Question

Complete hydrolysis of a \(100.00-\mathrm{g}\) sample of a peptide gave the following amounts of individual amino acids (molar masses, in \(\mathrm{g} / \mathrm{mol}\), appear in parentheses $$ \begin{aligned} &\begin{array}{ll} 3.70 \mathrm{~g} \text { of valine }(117.15) & 6.90 \mathrm{~g} \text { of proline }(115.13) \end{array}\\\ &7.30 \mathrm{~g} \text { of serine }(105.10) \quad 86.00 \mathrm{~g} \text { of arginine }(174.21) \end{aligned} $$ (a) Why does the total mass of amino acids exceed the mass of peptide? (b) What are the relative numbers of amino acids in the peptide? (c) What is the minimum molar mass of the peptide?

Short answer

(a) Addition of water (3.90 g) during hydrolysis. (b) Valine: 1, Proline: 2, Serine: 2, Arginine: 16. (c) 3344.97 g/mol.

Step-by-step solution

- Calculate the Moles of Each Amino Acid

Divide the mass of each amino acid by its respective molar mass to find the number of moles of each amino acid. Use the formula: \[ \text{Moles} = \frac{\text{Mass}}{\text{Molar Mass}} \] For Valine: \[ \text{Moles of Valine} = \frac{3.70 \text{ g}}{117.15 \text{ g/mol}} = 0.0316 \text{ mol} \] For Proline: \[ \text{Moles of Proline} = \frac{6.90 \text{ g}}{115.13 \text{ g/mol}} = 0.0599 \text{ mol} \] For Serine: \[ \text{Moles of Serine} = \frac{7.30 \text{ g}}{105.10 \text{ g/mol}} = 0.0695 \text{ mol} \] For Arginine: \[ \text{Moles of Arginine} = \frac{86.00 \text{ g}}{174.21 \text{ g/mol}} = 0.4937 \text{ mol} \]

- Calculate Total Mass of Amino Acids

The total masses of the amino acids are summed up: \[ \text{Total Mass of Amino Acids} = 3.70 \text{ g} + 6.90 \text{ g} + 7.30 \text{ g} + 86.00 \text{ g} = 103.90 \text{ g} \]

- Compare Total Mass of Amino Acids to Peptide Mass

Notice that the total mass of the amino acids (103.90 g) is greater than the mass of the peptide (100.00 g). The difference is due to the addition of water molecules during hydrolysis which adds to the mass. Thus, \[ \text{Mass of Added Water} = 103.90 \text{ g} - 100.00 \text{ g} = 3.90 \text{ g} \]

- Determine Relative Numbers of Each Amino Acid in Peptide

To find the relative numbers, divide the moles of each amino acid by the smallest number of moles among them, which is 0.0316 mol (Valine): For Valine: \[ \frac{0.0316}{0.0316} = 1 \] For Proline: \[ \frac{0.0599}{0.0316} = 1.9 \rightarrow 2 \] For Serine: \[ \frac{0.0695}{0.0316} = 2.2 \rightarrow 2 \] For Arginine: \[ \frac{0.4937}{0.0316} = 15.6 \rightarrow 16 \] Thus, the relative amounts are: Valine: 1, Proline: 2, Serine: 2, Arginine: 16.

- Calculate Minimum Molar Mass of Peptide

To find the minimum molar mass, multiply the smallest molar ratio of each amino acid by its molar mass and sum them up: \[ \text{Minimum Molar Mass} = 1 \times 117.15 \text{ g/mol} + 2 \times 115.13 \text{ g/mol} + 2 \times 105.10 \text{ g/mol} + 16 \times 174.21 \text{ g/mol} \] \[ = 117.15 + 230.26 + 210.20 + 2787.36 = 3344.97 \text{ g/mol} \] Thus, the minimum molar mass of the peptide is 3344.97 g/mol.

Key concepts

amino acid molar masses

Amino acids are the building blocks of peptides and proteins, and each has a specific molar mass. The molar mass of a compound is the mass of one mole of that compound, measured in grams per mole (g/mol). For example, in the given problem, valine has a molar mass of 117.15 g/mol, proline has a molar mass of 115.13 g/mol, serine has a molar mass of 105.10 g/mol, and arginine has a molar mass of 174.21 g/mol. Understanding these values is essential because they allow us to convert between the mass of an amino acid and the number of moles of that amino acid, which is crucial for various calculations in chemistry and biochemistry.

amino acid mole calculation

To understand the composition of a peptide, we first need to calculate the number of moles of each amino acid present. This is done using the formula:

\[ \text{Moles} = \frac{\text{Mass}}{\text{Molar Mass}} \]

For example, to find the moles of valine:

• Divide the mass of valine (3.70 g) by its molar mass (117.15 g/mol):
• \(\text{Moles of Valine} = \frac{3.70 \text{ g}}{117.15 \text{ g/mol}} = 0.0316 \text{ mol}\)

Repeat this process for the other amino acids:

• Proline: \(\text{Moles of Proline} = \frac{6.90 \text{ g}}{115.13 \text{ g/mol}} = 0.0599 \text{ mol}\)
• Serine: \(\text{Moles of Serine} = \frac{7.30 \text{ g}}{105.10 \text{ g/mol}} = 0.0695 \text{ mol}\)
• Arginine: \(\text{Moles of Arginine} = \frac{86.00 \text{ g}}{174.21 \text{ g/mol}} = 0.4937 \text{ mol}\)

This calculation allows us to understand the quantity of each amino acid present in the sample, which is critical for determining their relative composition in the peptide.

minimum molar mass of peptide

The minimum molar mass of a peptide is an important concept when determining the smallest possible size of the peptide based on its constituent amino acids. To calculate the minimum molar mass, follow these steps:

• First, determine the relative numbers of each amino acid in the peptide by dividing their moles by the smallest number of moles present:
• For Valine: \(\frac{0.0316}{0.0316} = 1\)
• For Proline: \(\frac{0.0599}{0.0316} = 1.9 \rightarrow 2\)
• For Serine: \(\frac{0.0695}{0.0316} = 2.2 \rightarrow 2\)
• For Arginine: \(\frac{0.4937}{0.0316} = 15.6 \rightarrow 16\)

This gives us the ratio of 1 Valine: 2 Proline: 2 Serine: 16 Arginine.
Next, multiply these ratios by their respective molar masses and sum them up:

• \(1 \times 117.15 \text{ g/mol} = 117.15 \text{ g/mol}\)
• \(2 \times 115.13 \text{ g/mol} = 230.26 \text{ g/mol}\)
• \(2 \times 105.10 \text{ g/mol} = 210.20 \text{ g/mol}\)
• \(16 \times 174.21 \text{ g/mol} = 2787.36 \text{ g/mol}\)

Finally, add these values together to find the minimum molar mass of the peptide:

\[ \text{Minimum Molar Mass} = 117.15 + 230.26 + 210.20 + 2787.36 = 3344.97 \text{ g/mol} \]
This calculation is crucial for biochemists and molecular biologists in understanding the structure and function of peptides.