Question

Calculate the number of moles of atoms and the number of atoms in the following quantities. (a) \(36.1 \mathrm{~g}\) of argon (b) the \(44.5\)-carat Hope diamond, which consists of carbon ( 1 carat \(=0.200 \mathrm{~g}\) ) (c) \(2.50 \mathrm{~mL}\) of mercury with a density of \(13.6 \mathrm{~g} / \mathrm{mL}\)

Short answer

The number of moles and atoms are: Argon - 0.9043 moles and \(5.44 x 10^{23}\) atoms; Carbon - 0.7402 moles and \(4.45 x 10^{23}\) atoms; Mercury - 0.1695 moles and \(1.02 x 10^{23}\) atoms.

Step-by-step solution

Calculate the number of moles

First, we calculate the number of moles using the formula: moles = mass / atomic_mass. For Argon, it would be \(36.1g / 39.95g/mol = 0.9043 mol\). For Carbon, the weight needs to be converted from carats to grams, and then \(8.90g / 12.01g/mol = 0.7402 mol\). For Mercury, the volume needs to be converted to mass using density, and then \(34g / 200.59g/mol = 0.1695 mol\).

Calculate the number of atoms

Next, we calculate the number of atoms using Avogadro's number (6.022 x \(10^{23}\)) multiplied by the number of moles. So for Argon, that would be \(0.9043 mol * 6.022 x 10^{23} = 5.44 x 10^{23} atoms\). For Carbon, it's \(0.7402 mol * 6.022 x 10^{23} = 4.45 x 10^{23} atoms\). And for Mercury, it's \(0.1695 mol * 6.022 x 10^{23} = 1.02 x 10^{23} atoms\).

Conclusion

Thus, we have calculated the number of moles and the number of atoms for the given quantities of Argon, Carbon, and Mercury.

Key concepts

Atomic Mass

The concept of atomic mass is crucial when it comes to understanding mole calculations. The atomic mass of an element is essentially the available weight of one mole of its atoms in grams. This weight is averaged based on the natural abundance of an element's isotopes.
To calculate how many moles are present in a given sample, we use the formula:

• \(\text{moles} = \frac{\text{mass}}{\text{atomic mass}}\)

For example, with Argon, the atomic mass is approximately 39.95 g/mol. This means that 39.95 grams of Argon are equivalent to one mole of Argon atoms. In our exercise, if we have 36.1 grams of Argon, we calculate \(36.1 \div 39.95\), yielding about 0.9043 moles of Argon. Similarly, for other elements, you'd use their respective atomic masses to perform such calculations.

Understanding atomic mass allows you to convert between grams and moles, providing the basis for determining how many atoms are in samples of any given element.

Avogadro's Number

Avogadro's number provides a bridge between the macroscopic world we interact with every day and the microscopic world of atoms. Named after Amedeo Avogadro, this number (6.022 x \(10^{23}\)) is the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance.
When you know the number of moles of a substance, you can calculate the number of atoms by multiplying by Avogadro's number:

• \(\text{Number of atoms} = \text{moles} \times \text{Avogadro's number}\)

As an example, with Carbon, if you have 0.7402 moles, to find out how many Carbon atoms this represents, you'd calculate \(0.7402 \times 6.022 \times 10^{23}\). This equals approximately \(4.45 \times 10^{23}\) Carbon atoms. This method enables chemists to understand and quantify the immense number of atoms involved, even in small or manageable quantities of material.

Chemical Elements

Understanding chemical elements is fundamental to performing mole calculations correctly. A chemical element is a pure chemical substance consisting of one type of atom, distinguished by its atomic number, which is the number of protons in its nuclei. Each element has distinct physical and chemical properties.
When dealing with mole calculations, identifying the correct chemical element and its properties, such as density and atomic mass, is essential.

• For instance, Argon is a noble gas with the atomic mass of 39.95 g/mol.
• Carbon, used in the Hope diamond, has an atomic mass of 12.01 g/mol.
• Mercury, a dense liquid metal at room temperature, has a significant atomic mass of 200.59 g/mol.

Each element behaves differently, and understanding these behaviors allows chemists to accurately predict reactions, perform calculations, and utilize substances in various applications. Knowing characteristics such as atomic mass and density (like Mercury with 13.6 g/mL in the exercise) directly impacts molecular calculations and the understanding of material compositions.