Question

Dimethylnitrosamine, \(\left(\mathrm{CH}_{3}\right)_{2} \mathrm{N}_{2} \mathrm{O}\) is a carcinogenic (cancer causing) 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 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. Molar mass of \((\mathrm{CH}_{3})_{2} \mathrm{N}_{2} \mathrm{O}\) = \(74.16 \,\mathrm{g/mol}\) b. No. of moles in 250 mg dimethylnitrosamine = \(3.37 \times 10^{-3} \,\mathrm{moles}\) c. Mass of 0.050 mole of dimethylnitrosamine = \(3.71 \,\mathrm{g}\) d. Number of hydrogen atoms in 1.0 mole of dimethylnitrosamine = \(3.61 \times 10^{24} \, \mathrm{H\, atoms}\) e. Mass of \(1.0 \times 10^{6}\) molecules of dimethylnitrosamine = \(1.23 \times 10^{-16} \,\mathrm{g}\) f. Mass of one molecule of dimethylnitrosamine = \(1.23 \times 10^{-22} \,\mathrm{g}\)

Step-by-step solution

Find the molar mass of individual elements

We need to find the molar mass of C, H, N, and O. - Carbon (C): 12.01 g/mol - Hydrogen (H): 1.01 g/mol - Nitrogen (N): 14.01 g/mol - Oxygen (O): 16.00 g/mol

Determine the molar mass of dimethylnitrosamine

Using the molar masses of the individual elements, we can find the molar mass of the whole compound. Molar mass of \((\mathrm{CH}_{3})_{2} \mathrm{N}_{2} \mathrm{O}\) = \([(2 × 12.01) + (6 × 1.01) + (2 × 14.01) + 16.00] \,\mathrm{g/mol}\) = \(74.16 \,\mathrm{g/mol}\) b. Calculate the number of moles in 250 mg of dimethylnitrosamine.

Convert mass to moles

To convert the mass of dimethylnitrosamine to moles, we will use the molar mass we found earlier (74.16 g/mol). No. of moles = \(\frac{250 \,\mathrm{mg}}{74.16 \,\mathrm{g/mol}}\) = \(\frac{0.250 \,\mathrm{g}}{74.16 \,\mathrm{g/mol}}\) = \(3.37 \times 10^{-3}\) moles c. Calculate the mass of 0.050 mole of dimethylnitrosamine.

Convert moles to mass

We can use the molar mass we found earlier and multiply it by the number of moles given to find the mass. Mass = No. of moles × Molar mass = \(0.050 \,\mathrm{mol} \times 74.16 \,\mathrm{g/mol}\) = \(3.71 \,\mathrm{g}\) d. Find the number of hydrogen atoms in 1.0 mole of dimethylnitrosamine.

Determine the number of H atoms per molecule

There are 2 CH3 groups per molecule, with each group containing 3 H atoms. So, the total number of H atoms per molecule is 2 × 3 = 6.

Calculate the total number of H atoms in 1.0 mole

Now, multiply the number of H atoms per molecule by Avogadro's number (6.022 × 10^23) to find the total number of hydrogen atoms in 1.0 mole of dimethylnitrosamine. No. of H atoms = 6 × 1.0 mole × 6.022 × 10^23 = \(3.61 \times 10^{24}\) H atoms e. Calculate the mass of \(1.0 \times 10^{6}\) molecules of dimethylnitrosamine.

Convert number of molecules to moles

First, find the number of moles using Avogadro's number. No. of moles = \(\frac{1.0 \times 10^{6}}{6.022 \times 10^{23}}\) = \(1.66 \times 10^{-18}\) moles

Calculate the mass

We can now use the molar mass to find the mass of \(1.0 \times 10^{6}\) molecules. Mass = No. of moles × Molar mass = \(1.66 \times 10^{-18} \,\mathrm{mol} \times 74.16 \,\mathrm{g/mol}\) = \(1.23 \times 10^{-16} \,\mathrm{g}\) f. Find the mass of one molecule of dimethylnitrosamine.

