Question

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

Short answer

76.08 grams of lithium contain the same number of atoms as 1 kg of zirconium.

Step-by-step solution

Find moles of zirconium

First, we need to find the moles of zirconium in 1 kg (1000 g) of zirconium. We can use the molar mass of zirconium, which is \(91.22 \mathrm{~g/mol}\), for this conversion. Moles of Zr = (1000 g Zr) / (91.22 g/mol) Moles of Zr = 10.96 mol So, there are 10.96 moles of zirconium in 1 kg of zirconium.

Find the number of Zr atoms

Next, we need to find the number of atoms of zirconium in 10.96 moles of zirconium. We can use Avogadro's number, which is \(6.022 \times 10^{23} \mathrm{atoms/mol}\), for this conversion. Number of Zr atoms = (10.96 mol) x (6.022 x 10^23 atoms/mol) Number of Zr atoms = 6.60 x 10^24 atoms

Convert atoms to moles of lithium

In this step, we will use the number of zirconium atoms to find the moles of lithium that contain the same number of atoms. Moles of Li = (6.60 x 10^24 Li atoms) / (6.022 x 10^23 atoms/mol) Moles of Li = 10.96 mol So, there are 10.96 moles of lithium that contain the same number of atoms as 1 kg of zirconium.

Convert moles of lithium to grams

Finally, we will convert the moles of lithium to grams using the molar mass of lithium, which is \(6.94 \mathrm{~g/mol}\). Mass of Li = (10.96 mol Li) x (6.94 g/mol) Mass of Li = 76.08 g Hence, 76.08 grams of lithium contains the same number of atoms as 1 kg of zirconium.

Key concepts

Molar Mass

Molar mass is a fundamental concept in stoichiometry. It represents the mass of one mole of a substance and is expressed in grams per mole (g/mol). Understanding molar mass is vital for conversions between mass and moles in chemical equations.

For any element, the molar mass is numerically equivalent to its atomic mass but with different units. For example, zirconium has a molar mass of 91.22 g/mol. This means that one mole of zirconium atoms weighs 91.22 grams.

To convert grams to moles, divide the mass of the substance by its molar mass. Conversely, to convert moles to grams, multiply the number of moles by the molar mass. This conversion is essential for solving problems like finding how much lithium has the same number of atoms as a given mass of zirconium.

Avogadro's Number

Avogadro's number, approximately \(6.022 \times 10^{23}\), is the number of atoms, molecules, or particles in one mole of a substance. Named after Amedeo Avogadro, this number provides the link between macroscopic quantities of material and the countable number of atoms or molecules.

In practical applications, like the exercise we discussed, Avogadro's number allows the conversion of moles of an element into actual atoms. This is crucial when calculating the number of atoms in a given mass of substance.

• For example, knowing the moles from the zirconium exercise helped in calculating the exact number of atoms using Avogadro's number.

Without Avogadro's number, it would be impossible to bridge the gap between the immensely large count of atoms and the tangible mass measurements we use in labs and calculations.

Moles Conversion

Moles conversion is the backbone of many calculations in chemistry. It involves changing quantities of substances from moles to grams, and to atoms or molecules and vice versa. This ensures precise measurements in reactions and synthesis.

In the step-by-step solution of the exercise, the number of moles of zirconium was converted to moles of lithium using the number of atoms, thanks to Avogadro's number.

The basic conversion factors include:

• Moles to grams: multiply by molar mass.
• Moles to atoms/molecules: multiply by Avogadro's number.
• Atoms/molecules to moles: divide by Avogadro's number.

This flexibility provides the tools needed to handle complex chemical problems by simplifying unknown quantities into manageable calculations.

Atomic Mass

Atomic mass is the mass of an atom expressed in atomic mass units (amu). It signifies the relative weight of atoms and is essentially the sum of protons and neutrons in the atomic nucleus.

Unlike molar mass, which is grams per mole, atomic mass is scaled to a dimensionless number that reflects the mass relative to other atoms.

When working with elements, the atomic mass is pivotal in understanding stoichiometric calculations, essentially serving as a conversion factor between elements when considering reactions.

• For example, in determining how much lithium has the same number of atoms as zirconium, knowledge of their atomic masses allowed the conversion from grams to moles and vice versa.

Atomic mass provides the analytical weight perspective needed to execute precise and meaningful chemical conversions.