Question

\(1.4\) moles of phosphorus trichloride are present in a sample. How many atoms are there in the sample? (a) \(5.6\) (b) 34 (c) \(2.4 \times 10^{23}\) (d) \(3.372 \times 10^{24}\)

Short answer

3.37232 × 10^24 atoms, which can be rounded to 3.372 × 10^24 atoms (Option d).

Step-by-step solution

Understanding the exercise

The task is to determine the total number of atoms in a sample containing 1.4 moles of phosphorus trichloride (PCl3). Recall that 1 mole of any substance contains Avogadro's number of molecules, which is approximately 6.022 x 10^23 molecules.

Calculate the number of molecules in the sample

Firstly, calculate the number of PCl3 molecules in 1.4 moles by multiplying the number of moles by Avogadro's number:\[1.4 \text{ moles} \times 6.022 \times 10^{23} \text{ molecules/mole}\]

Find the total number of molecules of phosphorus trichloride

The total number of PCl3 molecules in the sample is given by the multiplication done in the previous step:\[1.4 \times 6.022 \times 10^{23} = 8.4308 \times 10^{23} \text{ PCl3 molecules} \]

Determine the number of atoms in each PCl3 molecule

Each molecule of PCl3 contains one phosphorus atom and three chlorine atoms, summing up to four atoms per molecule.

Calculate the total number of atoms in the sample

To find the total number of atoms in the sample, multiply the number of PCl3 molecules by the number of atoms per molecule:\[8.4308 \times 10^{23} \text{ PCl3 molecules} \times 4 \text{ atoms/molecule} = 3.37232 \times 10^{24} \text{ atoms}\]

Key concepts

Avogadro's Number

Imagine you have a tiny world where particles are so small, counting them individually seems impossible. That's when Avogadro's Number comes to the rescue. It's a consistent value that provides a link between the microscopic world of atoms and molecules and the macroscopic world that we can measure in a lab.

Avogadro's Number, which is approximately \(6.022 \times 10^{23}\), is defined as the number of particles (atoms, molecules, ions, or electrons) in one mole of a substance. The number itself is named after the Italian scientist Amedeo Avogadro, who is credited with first recognizing the importance of this constant. One mole, therefore, is like a set pack of particles, always containing Avogadro's Number, just like a dozen always contains 12 items.

This massive number allows chemists to count out precise quantities of atoms or molecules by simply weighing out a substance on a balance. For instance, if a sample contains a mole of water, it would have \(6.022 \times 10^{23}\) water molecules.

Mole Concept

In the realm of chemistry, the mole is a fundamental unit in the International System of Units, similar to fundamental units like the pair for two shoes or a dozen for twelve eggs. However, a mole is much, much bigger—it's a unit that signifies a grand number of particles.

The mole concept is a bridging technique that allows chemists to work with the submicroscopic world of atoms and molecules in a manageable way. When we say 'a mole' of any substance, we're actually referring to a set number of units of that substance—specifically, Avogadro's Number of units.

The beauty of the mole concept is that it applies universally to all substances. A mole of carbon contains the same Avogadro's Number of atoms as a mole of hydrogen does of molecules, despite these substances having different weights and properties. It's this universality that makes the mole concept indispensable in stoichiometry calculations and understanding chemical reactions.

Stoichiometry

Stoichiometry is like the recipe for chemistry—it defines the quantitative relationships between the substances involved in chemical reactions. This is where all those numbers, coefficients, and the mole concept come together to make sure you have just the right amount of each reactant to produce your desired products.

Think of it as baking; you need the right proportions of ingredients to get the perfect loaf of bread. In chemistry, if you don't balance your reactants properly, you might not get any product at all! Stoichiometry takes into account the conservation of mass by ensuring that the amount of atoms of each element is the same on both sides of a reaction.

Using stoichiometry, you can predict how much product will result from a given amount of reactants, or conversely, how much of each reactant you need to produce a certain amount of product. This makes stoichiometry a crucial part of both theoretical calculations and practical lab work in chemistry.

Chemical Formula

The chemical formula is the shorthand notation that gives us a clear picture of the composition of a compound. It's like a friend telling you their address—the numbers and letters let you know exactly where they live and how to get there. In chemistry, the letters represent the types of atoms present while the numbers show how many of each atom come together to form the compound.

For example, the chemical formula for water, H2O, reveals that each molecule of water is made up of two hydrogen atoms and one oxygen atom. Similarly, in our exercise's compound phosphorus trichloride, represented as PCl3, we see that one phosphorus atom is bonded to three chlorine atoms.

This notation is essential as it not only informs about the elements present in a compound but also indicates the proportions in which these elements combine, playing a critical role in understanding and predicting the outcomes of chemical reactions.