Calculate the mass of one molecule

The mass of one molecule can be found by dividing the mass of \(1.0 \times 10^6\) molecules by \(1.0 \times 10^6\). Mass of one molecule = \(\frac{1.23 \times 10^{-16} \,\mathrm{g}}{1.0 \times 10^{6}}\) = \(1.23 \times 10^{-22} \,\mathrm{g}\)

Key concepts

Molar Mass Calculation

To find the molar mass of a compound, you first need to identify all the elements present in the molecule and their respective quantities. In the case of dimethylnitrosamine, represented by the formula \((\mathrm{CH}_{3})_{2} \mathrm{N}_{2} \mathrm{O}\), it consists of:

• Carbon (C) - 2 atoms
• Hydrogen (H) - 6 atoms
• Nitrogen (N) - 2 atoms
• Oxygen (O) - 1 atom

Next, you use the molar masses for each element, usually given in grams per mole (g/mol). These values are standard and can be found on the periodic table:

• Carbon (C): 12.01 g/mol
• Hydrogen (H): 1.01 g/mol
• Nitrogen (N): 14.01 g/mol
• Oxygen (O): 16.00 g/mol

Now, calculate the total molar mass by multiplying the number of atoms of each element by its molar mass and adding them all together:\[\text{Molar Mass of } (\mathrm{CH}_3)_2\mathrm{N}_2\mathrm{O} = (2 \times 12.01) + (6 \times 1.01) + (2 \times 14.01) + (1 \times 16.00) = 74.16\, \mathrm{g/mol}\]

Moles and Mass Conversion

The mole is a fundamental unit in chemistry that indicates the amount of a substance. To find how many moles are in a given mass, you can use the formula:\[\text{Number of moles} = \frac{\text{mass (in grams)}}{\text{molar mass (g/mol)}}\]For example, to find how many moles are in 250 mg of dimethylnitrosamine, first convert the mass from milligrams to grams (since the molar mass is in grams per mole):

• 250 mg = 0.250 g

Then, use the molar mass of dimethylnitrosamine (74.16 g/mol):\[\text{Number of moles} = \frac{0.250\, \mathrm{g}}{74.16\, \mathrm{g/mol}} = 3.37 \times 10^{-3}\, \text{moles}\]To find the mass from moles, you reverse the process:

• Multiply the number of moles by the molar mass.

For instance, the mass of 0.050 moles would be:\[\text{Mass} = 0.050\, \text{moles} \times 74.16\, \mathrm{g/mol} = 3.71\, \mathrm{g}\]

Avogadro's Number

Avogadro's number is a constant that is crucial in chemistry for relating the number of individual molecules or atoms to the number of moles. Defined as 6.022 \times 10^{23}, this number indicates how many molecules are in one mole of a substance.For example, if you need to determine the number of hydrogen atoms in one mole of dimethylnitrosamine:

• The molecule \((\mathrm{CH}_3)_2\mathrm{N}_2\mathrm{O}\) contains 6 hydrogen atoms.
• In one mole of the compound, you have Avogadro's number worth of molecules.
• Thus, the number of hydrogen atoms is calculated by multiplying Avogadro's number by the number of hydrogen atoms in one molecule:\[ 6 \times 6.022 \times 10^{23} = 3.61 \times 10^{24}\, \text{atoms} \]

Chemical Composition Analysis

Chemical composition analysis involves examining a compound's constitution in terms of its atomic or molecular components. Understanding this is essential to determine the mass or amount of specific molecules based on a known quantity.To find the mass of \(1.0 \times 10^{6}\) molecules of dimethylnitrosamine, you first convert the number of molecules into moles using Avogadro's number:\[\text{Number of moles} = \frac{1.0 \times 10^{6}}{6.022 \times 10^{23}} = 1.66 \times 10^{-18}\, \text{moles}\]Then, use the molar mass to find the total mass:\[\text{Mass} = 1.66 \times 10^{-18}\, \text{moles} \times 74.16\, \mathrm{g/mol} = 1.23 \times 10^{-16}\, \mathrm{g}\]Finally, to find the mass of a single molecule, divide the mass for \(1.0 \times 10^{6}\) molecules by \(1.0 \times 10^{6}\):\[\text{Mass of one molecule} = \frac{1.23 \times 10^{-16} \mathrm{g}}{1.0 \times 10^{6}} = 1.23 \times 10^{-22}\, \mathrm{g}\]This analysis allows you to understand how the mass relates to the number of molecules or atoms present, offering insights into the proportion and makeup of chemical compounds.