Problem 22 - Basic Principles of Organic Chemistry

Question

Suppose there is \(1 \%\) formation in each step of the wrong isomer of the acylating component in an otherwise quantitative 100 -step peptide synthesis. What is the yield of the desired polypeptide isomer?

Step-by-Step Solution

Answer

The yield of the desired polypeptide isomer is approximately 36.6%.

Step 1: Understanding the Problem

In a 100-step peptide synthesis, there is a 1% chance of forming the wrong isomer at each step. Our task is to determine the yield of the desired isomer after all 100 steps are completed.

Step 2: Identifying Probability Values

Each step has a 1% chance of forming the wrong isomer and therefore a 99% chance of forming the correct one. This can be expressed as probabilities: Probability of correct isomer per step = 0.99 Probability of wrong isomer per step = 0.01.

Step 3: Calculating the Yield

Since each step is independent, we multiply the probability of forming the correct isomer per step across all 100 steps. This gives us the yield:\[0.99^{100}\]

Step 4: Performing the Calculation

Calculate the result of the expression \(0.99^{100}\) using a calculator:\[0.99^{100} \ \approx 0.366\]This represents approximately 36.6% yield of the correct isomer.

Key Concepts

Isomer Formation

In peptide synthesis, an isomer refers to molecules with the same molecular formula but different structural arrangements. During peptide synthesis, especially in long chains involving many steps, there is always a chance of forming the wrong isomer. This can lead to slight but significant changes in the properties of the final polypeptide.

In this problem, we are dealing with a controlled synthesis process where there's a 1% chance in each step of the acylation process that the wrong isomer is formed. While this might seem small at first glance, over many steps—like the 100 steps in this question—these small percentages can compound, significantly affecting the final yield of the desired polypeptide. Knowing how to manage and predict these formations is crucial in chemistry, particularly when aiming for high purities and precise folding in biotechnological and pharmaceutical applications.

Probability Calculation

To understand the final yield of the desired polypeptide, we need to understand how probabilities work in a sequence of independent events. Each step in a peptide synthesis is considered an independent event, and calculating the overall probability requires multiplying the probabilities of individual steps.

Here, the correct isomer formation probability per step is 0.99 because there is only a 1% chance of forming the wrong isomer. Therefore, the formula to calculate the likelihood of correctly forming the desired polypeptide over 100 steps is \[ 0.99^{100} \] This shows how precise and stable each step must be to ensure a good overall yield. Understanding these calculations can help you predict and optimize the outcome in multi-step chemical syntheses, crucial for effective experiment design and resource allocation.

Polypeptide Yield

Yield in chemical reactions refers to the amount of product obtained compared to the maximum possible amount, expressed as a percentage. It's a crucial factor in evaluating the efficiency of a reaction, especially in processes requiring high precision, like peptide synthesis.

For this exercise, the calculation of \[ 0.99^{100} \] translates to an actual yield of about 36.6%. This means that after implementing 100 steps of peptide synthesis, only 36.6% of the polypeptides are of the correct isomer. This relatively low yield underscores the challenge in achieving high fidelity synthesis as the number of steps increases.

• It's important to remember that each error in isomer formation reduces the final yield.
• Appropriate adjustments to the synthesis process can improve overall efficiency.
• Yield data helps chemists make informed decisions about resource investments and batch processing.

Achieving a high yield is often a balance of optimizing individual steps, controlling errors, and sometimes accepting realistic production limits due to synthesis complexity.