Question

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?

Short answer

a. The molar mass of \((CH_3)_2N_2O\) is \(74.08\ \text{g/mol}\). b. There are \(3.37 \times 10^{-3}\) moles of dimethylnitrosamine in 250 mg. c. The mass of 0.050 moles of dimethylnitrosamine is \(3.70\ \text{g}\). d. There are \(1.205 \times 10^{24}\) hydrogen atoms in 1.0 mole of dimethylnitrosamine. e. The mass of \(1.0 \times 10^6\) molecules of dimethylnitrosamine is \(1.23 \times 10^{-16}\ \text{g}\). f. The mass of one molecule of dimethylnitrosamine is \(1.23 \times 10^{-25}\ \text{g}\).

Step-by-step solution

To find the molar mass of \((CH_3)_2N_2O\), we will first determine the number of each atom in the molecule and then multiply those by their respective atomic masses. There are 2 CH3 groups in the molecule, which have an atomic mass of 12.01 g/mol for carbon and 1.008 g/mol for hydrogen. There are 2 nitrogen atoms, with an atomic mass of 14.01 g/mol. Finally, there is 1 oxygen atom, with an atomic mass of 16.00 g/mol. Molar mass of \((CH_3)_2N_2O\): \(2 \times (12.01 + 3 \times 1.008) + 2 \times 14.01 + 16.00\) #b. Calculate moles of dimethylnitrosamine in 250 mg.#

First, convert 250 mg of dimethylnitrosamine to grams: \(250\ \text{mg} = 250 \times 10^{-3}\ \text{g}\) Next, use the molar mass from part a to find the number of moles of dimethylnitrosamine: Moles \(= \frac{ mass}{molar\ mass}\) #c. Find the mass of 0.050 moles of dimethylnitrosamine.#

Use the molar mass from part a and the given moles to calculate the mass of dimethylnitrosamine: Mass \(= \text{moles} \times \text{molar mass}\) #d. Calculate the number of hydrogen atoms in 1.0 mole of dimethylnitrosamine.#

In 1.0 mole of \((CH_3)_2N_2O\), there are 2 CH3 groups, each containing 3 hydrogen atoms. So, the total number of hydrogen atoms in 1 mole of dimethylnitrosamine is: Hydrogen atoms = Mole of dimethylnitrosamine x 2 CH3 groups x 3 hydrogen atoms per CH3 group #e. Find the mass of 1.0 x 10^6 molecules of dimethylnitrosamine.#

First, we need to find the number of molecules in 1 mole by using Avogadro's number (\(6.022\times 10^{23}\) molecules/mole). Next, use the molecules given and the number of molecules per mole to find the moles of dimethylnitrosamine: Moles \(= \frac{ molecules}{Avogadro's\ number}\) Finally, use the molar mass from part a and the moles found above to calculate the mass of dimethylnitrosamine. #f. Find the mass in grams of one molecule of dimethylnitrosamine.#

First, find the mass of one mole of dimethylnitrosamine using the molar mass calculated in part a. Next, divide the molar mass by Avogadro's number to find the mass of one molecule: Mass of one molecule \(= \frac{ mass\ of\ one\ mole}{Avogadro's\ number}\)

Key concepts

The Mole Concept

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

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.

Atomic 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.