Question

Calculate the number of grams of lithium that contain the same number of atoms as \(1.00 \mathrm{kg}\) of zirconium.

Short answer

The number of grams of lithium that contain the same number of atoms as 1.00 kg of zirconium is \(76.06\, \text{g}\).

Step-by-step solution

Find the molar mass of lithium and zirconium

Using the periodic table, we find out that the molar mass of lithium (Li) is 6.94 g/mol and that of zirconium (Zr) is 91.22 g/mol.

Find the moles of zirconium in 1.00 kg

To find the moles of zirconium, we need to convert the mass given in kilograms to grams. 1.00 kg = 1000 g. Next, we can use the molar mass of zirconium to determine the number of moles: \[ \text{moles of Zr} = \frac{\text{mass of Zr}}{\text{molar mass of Zr}} = \frac{1000\,\text{g}}{91.22\,\text{g/mol}} = 10.96\,\text{mol} \hspace{20pt} (1) \]

Determine the moles of lithium that contain the same number of atoms

Since we want the same number of atoms in both samples, the moles of lithium will be the same as the moles of zirconium. Therefore, based on equation (1), there are 10.96 moles of lithium.

Find the mass of lithium in grams

Now that we know the moles of lithium, we can use the molar mass of lithium to determine the mass of lithium: \[ \text{mass of Li} = \text{moles of Li} \times \text{molar mass of Li} = 10.96\,\text{mol} \times 6.94\,\text{g/mol} = 76.06\,\text{g} \] To summarize, the number of grams of lithium that contain the same number of atoms as 1.00 kg of zirconium is 76.06 g.

Key concepts

Molar Mass

Molar mass is a fundamental concept in chemistry, crucial in linking the mass of a substance to the amount of substance present, as expressed in moles. It represents the mass of one mole of a given substance and is usually expressed in grams per mole (g/mol).

In our example, the periodic table provides us with the molar masses of lithium (Li) and zirconium (Zr), where lithium has a molar mass of 6.94 g/mol and zirconium 91.22 g/mol.

This information enables chemists to convert between mass and moles, allowing for seamless calculation and conversion in chemical problems.

When tasked with calculating the mass that contains an equivalent number of atoms, understanding how to use molar mass is essential. By knowing the molar mass, we can transition from a macroscopic understanding (mass in grams) to a molecular level understanding (number of atoms/molecules via moles).

Moles Calculation

Calculating moles is an essential step in stoichiometry, providing a bridge between the chemical mass and the number of constituent particles like atoms or molecules.

The formula for calculating moles from mass is:

• \[\text{moles} = \frac{\text{mass in grams}}{\text{molar mass in g/mol}}\]

In our scenario, to find the moles of zirconium (Zr) in 1.00 kg, we first convert kilograms to grams, resulting in 1000 g.

We then apply the formula using zirconium's molar mass:

• \[\text{moles of Zr} = \frac{1000 \text{ g}}{91.22 \text{ g/mol}} \approx 10.96 \text{ mol}\]

This method is not only beneficial for zirconium but can be applied universally to calculate moles for any substance using its respective mass and molar mass.

Atoms Counting

Atoms counting, while it might seem daunting at first, becomes much more manageable using the concept of moles. Moles allow us to equate the number of particles in substances with their mass.

Avogadro's number, approximately \(6.022 \times 10^{23}\), is the key link as it represents the number of atoms or molecules present in one mole of any substance.

• By using Avogadro's constant, we ensure the same number of atoms in equivalent moles of different substances.

In the context of our exercise, the ability to equate moles of zirconium to moles of lithium means we implicitly have the same number of atoms. This understanding makes it feasible to determine the corresponding mass of a different substance that contains an equal number of atoms through stoichiometric calculations.