Question

Using the average atomic masses given inside the front cover of the text, calculate the number of atoms present in each of the following samples. a. \(1.50 \mathrm{g}\) of silver, \(\mathrm{Ag}\) b. 0.0015 mol of copper, \(\mathrm{Cu}\) c. \(0.0015 \mathrm{g}\) of copper, \(\mathrm{Cu}\) d. \(2.00 \mathrm{kg}\) of magnesium, \(\mathrm{Mg}\) e. 2.34 oz of calcium, \(\mathrm{Ca}\) f. \(2.34 \mathrm{g}\) of calcium, \(\mathrm{Ca}\) g. 2.34 mol of calcium, \(\mathrm{Ca}\)

Short answer

a. \(8.37 \times 10^{22}\) atoms of silver b. \(9.03 \times 10^{20}\) atoms of copper c. \(1.42 \times 10^{19}\) atoms of copper d. \(4.95 \times 10^{25}\) atoms of magnesium e. \(9.96 \times 10^{23}\) atoms of calcium f. \(3.51 \times 10^{22}\) atoms of calcium g. \(1.41 \times 10^{24}\) atoms of calcium

Step-by-step solution

a. Calculate the number of atoms in 1.50 g of silver (Ag)

1. Find the molar mass of silver: According to the periodic table, the molar mass of silver (Ag) is \(107.87 \: g/mol\). 2. Calculate the number of moles of silver: \(moles = \frac{mass}{molar\:mass} = \frac{1.50\:g}{107.87\:g/mol} = 0.0139 \: mol\) 3. Calculate the number of atoms of silver: \(number\:of\:atoms = moles \times Avogadro's\:number = 0.0139\:mol \times 6.022\times10^{23}\:atoms/mol = 8.37\times10^{22}\:atoms\)

b. Calculate the number of atoms in 0.0015 mol of copper (Cu)

1. Calculate the number of atoms of copper: \(number\:of\:atoms = moles \times Avogadro's\:number = 0.0015\:mol \times 6.022\times10^{23}\:atoms/mol = 9.03\times10^{20}\:atoms\)

c. Calculate the number of atoms in 0.0015 g of copper (Cu)

1. Find the molar mass of copper: According to the periodic table, the molar mass of copper (Cu) is \(63.55 \: g/mol\). 2. Calculate the number of moles of copper: \(moles = \frac{mass}{molar\:mass} = \frac{0.0015\:g}{63.55\:g/mol} = 2.36\times10^{-5} \: mol\) 3. Calculate the number of atoms of copper: \(number\:of\:atoms = moles \times Avogadro's\:number = 2.36\times10^{-5}\:mol \times 6.022\times10^{23}\:atoms/mol = 1.42\times10^{19}\:atoms\)

d. Calculate the number of atoms in 2.00 kg of magnesium (Mg)

1. Find the molar mass of magnesium: According to the periodic table, the molar mass of magnesium (Mg) is \(24.31 \: g/mol\). 2. Convert mass to grams: \(2.00\:kg = 2000\:g\) 3. Calculate the number of moles of magnesium: \(moles = \frac{mass}{molar\:mass} = \frac{2000\:g}{24.31\:g/mol} = 82.27 \: mol\) 4. Calculate the number of atoms of magnesium: \(number\:of\:atoms = moles \times Avogadro's\:number = 82.27\:mol \times 6.022\times10^{23}\:atoms/mol = 4.95\times10^{25}\:atoms\)

e. Calculate the number of atoms in 2.34 oz of calcium (Ca)

1. Find the molar mass of calcium: According to the periodic table, the molar mass of calcium (Ca) is \(40.08 \: g/mol\). 2. Convert mass to grams: \(2.34\:oz \times \frac{28.3495\:g}{1\:oz} = 66.3371\:g\) 3. Calculate the number of moles of calcium: \(moles = \frac{mass}{molar\:mass} = \frac{66.3371\:g}{40.08\:g/mol} = 1.654 \: mol\) 4. Calculate the number of atoms of calcium: \(number\:of\:atoms = moles \times Avogadro's\:number = 1.654\:mol \times 6.022\times10^{23}\:atoms/mol = 9.96\times10^{23}\:atoms\)

f. Calculate the number of atoms in 2.34 g of calcium (Ca)

1. Find the molar mass of calcium: According to the periodic table, the molar mass of calcium (Ca) is \(40.08 \: g/mol\). 2. Calculate the number of moles of calcium: \(moles = \frac{mass}{molar\:mass} = \frac{2.34\:g}{40.08\:g/mol} = 0.0583 \: mol\) 3. Calculate the number of atoms of calcium: \(number\:of\:atoms = moles \times Avogadro's\:number = 0.0583\:mol \times 6.022\times10^{23}\:atoms/mol = 3.51\times10^{22}\:atoms\)

g. Calculate the number of atoms in 2.34 mol of calcium (Ca)

1. Calculate the number of atoms of calcium: \(number\:of\:atoms = moles \times Avogadro's\:number = 2.34\:mol \times 6.022\times10^{23}\:atoms/mol = 1.41\times10^{24}\:atoms\)

Key concepts

Molar mass

Understanding molar mass is crucial when working with chemistry, as it is the bridge between macroscopic measurements and microscopic particles. The molar mass of a substance is the mass of one mole of its entities, whether they be atoms, molecules, or ions. To find the molar mass, look to the periodic table. Each element's atomic mass is listed, usually in atomic mass units (amu). For example, silver (Ag) has a molar mass of 107.87 g/mol and copper (Cu) has a molar mass of 63.55 g/mol. Molar mass allows us to convert from grams, which we can measure, to moles, which relate to the number of entities present.
To remember:

• Locate the atomic masses on the periodic table.
• Molar mass is expressed in grams per mole (g/mol).
• It's essential for converting between mass and moles in problem-solving.

Recognizing the significance of molar mass helps in calculating quantities accurately and understanding chemical reactions at a fundamental level.

Atom counting

Atom counting involves determining how many atoms are present in a given sample. The key to this is Avogadro's number, which is approximately 6.022 x 10^23 atoms/mole. This constant tells us how many atoms are in a mole of a substance. Let's consider a practical example:
1. Find the molar mass of your sample from the periodic table.
2. Measure the mass of your sample.
3. Use the formula: \[\text{moles} = \frac{\text{mass}}{\text{molar mass}}\]
Then, calculate the number of atoms: \[\text{number of atoms} = \text{moles} \times \text{Avogadro's number}\]

• Moles are converted into atoms by multiplying by Avogadro's number.
• This step-by-step approach helps in quantifying atoms in a macroscopic sample.
• Understanding these conversions is crucial in chemistry, as reactions happen at the atomic level.

Atom counting allows chemists to predict and measure chemical reactions effectively.

Chemistry problem-solving

Successfully solving chemistry problems often requires a step-by-step approach. A systematic method is valuable since chemistry problems often involve several intertwined concepts. Take the exercise above: each step requires turning given information into the form needed for subsequent calculations.
Here's how to ensure success in chemistry problem-solving:

• Identify what information is given and what you need to find.
• Use conversion factors like molar mass and Avogadro's number to bridge between different pieces of data.
• Write out all units to track conversions and verify results.

Break the problem into smaller, manageable parts, solve each part, and piece it back together for the final result. This builds strong problem-solving skills and deepens understanding of chemical principles. Mastering these methods not only helps in exams, but also in real-world applications where timely, accurate calculations are paramount.