Question

Mass of one atom of the element \(A\) is \(3.9854 \times 10^{-23} \mathrm{~g}\). How many atoms are contained in \(1 \mathrm{~g}\) of the element \(A\) ? (a) \(2.509 \times 10^{23}\) (b) \(6.022 \times 10^{23}\) (c) \(12.044 \times 10^{23}\) (d) None of these

Short answer

The number of atoms contained in 1 g of element A is approximately \(2.509 \times 10^{23}\), which corresponds to option (a).

Step-by-step solution

Identify Given Information

The given information includes the mass of one atom of element A, which is \(3.9854 \times 10^{-23} \textrm{g}\). We are asked to find the number of such atoms in \(1 \textrm{g}\) of element A.

Calculate the Number of Atoms

To find out the number of atoms of the element A in \(1 \textrm{g}\), use the formula: \[\text{Number of atoms} = \frac{\text{total mass}}{\text{mass of one atom}}\] Substitute the given values into the formula: \[\text{Number of atoms} = \frac{1 \textrm{g}}{3.9854 \times 10^{-23} \textrm{g}}\]

Perform the Division

Divide \(1 \textrm{g}\) by the mass of one atom to find the total number of atoms: \[2.5079 \times 10^{23} \approx 2.509 \times 10^{23}\] Therefore, the number of atoms in \(1 \textrm{g}\) of element A is approximately \(2.509 \times 10^{23}\).

Key concepts

Avogadro's number

Avogadro's number is a fundamental constant in chemistry representing the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. Its value is approximately \(6.022 \times 10^{23}\), which is the count of particles in a gram-molecular weight of a substance. This number is crucial because it allows chemists to count particles in a sample by simply weighing it.

When dealing with exercises like the one provided, understanding Avogadro’s number is essential because it helps ascertain the relationship between atomic scale masses and macroscopic scale masses. For instance, when the problem asks for the number of atoms in 1 gram of Element A, Avogadro’s number can be used to understand that 1 mole of any element contains exactly \(6.022 \times 10^{23}\) atoms, which then can be used as a conversion factor for atoms to grams.

Atomic Mass

Atomic mass, also known as atomic weight, is the average mass of atoms of an element, measured in atomic mass units (amu). It takes into account the distribution of isotopes in a given element and their respective atomic masses.

When solving problems related to the mass of single atoms, like in our exercise, the atomic mass allows us to convert between the mass of an individual atom and the mass of a mole of atoms (molar mass). This is instrumental in determining how many atoms are in a given sample by comparing it to the atomic mass. In our problem, the fact that the mass of one atom of element A is \(3.9854 \times 10^{-23} \textrm{g}\) is related to its atomic mass on a molar scale, enabling the calculation presented.

Mole Concept

The mole concept is a way to quantify the amount of substance in chemistry. One mole is defined as the amount of a substance that contains as many entities (atoms, ions, molecules, etc.) as there are atoms in 12 grams of the isotope carbon-12, which is Avogadro's number of particles.

This concept is particularly useful when converting between the mass of a substance and the number of particles it contains. In the exercise, understanding the mole allows us to compare the mass of one atom of element A to the mass of a single mole of these atoms to determine how many atoms are present in one gram. It provides a bridge between the atomic scale and the macroscopic scales that we deal with in practical measurements.

Stoichiometry

Stoichiometry is the part of chemistry that relates to the calculation of the quantities of reactants and products in chemical reactions. It involves using balanced chemical equations to work out the mole ratios of the compounds involved in the reactions.

While the original problem does not involve a chemical reaction, understanding stoichiometry can be extended to the concept of aligning proportional relationships in calculations involving atoms, moles, and mass. It's about the mathematical relationships that allow us to predict and quantify the results of chemical processes. For example, knowing the mole ratios and the mass of one atom could help determine the amount of a substance produced or required in a reaction, emphasizing the role of stoichiometry in exercises that ask for the quantity of atoms in a given mass of an element.