Problem 28 - Basic Principles of Organic Chemistry

Question

Hemoglobin, the protein responsible for carrying oxygen from the lungs to the body tissues, contains \(0.355 \%\) iron. Hydrolysis of \(100 \mathrm{~g}\) of hemoglobin gives \(1.48 \mathrm{~g}\) of tryptophan. Calculate the minimum molecular weight of hemoglobin that is consistent with these results.

Step-by-Step Solution

Answer

The minimum molecular weight of hemoglobin is 628.8 g/mol, consistent with its iron content and tryptophan yield.

Step 1: Calculate the Amount of Iron in 100g of Hemoglobin

First, we need to determine how much iron is present in 100g of hemoglobin. Since the percentage of iron is given as 0.355%, we calculate this by multiplying the total mass of hemoglobin by this percentage. Amount of iron = (0.355/100) × 100 g = 0.355 g.

Step 2: Relate Iron Mass to Molecular Weight

Next, we calculate how many moles of iron correspond to 0.355 g. The atomic weight of iron (Fe) is approximately 55.85 g/mol. Number of moles of iron = 0.355 g / 55.85 g/mol ≈ 0.00636 mol. Since hemoglobin contains iron in the form of heme groups that are essential for its function, one mole of hemoglobin should have at least one mole of iron. Therefore, the molecular weight must be consistent with one mole of hemoglobin having at least 0.00636 moles of iron.

Step 3: Calculate Minimum Molecular Weight Using Iron Content

To find the minimum molecular weight of hemoglobin, we assume there is at least one iron atom per hemoglobin molecule. Hence, the weight of hemoglobin per mole = 1 / 0.00636 mol = 157.2 g/mol. This is the molecular weight per iron atom, which represents the minimum molecular weight of hemoglobin. However, it is known from biochemistry that hemoglobin has four heme groups (each with an iron atom), so multiplying by 4 gives us the actual minimum molecular weight due to iron content = 157.2 g/mol × 4 = 628.8 g/mol.

Step 4: Use Tryptophan Content to Verify Molecular Weight

We calculate how much tryptophan must be present to account for 1.48 g in 100 g of hemoglobin. Assume all tryptophan belongs to the protein component. Molecular weight of tryptophan = C₁₁H₁₂N₂O₂ = 186.21 g/mol. Therefore, calculate the number of moles of tryptophan: Number of moles of tryptophan = 1.48 g / 186.21 g/mol ≈ 0.00795 mol. Using the idea that each molecule of hemoglobin can incorporate a full amount of tryptophan, we compare to iron attempt above and ensure that minimum molecular weight based on iron is viable with tryptophan content.

Step 5: Compare Calculated Molecular Weights

Multiply the moles of tryptophan by the iron-based molecular weight calculation. Use the tryptophan moles to suggest a proportion of mass: 628.8 g/mol × 0.00795 mol = 5.00 g, a portion lesser than the entire 100 g could be allocated possibly to side chains and amino acids including tryptophan slightly verifying indeed The minimum weight calculated previously did indeed include tryptophan.”

Key Concepts

Iron Content in Proteins

Iron plays a crucial role in proteins, especially in complex molecules like hemoglobin. Hemoglobin, known for transporting oxygen in our blood, contains iron in its heme groups. These heme groups allow hemoglobin to bind oxygen effectively. The fact that hemoglobin consists of about 0.355% iron gives us valuable data to estimate its molecular weight. By understanding the relationship between the iron content and hemoglobin, one can infer the essential role iron plays in the protein's structure and function. Calculating the amount of iron in a sample of hemoglobin involves basic percentage calculations. For example, determining how much iron is in 100 g of hemoglobin means using this iron percentage (0.355%) to find 0.355 g of iron. Knowing this, we can proceed to calculate how many moles of iron this corresponds to by dividing by the atomic weight of iron (55.85 g/mol). This provides us the basis to estimate the molecular weight of hemoglobin based on its iron content alone.

Molecular Weight Calculation

The molecular weight calculation is crucial for understanding the scale and composition of complex molecules like proteins. Hemoglobin's molecular weight determination begins by calculating how many moles of its key components, such as iron, are present in a given quantity.When it comes to hemoglobin, each molecule must have at least one mole of iron due to its integral role. Starting with the moles of iron calculated from a mere 0.355 g of iron in the 100 g of hemoglobin, we find \[ \frac{0.355\, \text{g}}{55.85\, \text{g/mol}} = 0.00636\, \text{mol} \].Now, to find the molecular weight per mole of hemoglobin with its iron content, divide 1 by this mole number: \[ \frac{1}{0.00636\, \text{mol}} = 157.2\, \text{g/mol} \].Yet, since hemoglobin contains four iron atoms only in its structure, this result must be multiplied by four, yielding a realistic minimum molecular weight of \[ 628.8\, \text{g/mol} \]. This accounts for the four heme groups in hemoglobin. Adjusting the estimation through known biochemical data ensures the calculated minimal molecular weight aligns with actual protein structures.

Tryptophan in Hemoglobin

Tryptophan is an important amino acid within the hemoglobin structure. Insight into tryptophan's concentration helps validate molecular weight calculations for hemoglobin. In the context of this exercise, tryptophan content analysis is part of verifying the hemoglobin's calculated measurement using known standards.The molecular weight of tryptophan (\[ \text{C}_{11}\text{H}_{12}\text{N}_2\text{O}_2 \]) is \[ 186.21 \text{ g/mol} \]. By deducing the moles of tryptophan in hemoglobin (1.48 g in 100 g of hemoglobin),we calculate: \[ \frac{1.48\, \text{g}}{186.21\, \text{g/mol}} = 0.00795\, \text{mol} \]. This observation allows for a cross-check with the molecular weight deduced from iron.By examining how much tryptophan can combine in hemoglobin, a strong relationship becomes apparent. The compatibility of these numbers reinforces the calculated minimum molecular weight from iron content, showing that tryptophan and other amino acids are consistently accounted for within this measurement.