/mol = 237.95 \, g/mol\) This is the molecular weight of \(CoCl_{2} \cdot 6H_{2}O\).

Step 3: Apply Avogadro’s number

The task is to have as many molecules as Avogadro's number, which is \(6.02 \times 10^{23}\). Since, by definition, one mole of a substance contains \(6.02 \times 10^{23}\) particles, exactly one mole of \(CoCl_{2} \cdot 6H_{2}O\) is required to get Avogadro's number of particles.

Step 4: Find mass needed

Since the molar mass of \(CoCl_{2} \cdot 6H_{2}O\) is the amount of grams per mole, the amount of grams equal to one mole is the molar mass. Therefore, to get Avogadro's number of particles, exactly 237.95 grams of \(CoCl_{2} \cdot 6H_{2}O\) are needed.

Key Concepts

Avogadro's number

Avogadro's number is a fundamental concept in chemistry that helps in understanding the amount of particles in a mole of a substance. It is defined as the number of atoms, ions, or molecules contained in one mole of a substance and has a value of approximately \(6.02 \times 10^{23}\). This large number allows chemists to convert between the mass of substances and the number of molecules or atoms they contain.
For example, when dealing with compounds like \(CoCl_2 \cdot 6H_2O\), knowing Avogadro's number helps us understand how many individual molecules make up a given mass. It acts as a bridge between the atomic scale and the macroscopic scale we can easily measure in a lab. When a problem requires "Avogadro's number of particles," it means you're dealing with a mole's worth of the substance's individual units.

Stoichiometry

Stoichiometry involves calculations based on chemical equations. It allows us to predict the amounts of reactants and products in a chemical reaction. This is crucial for converting measurements between mass, moles, and molecules.

Here's how stoichiometry was applied in the exercise:

• First, the molecular weight of \(CoCl_2 \cdot 6H_2O\) was calculated by adding the atomic weights of its constituent atoms.
• This molecular weight (237.95 g/mol) was then used to determine how many grams equate to Avogadro’s number of molecules.

This relationship provides the necessary conversion from mass to moles, allowing one to determine how many grams are needed to equate to a specific number of molecules (in this case, Avogadro’s number). Stoichiometry provides clear, step-by-step pathways through these conversions.

Mole concept

The mole concept is central in chemistry, allowing for a convenient method of expressing amounts of a chemical substance. One mole is defined as exactly 6.02 \times 10^{23} particles, be it atoms or molecules. The mass of one mole of a substance, expressed in grams, is numerically equal to its molecular or atomic mass expressed in atomic mass units (amu).

This is how the mole concept was applied in the solution:

• By calculating the molecular weight of \(CoCl_2 \cdot 6H_2O\) as 237.95 g/mol, the solution equates this mass to contain one mole or Avogadro's number of particles.
• Hence, measuring out 237.95 grams of \(CoCl_2 \cdot 6H_2O\) ensures you have exactly one mole, or \(6.02 \times 10^{23}\), of its molecules.

The mole concept simplifies the passage between the microscopic world of atoms and the macroscopic world of grams that we utilize in the laboratory.