Question

One chocolate chip used in making chocolate chip cookies has a mass of \(0.324 \mathrm{~g}\). (a) How many chocolate chips are there in one mole of chocolate chips? (b) If a cookie needs 15 chocolate chips, how many cookies can one make with a billionth \(\left(1 \times 10^{-9}\right)\) of a mole of chocolate chips? (A billionth of a mole is scientifically known as a nanomole.)

Short answer

(b) How many cookies can be made with a nanomole of chocolate chips if each cookie needs 15 chocolate chips? Answer: (a) In one mole of chocolate chips, there are \(6.022\times 10^{23}\times 1000\) chocolate chips. (b) With a nanomole of chocolate chips, one can make \(\frac{6.022\times 10^{23}\times 1000\times 10^{-9}}{15}\) cookies.

Step-by-step solution

Calculate the number of chocolate chips in one mole using Avogadro's number

To calculate the number of chocolate chips in one mole, we can use the mole concept. First, we have to convert the given mass of one chocolate chip into moles using the mass-mole relationship: 1 mole = \(6.022\times 10^{23}\) chocolate chips Given, Mass of 1 chocolate chip = \(0.324 ~g\) To find the moles in one chocolate chip, we can use the following relationship: 1 mole = \(6.022\times 10^{23}\) chocolate chips = 324 g Now dividing the total weight of the chocolate chips (324 g) by the weight of one chocolate chip: \(\frac{324 ~g}{0.324 ~g}\) = 1000 moles So, there are \(6.022\times 10^{23}\times 1000\) chocolate chips in one mole.

Calculate the number of chocolate chips in a nanomole

Now we will calculate how many chocolate chips are in one nanomole (a billionth of a mole) using the given conversion factor: 1 nanomole = \(10^{-9}\) moles So, the number of chocolate chips in one nanomole is: \(6.022\times 10^{23}\times 1000\times 10^{-9}\)

Calculate the number of cookies that can be made with a nanomole of chocolate chips

Now that we have the number of chocolate chips in a nanomole, we can calculate the number of cookies that can be made with that amount. Given that each cookie needs 15 chocolate chips: Number of cookies = \(\frac{6.022\times 10^{23}\times 1000\times 10^{-9}}{15}\) (a) So, in one mole of chocolate chips, there are \(6.022\times 10^{23}\times 1000\) chocolate chips. (b) With a nanomole of chocolate chips, one can make: \(\frac{6.022\times 10^{23}\times 1000\times 10^{-9}}{15}\) cookies.

Key concepts

Avogadro's Number

Avogadro's number is a fundamental concept in chemistry that defines the amount of particles, such as atoms, molecules, or in this whimsical example, chocolate chips, contained in one mole. Named after the scientist Amedeo Avogadro, this constant is invaluable for converting between the number of particles and the amount of substance in moles.

The value of Avogadro's number is approximately \(6.022 \times 10^{23}\). This immense number expresses the number of entities in a mole and enables the mass-mole relationship, which is essential for stoichiometric calculations in chemistry. A mole can be thought of like a 'chemist's dozen'—just as a dozen refers to 12 items, a mole refers to \(6.022 \times 10^{23}\) particles, regardless of what those particles are.

Understanding Avogadro's number aids in visualizing the scale of chemical reactions, where even tiny masses can involve a vast number of particles due to the incredibly small size of atoms and molecules.

Mass-Mole Relationship

The mass-mole relationship is a cornerstone in the study of chemistry, allowing scientists to connect the mass of a substance to the number of moles it contains. This is key to performing stoichiometric calculations, which are used to predict the amounts of reactants and products involved in chemical reactions.

To make use of the mass-mole relationship, one must understand molar mass—the mass of one mole of a substance, usually expressed in grams per mole (g/mol). For instance, in the exercise with chocolate chips, the mass of one mole of chocolate chips is 324 g, based on the given mass of a single chip. By dividing the mass of a sample by the molar mass, you can determine the number of moles present.

• To find the moles of a substance: \( \text{{moles}} = \frac{{\text{{mass of the sample (g)}}}}{{\text{{molar mass (g/mol)}}}} \)
• To convert from moles to mass: \( \text{{mass}} = \text{{moles}} \times \text{{molar mass}} \)

This relationship is necessary for quantifying the reactants and products in a balanced chemical equation and plays a vital role in laboratory measurements and experiments.

Nanomole Conversion

A nanomole is a subunit of a mole that represents a much smaller number of entities, specifically one billionth (\(10^{-9}\)) of a mole. Converting between moles and nanomoles is an important skill when dealing with quantities of a substance at the nanoscale, such as in cases where precise, minuscule amounts of chemicals are used.

Conversion between moles and nanomoles is straightforward:

• 1 nanomole = \(10^{-9}\) moles
• 1 mole = \(10^{9}\) nanomoles

This conversion allows one to scale up or down between the macroscopic and microscopic levels. In the textbook exercise, understanding nanomole conversions is key for calculating how many cookies can be made from a tiny fraction of a mole of chocolate chips. In practical terms, it helps when dealing with highly concentrated or highly dilute substances in both research and manufacturing industries, especially in the fields of pharmacology and nanotechnology, where precise dosages and particle counts are crucial.