Question

Consider a sample of silver weighing 300.0 g. How many atoms of silver are present in the sample? What mass of copper would you need for the copper sample to contain the same number of atoms as the silyer sample?

Short answer

The silver sample contains approximately \(1.676 \times 10^{24}\) atoms of silver. To have the same number of atoms in a copper sample, you would need a mass of about 176.9 g of copper.

Step-by-step solution

Find the moles of silver atoms in the sample

To find the moles of silver in the sample, we will use the molar mass of silver, which is 107.87 g/mol. We can find the moles using the following formula: Moles (Ag) = (Mass of sample) / (Molar mass of Ag)

Calculate the moles of silver

Using the formula in step 1, we can find the moles of silver in the sample: Moles (Ag) = 300.0 g / 107.87 g/mol = 2.782 moles

Calculate the number of silver atoms in the sample

Using Avogadro's number, \(6.022 \times 10^{23}\) atoms/mol, we can find the number of atoms in the silver sample: Number of atoms (Ag) = Moles (Ag) × Avogadro's number

Find the number of silver atoms

Using the formula in step 3, we can find the number of silver atoms: Number of atoms (Ag) = 2.782 moles × \(6.022 \times 10^{23}\) atoms/mol ≈ \(1.676 \times 10^{24}\) atoms

Calculate the mass of the copper sample containing the same number of atoms

We are given that the copper sample will contain the same number of atoms as the silver sample. To find the mass of the copper sample, we will use the molar mass of copper, which is 63.55 g/mol, and Avogadro's number. We can find the moles of copper using the following formula: Moles (Cu) = Number of atoms (Cu) / Avogadro's number And then find the mass of the copper sample using the following formula: Mass (Cu) = Moles (Cu) × Molar mass of Cu

Calculate the moles of copper

Using the formula in step 5, we can find the moles of copper: Moles (Cu) = \(1.676 \times 10^{24}\) atoms / \(6.022 \times 10^{23}\) atoms/mol ≈ 2.782 moles

Calculate the mass of the copper sample

Using the formula in step 5, we can find the mass of the copper sample: Mass (Cu) = 2.782 moles × 63.55 g/mol ≈ 176.9 g So, the mass of copper needed for the copper sample to contain the same number of atoms as the silver sample is approximately 176.9 g.

Key concepts

Moles Calculation

When you're working with atoms and molecules, it’s often handy to deal with the concept of "moles". Imagine dealing with individual atoms; it would be like counting grains of sand on a beach.

Luckily, the mole makes this much easier! One mole represents a quantity of particles—usually atoms or molecules—just like a dozen refers to 12 items. Mathematically, it's expressed as:

• Number of Moles = \( \frac{\text{Mass of the Sample}}{\text{Molar Mass}} \)

In the exercise, we calculate the moles of silver in the given sample using this formula. With the mass of silver at 300.0 g and its molar mass as 107.87 g/mol, dividing these gives you approximately 2.782 moles.

Avogadro's Number

To count individual atoms or molecules, Avogadro's number becomes invaluable. It's a huge number: \(6.022 \times 10^{23}\), and it represents how many particles are in one mole of a substance.

Think of Avogadro's number as a bridge between the atomic scale and macroscopic measurements like grams.

To determine the number of atoms in a substance, you multiply the number of moles by Avogadro's number. So, once you find the moles of silver (which was approximately 2.782), multiply this by \(6.022 \times 10^{23}\) to find out how many atoms are present. This gives you about \(1.676 \times 10^{24}\) silver atoms.

Molar Mass

Molar mass is key when converting between mass and moles. Simply put, it's the mass of one mole of a substance, measured in grams per mole (g/mol). Each element has its own unique molar mass, which can often be found on the periodic table.

For example, silver has a molar mass of 107.87 g/mol, while copper's molar mass is 63.55 g/mol. Knowing these values enables the calculation of moles when given a specific sample mass.

By dividing the sample's mass by the molar mass, you determine how many moles exist in that sample. This is illustrated in step one and five of the exercise, where we work this out for both silver and copper.

Chemical Atoms Counting

When asked to find how many atoms are in a sample, precision is key. Counting atoms directly isn't feasible, so we use the concepts of moles and Avogadro's number for our calculations.

First, calculate the moles from the given mass using the particular element's molar mass. Then, the quintessential step is multiplying the number of moles by Avogadro's number, which gives the count of atoms.

In the example provided, you calculate the number of silver atoms and then use that knowledge to work backwards for copper. You find out how many moles of copper match the same number of atoms and subsequently, what mass that correlates to, giving you a well-rounded understanding of chemical atoms counting.