Problem 116 - A Complete Resource Book in Chemistry for JEE Main

Question

A purified pepsin was subjected to amino acid analysis. The amino acid present in the smallest amount was lysine, \(\mathrm{C}_{6} \mathrm{H}_{14} \mathrm{~N}_{2} \mathrm{O}_{2}\) and the amount of lysine was found to be \(0.431 \mathrm{~g}\) per \(100 \mathrm{~g}\) of protein. The minimum molecular mass of protein is (a) \(34 \mathrm{u}\) (b) \(3400 \mathrm{u}\) (c) \(34,000 \mathrm{u}\) (d) \(3400,000 \mathrm{u}\)

Step-by-Step Solution

Answer

The minimum molecular mass of the protein is 34,000 u (option c).

Step 1: Calculate Moles of Lysine

First, we need to calculate the moles of lysine in 0.431 g. The molar mass of lysine, \( \mathrm{C}_{6} \mathrm{H}_{14} \mathrm{N}_{2} \mathrm{O}_{2} \), is \(6(12.01) + 14(1.01) + 2(14.01) + 2(16.00) = 146.19 \) g/mol. Thus, the moles of lysine are calculated as:\[ \text{Moles of lysine} = \frac{0.431 \text{ g}}{146.19 \text{ g/mol}} \approx 0.00295 \text{ moles} \]

Step 2: Determine Minimum Protein Mass

In terms of minimum protein mass, we assume that there is at least one lysine molecule present in the protein. Therefore, since the given smallest amount is per 100 g of protein, the calculation is done as follows:\[ \text{Minimum protein mass} = \frac{100 \text{ g of protein}}{0.00295 \text{ moles of lysine}} \approx 33,898.31 \text{ g/mol} \]

Step 3: Select Closest Answer

Given the choices, the closest value to our calculated minimum protein mass of \(33,898.31 \text{ u} \) is \(34,000 \text{ u} \). Therefore, the protein's minimum molecular mass corresponds to option (c).

Key Concepts

Amino Acid Analysis

Understanding amino acid analysis is key in determining the structure and composition of proteins. This process helps identify and quantify the amino acids that make up a protein. During amino acid analysis, proteins are typically hydrolyzed into their constituent amino acids. Then, these amino acids are separated, identified, and often quantified using methods like High-Performance Liquid Chromatography (HPLC).
This analysis is crucial as it provides insights into the protein's sequence and functional properties. Knowing the specific types and amounts of amino acids, such as lysine, allows scientists to infer the protein's molecular weight and potential biological roles. For instance, in the case of pepsin, identifying lysine as the least abundant amino acid aids in estimating the protein's minimal molecular mass by ensuring all amino acids, including the rarest ones, are accounted for in calculations.

Lysine Calculation

Calculating the amount of lysine within a protein sample is an essential step. Lysine is an important amino acid composed of the elements: carbon (C), hydrogen (H), nitrogen (N), and oxygen (O). Its molecular formula is \(\mathrm{C}_{6} \mathrm{H}_{14} \mathrm{N}_{2} \mathrm{O}_{2}\).
To calculate how many moles of lysine are present in a given mass, we use its molar mass. Here’s how it is figured:

• Calculate each element's contribution to the molar mass:
  - Carbon: 6 atoms × 12.01 g/mol = 72.06 g/mol
  - Hydrogen: 14 atoms × 1.01 g/mol = 14.14 g/mol
  - Nitrogen: 2 atoms × 14.01 g/mol = 28.02 g/mol
  - Oxygen: 2 atoms × 16.00 g/mol = 32.00 g/mol

Summing these gives lysine a molar mass of 146.19 g/mol.
By dividing the mass of lysine we have (0.431 g) by its molar mass, we find the moles of lysine. This enables accurate analysis and comparison to the protein mass.

Molar Mass

The concept of molar mass is pivotal in chemical calculations and understanding how it relates to proteins is no different. Molar mass, typically expressed in grams per mole (g/mol) or unified atomic mass units (u), represents the mass of one mole of a substance. It is instrumental when equating the quantity of amino acids in a protein to infer the protein's size.
When we calculate the molar mass of a protein, we are estimating its minimal molecular mass by assuming the presence of all constituent amino acids, even those least abundant, like lysine. In this context, the minimal molecular mass is calculated by assuming just one lysine molecule per protein unit. This ensures that anything pertainable to the protein is included in the calculations.
Given lysine's molar determination reveals the protein's minimal molar mass to approximate 33,898.31 u, which matches closely with the selection of 34,000 u, it reflects how interconnected the concepts of amino acid analysis and the molar mass are for elucidating protein characteristics.