Problem 76 - Chemistry The Molecular Science

Question

You have a pure sample of the antiseptic aminacrine, \(\mathrm{C}_{13} \mathrm{H}_{10} \mathrm{~N}_{2}\) (a) Calculate the mass in grams of 0.06500 mol aminacrine. (b) Calculate the number of aminacrine molecules in a \(0.2480-\mathrm{g}\) sample. (c) Calculate the number of nitrogen atoms in this \(0.2480-\mathrm{g}\) sample. (d) Calculate the mass of \(\mathrm{N}\) in 100 . \(\mathrm{g}\) aminacrine.

Step-by-Step Solution

Answer

(a) 12.625 g, (b) \(7.684 \times 10^{20}\) molecules, (c) \(1.537 \times 10^{21}\) nitrogen atoms, (d) 14.42 g nitrogen.

Step 1: Calculate Molar Mass

To solve the problems, we first need to determine the molar mass of aminacrine. Aminacrine has the formula: \(\mathrm{C}_{13}\mathrm{H}_{10}\mathrm{N}_{2}\). We calculate the molar mass by summing the atomic masses (found in the periodic table) of each element in the molecule:\[M = 13(12.01) + 10(1.008) + 2(14.01)= 156.13 + 10.08 + 28.02 = 194.23 \, \text{g/mol}\]

Step 2: Mass of Aminacrine

For part (a), we need to calculate the mass of 0.06500 mol of aminacrine using the molar mass found in Step 1:\[\text{Mass} = 0.06500 \times 194.23 \, \text{g/mol} = 12.62495 \, \text{g}\]

Step 3: Calculate Molecules in a Sample

For part (b), we first need to find the number of moles in a 0.2480-g sample of aminacrine using the molar mass:\[\text{Moles} = \frac{0.2480}{194.23} = 0.001276 \, \text{mol}\]Next, use Avogadro's number (\(6.022 \times 10^{23}\) molecules/mol) to find the number of molecules:\[\text{Molecules} = 0.001276 \times 6.022 \times 10^{23} = 7.684 \times 10^{20} \text{ molecules} \]

Step 4: Calculate Nitrogen Atoms

For part (c), calculate the number of nitrogen atoms in the 0.2480-g sample. Since there are 2 nitrogen atoms per molecule of aminacrine, multiply the number of molecules by 2:\[\text{Nitrogen atoms} = 2 \times 7.684 \times 10^{20} = 1.537 \times 10^{21} \text{ atoms}\]

Step 5: Mass of Nitrogen in Aminacrine

For part (d), calculate the mass of nitrogen in 100.0 g of aminacrine. First, find the fraction of the molecule's mass contributed by nitrogen. In \(\mathrm{C}_{13}\mathrm{H}_{10}\mathrm{N}_{2}\), nitrogen's total mass contribution is \(2 \times 14.01 = 28.02\) g. The total molar mass is 194.23 g/mol, so the mass fraction for nitrogen is:\[\text{Fraction of N} = \frac{28.02}{194.23}\]Finally, the mass of nitrogen in 100 g of aminacrine is:\[\text{Mass of N} = 100.0 \times \frac{28.02}{194.23} = 14.42 \, \text{g}\]

Key Concepts

Moles

A mole is a fundamental concept in chemistry, serving as a bridge between the atomic scale and the macroscopic world. It allows chemists to count particles, such as atoms or molecules, by weighing them. One mole corresponds to exactly 6.022 x 10²³ particles, a number known as Avogadro's number.

The reason we use moles is due to the incredibly small size of atoms and molecules. Counting them individually is impractical, so the mole provides a manageable way to express amounts of a chemical substance.

In the exercise, utilizing the concept of moles, we were able to determine the mass of aminacrine. We used the molar mass, 194.23 g/mol, to convert moles to grams, and vice versa. Similar mole-to-mass conversions can tackle many chemical calculations.

To calculate the number of moles from a given mass:

• Divide the mass of the substance by its molar mass.

Avogadro's Number

Avogadro's Number, denoted as 6.022 x 10²³, is crucial for converting between moles and number of molecules or atoms. Named after Amedeo Avogadro, it represents the number of atoms or molecules in one mole of a substance.

In practical terms, Avogadro's Number helps to relate the extremely small world of atoms and molecules to the quantities you might handle in a lab. For example, using Avogadro's Number, chemists can determine how many atoms are present in a sample that weighs as little as a few grams.

In our problem, Avogadro's Number was used to determine the number of aminacrine molecules present in a 0.2480-g sample. By knowing how many moles of aminacrine were in the sample, we multiplied by Avogadro's Number to find the exact count of molecules.

To use Avogadro’s number:

• Multiply the number of moles by 6.022 x 10²³ to find the number of molecules or atoms.

Atomic Mass

Atomic mass refers to the mass of a single atom, usually expressed in atomic mass units (amu). The atomic mass is found on the periodic table and is derived mainly from the protons and neutrons in an atom’s nucleus.

Calculating molar mass involves adding up the atomic masses of all individual atoms present in a molecule. In the exercise, we calculated the molar mass of aminacrine by summing the atomic masses from the formula \( \mathrm{C}_{13}\mathrm{H}_{10}\mathrm{N}_{2} \). This provided a mole-to-grams conversion.

Molar mass allows us to use the concept of moles effectively:

• Determine the mass of one mole of a substance, essential for any stoichiometric calculations.

Atomic mass forms the foundation for many other chemical calculations, linking directly to both moles and Avogadro's Number for quantifying substances.