Question

From \(2 \mathrm{mg}\) calcium, \(1.2 \times 10^{19}\) atoms are removed. The number of g-atoms of calcium left is \((\mathrm{Ca}=40)\) (a) \(5 \times 10^{-5}\) (b) \(2 \times 10^{-5}\) (c) \(3 \times 10^{-5}\) (d) \(5 \times 10^{-6}\)

Short answer

The number of g-atoms of calcium left is (b) \(2 \times 10^{-5}\).

Step-by-step solution

Determine the molar mass of calcium

The molar mass of calcium (Ca) is given as 40 grams per mole (g/mol). This will be used to convert mass to moles and atoms to moles.

Convert mass of Ca to moles

To convert the initial mass of 2 mg of calcium to moles, divide by the molar mass (in grams) and convert milligrams to grams: \[\begin{equation}\text{Initial moles of Ca} = \frac{2\times10^{-3}\text{ g}}{40\text{ g/mol}}\end{equation}\].

Calculate the initial number of atoms

Using Avogadro's number (\(6.022 \times 10^{23}\) atoms/mol), convert the initial moles of Ca to atoms.

Remove the given number of atoms

Subtract the number of removed atoms (\(1.2 \times 10^{19}\)) from the initial number of atoms to find the remaining number of atoms.

Convert the remaining atoms back to moles

Use Avogadro's number to convert the remaining number of atoms to moles of Ca.

Find the number of gram-atoms left

The number of moles calculated in the previous step is also the number of gram-atoms of Ca left as by definition 1 mole = 1 gram-atom. This is the final answer.

Key concepts

Molar Mass of Calcium

Understanding the molar mass of an element is essential for converting between mass and moles. The molar mass of calcium (Ca), a common alkaline earth metal, is 40 grams per mole (g/mol). This measurement is derived from the average atomic mass of all isotopes of calcium as they occur naturally, which is then expressed in grams for every mole of the element.

When working with chemical quantities, scientists use the molar mass as a conversion factor. It relates the mass of a substance to the number of moles, and hence the number of particles (atoms, molecules, or ions) in that given mass. For example, if you have a 40 g sample of calcium, you actually possess 1 mole of calcium atoms by definition, since the molar mass of calcium is 40 g/mol. This uniformity allows for precise calculations in chemistry, such as understanding how much of a substance reacts with another in a chemical reaction.

Avogadro's Number

Avogadro's number, named after the Italian scientist Amedeo Avogadro, is a fundamental constant in chemistry that provides the link between the macroscopic and microscopic worlds. This number, designated as \(6.022 \times 10^{23}\), represents the number of particles (typically atoms or molecules) in one mole of a substance.

This constant is used to translate moles, a measure that is convenient for chemical reactions and stoichiometry, to the actual number of particles. For instance, if a chemist says there is 1 mole of calcium atoms, they are inherently stating that there are \(6.022 \times 10^{23}\) calcium atoms present. Avogadro's number allows for the calculation of specific quantities of substances required in chemical reactions and forms the basis for mole-to-atoms conversions, an essential concept in both theoretical and practical chemistry.

Moles to Atoms Conversion

Converting moles to atoms involves utilizing Avogadro's number as a conversion factor. The process is straightforward: take the number of moles and multiply it by \(6.022 \times 10^{23}\) to find the corresponding number of atoms.

For example, if you're given the moles of calcium, as with the problem at hand, you multiply by Avogadro's number to determine how many calcium atoms are in the sample. Conversely, if you start with atoms and want to determine moles, you divide the number of atoms by Avogadro's number. This two-way conversion is essential in chemistry, where precise amounts of substances are routinely calculated for reactions, forming the basis of chemical stoichiometry. It's this kind of conversion that helps in visualizing and measuring quantities at the atomic level in a more tangible macroscopic scale, bridging the gap between the size scales of chemistry.