Dimethylnitrosamine, \(\left(\mathrm{CH}_{3}\right)_{2} \mathrm{N}_{2} \mathrm{O},\) is a carcinogenic (cancercausing) substance that may be formed in foods, beverages, or gastric juices from the reaction of nitrite ion (used as a food preservative) with other substances. a. What is the molar mass of dimethylnitrosamine? b. How many moles of \(\left(\mathrm{CH}_{3}\right)_{2} \mathrm{N}_{2} \mathrm{O}\) molecules are present in \(250 \mathrm{mg}\) dimethylnitrosamine? c. What is the mass of 0.050 mole of dimethylnitrosamine? d. How many atoms of hydrogen are in 1.0 mole of dimethylnitrosamine? e. What is the mass of \(1.0 \times 10^{6}\) molecules of dimethylnitrosamine? f. What is the mass in grams of one molecule of dimethylnitrosamine?

Understanding the mole concept is foundational in chemistry for counting atoms, molecules, ions, and other chemical entities. A mole is simply a unit used to measure the quantity of anything at the atomic or molecular scale. It's just like how a 'dozen' refers to 12 items, whether they are eggs or cookies.

When dealing with substances at the microscopic level, it's not practical to count each particle individually. This is where the mole comes in. It allows chemists to work with amounts of substances in a comprehensible way. One mole of any substance contains exactly the same number of entities as there are atoms in exactly 12 grams of carbon-12. This number, known as Avogadro's number, is about 6.022 x 10^23.

To apply the mole concept in calculations, first, we need the molar mass of a substance, which is the mass of one mole of that substance, measured in grams. The molar mass of a compound can be calculated by adding up the atomic masses of each atom in the compound, according to the number of each type of atom present in the formula unit of the substance. For instance, in the dimethylnitrosamine example, the molar mass calculation sums the masses of carbon, hydrogen, nitrogen, and oxygen atoms to find the overall molar mass of one mole of that compound.

Avogadro's number is a constant that represents the number of particles contained in one mole. Its value, approximately 6.022 x 10^23, is an essential conversion factor in chemistry. It bridges the gap between the microscopic world of atoms and the macroscopic world that we can measure. For instance, if you have a mole of dimethylnitrosamine, you have 6.022 x 10^23 molecules of it.

Whenever you're interested in finding out the number of entities (like molecules in our exercise example), you refer to Avogadro's number. To calculate the number of molecules in a given mass of substance, you can divide the mass of the sample by its molar mass to get the moles, and then multiply that by Avogadro's number to obtain the number of molecules. Conversely, to find out the mass for a certain number of molecules, you convert the molecule count to moles by dividing by Avogadro's number, and then multiply by the molar mass.

Each element has a unique atomic mass, also referred to as atomic weight, which is the average mass of atoms of an element measured in atomic mass units (amu). It's averaged because natural elements are usually a mixture of isotopes, which are atoms with the same number of protons but different numbers of neutrons.

The atomic mass of each element can be found on the periodic table and is key to calculating molar masses of compounds. You add up the atomic masses of all the atoms in a formula unit based on the compound's chemical formula.

For instance, the molar mass of dimethylnitrosamine is calculated by multiplying the atomic mass of carbon, hydrogen, nitrogen, and oxygen with the number of each of those atoms present in a molecule, and then summing those values. The atomic mass becomes particularly useful when converting between mass and moles, such as determining how many hydrogen atoms are in 1.0 mole of dimethylnitrosamine.