Problem 7 - General Chemistry: Principles and Modern Applications

Question

The amino acid methionine, which is essential in human diets, has the molecular formula \(\mathrm{C}_{5} \mathrm{H}_{11} \mathrm{NO}_{2} \mathrm{S}\) Determine (a) its molecular mass; (b) the number of moles of \(\mathrm{H}\) atoms per mole of methionine; (c) the number of grams of \(C\) per mole of methionine; (d) the number of \(C\) atoms in 9.07 mol methionine.

Step-by-Step Solution

Answer

The molecular mass of methionine is 149.21 amu, there are 11 moles of H per mole of methionine, there are 60.05 grams of carbon per mole of methionine, and there are \(2.73 × 10^{25}\) carbon atoms in 9.07 moles of methionine.

Step 1: Calculate the molecular mass

To calculate the molecular mass (in atomic mass units), one must add up the atomic masses of all the atoms present in the molecule. This can be achieved by looking up the atomic masses of \(\mathrm{C}\), \(\mathrm{H}\), \(\mathrm{N}\), \(\mathrm{O}\) and \(\mathrm{S}\), in the periodic table, then multiplying by the number of each atom in the molecular formula and summing up these values. For methionine with a formula of \(\mathrm{C}_{5} \mathrm{H}_{11} \mathrm{NO}_{2} \mathrm{S}\): Atomic masses are \(\mathrm{C}\) = 12.01 amu, \(\mathrm{H}\) = 1.008 amu, \(\mathrm{N}\) = 14.007 amu, \(\mathrm{O}\) = 15.999 amu, \(\mathrm{S}\) = 32.06 amu. Therefore, the molecular mass is \(5 × 12.01 + 11 × 1.008 + 1 × 14.007 + 2 × 15.999 + 1 × 32.06\) = 149.21 amu.

Step 2: Determine the number of moles of H per mole of methionine

There are as many moles of an atom in a compound as the subscript after that atom in the molecular formula. In this case, the molecular formula tells us that there are 11 moles of H in every mole of methionine.

Step 3: Calculate the grams of C per mole of methionine

To calculate the number of grams of carbon (C) per mole of methionine, multiply the number of moles of C in methionine (which is 5 as per the molecular formula) by the atomic mass of carbon (12.01 g/mol). Therefore, the number of grams of carbon per mol of methionine is \(5 × 12.01 = 60.05\) g/mol.

Step 4: Calculate the number of C atoms in 9.07 mol methionine

Since there are 5 moles of carbon for every mole of methionine, there would be \(5 × 9.07 = 45.35\) moles of carbon in 9.07 mol methionine. Using Avogadro's number (\(6.022 × 10^{23}\) atoms/mole), we get the total number of carbon atoms to be \(45.35 × (6.022 × 10^{23})\), which rounds off to \(2.73 × 10^{25}\) carbon atoms.

Key Concepts

Molecular Mass Calculation

Calculating the molecular mass of a compound is an important step in understanding its bigger picture. Molecular mass, sometimes referred to as molecular weight, is the sum of the atomic masses of all the atoms in a molecule. Each element has a specific atomic mass found on the periodic table. To find the molecular mass of methionine, you add up the atomic masses of carbon (\( C \)), hydrogen (\( H \)), nitrogen (\( N \)), oxygen (\( O \)), and sulfur (\( S \)), based on its formula \(\mathrm{C}_5\mathrm{H}_{11}\mathrm{NO}_2\mathrm{S} \). This means:

• 5 carbon atoms: \(5 \times 12.01 \, \text{amu} = 60.05 \, \text{amu} \)
• 11 hydrogen atoms: \(11 \times 1.008 \, \text{amu} = 11.088 \, \text{amu} \)
• 1 nitrogen atom: \(1 \times 14.007 \, \text{amu} = 14.007 \, \text{amu}\)
• 2 oxygen atoms: \(2 \times 15.999 \, \text{amu} = 31.998 \, \text{amu} \)
• 1 sulfur atom: \(1 \times 32.06 \, \text{amu} = 32.06 \, \text{amu}\)

When you total these, the molecular mass of methionine is \(149.21 \, \text{amu} \). This calculation is foundational for further analysis such as chemical reactions and stoichiometry.

Moles in Chemistry

In chemistry, the mole is a basic unit used to quantify the amount of substance. One mole contains exactly \(6.022 \times 10^{23} \) entities (Avogadro's number), which could be atoms, molecules, or other particles. This concept helps chemists understand and work with the incredibly large numbers of tiny atoms and molecules. When dealing with a molecular formula, like methionine's \(\mathrm{C}_5\mathrm{H}_{11}\mathrm{NO}_2\mathrm{S} \), the subscripts indicate the number of moles of each type of atom present in one mole of the compound. For example, 11 hydrogen atoms mean 11 moles of hydrogen are in one mole of methionine. This allows chemists to perform stoichiometric calculations and predict the outcomes of reactions based on mole ratios.

Avogadro's Number

Avogadro's number is a constant that is fundamental in chemistry. It provides the link between the macroscopic scale that we can observe and the microscopic scale of atoms and molecules that we calculate. Avogadro's number \(6.022 \times 10^{23} \) acts as a conversion factor between the number of moles and number of atoms or molecules. For methionine, if you want to find out how many carbon atoms are present in a given amount of the substance, you would use Avogadro’s number. For instance, 9.07 moles of methionine contain 45.35 moles of carbon (since there are 5 carbon atoms per methionine molecule), and to find the exact count of carbon atoms, multiply by Avogadro's number, resulting in \(45.35 \times 6.022 \times 10^{23} \approx 2.73 \times 10^{25} \) carbon atoms. Understanding Avogadro's number is key to interpreting chemical quantities.

Chemical Formula Interpretation

Chemical formulas like \(\mathrm{C}_5\mathrm{H}_{11}\mathrm{NO}_2\mathrm{S} \) for methionine provide a wealth of information about the compound's composition. These formulas give the exact number of each element's atoms in one molecule of the substance and help in determining relationships among them. From the formula, one can deduce the number of atoms, calculate molecular mass, and understand molar ratios. Each subscript represents how many atoms of that element appear in a molecule and inherently suggests how moles of elements relate to each other within a mole of compound. Therefore, knowing how to interpret chemical formulas is essential for performing calculations such as determining the mass of a particular element in a compound or predicting how much of a substance is needed for a chemical reaction.