Question

Graphical Equilibrium Stage Calculations for Extraction of a Peptide The equilibrium partitioning of a peptide between an aqueous feed phase and an organic solvent extract phase has been found to be nonlinear and can be represented by the following equation: $$ y=1.47 \ln x+3.96 $$ where \(y\) and \(x\) are concentrations of the peptide in the extract and aqueous feed (or raffinate) phases, respectively, in grams per liter. It is desired to extract \(95 \%\) of the peptide from a feed stream having a peptide concentration of \(4.0 \mathrm{~g} / \mathrm{liter}\). For a feed stream at a flow rate of \(5.0 \mathrm{liters} / \mathrm{min}\) and an extract stream at a flow rate of \(3.3 \mathrm{liters} / \mathrm{min}\), graphically estimate how many equilibrium stages will be required for countercurrent flow of the phases. What is the concentration of the peptide in the exit extract stream? As the concentration of the peptide in the raffinate decreases, does the partitioning of the peptide into the extract become more or less favorable?

Short answer

Through the graphical analysis, assuming the necessary accuracy in the count of equilibrium stages, we find that the peptide concentration in the exit extract stream can be derived from the last intersection with the operating line. In the decreasing peptide concentration in the raffinate phase, it is observed that the partitioning of the peptide into the extract phase becomes more favorable.

Step-by-step solution

Decipher the Given Equation

The equation \(y = 1.47 \ln x + 3.96\) has been provided which shows the equilibrium partitioning of the peptide between the extract and raffinate phases with \(y\) referring to the concentration in the extract phase and \(x\) referring to the concentration in the feed or raffinate phase. Both concentrations are in grams per liter.

Determine the Desired Extraction Value

It is desired to extract 95% of the peptide from the feed stream, which has an initial concentration of \(4.0 \, \mathrm{g/liter}\). So, the desired concentration in the raffinate phase can be calculated by (1 - 0.95) * initial concentration = \(0.05 * 4.0 \, \mathrm{g/liter}\) = \(0.2 \, \mathrm{g/liter}\).

Graph the Equilibrium Curve

Develop a graph of the equilibrium curve using the equation \(y = 1.47 \ln x + 3.96\) for a plot of \(y\) (the concentration in the extract phase) against \(x\) (the concentration in the raffinate phase). Mark points on this curve corresponding to the initial concentration (4.0) and the desired concentration (0.2) in the raffinate phase.

Establish the Operating Line

The operating line is drawn from the point corresponding to the initial concentration in the feed phase and bisects the distance between the feed and the extract. It intersects the equilibrium curve at the point where the concentration in both phases is equal.

Determine the Number of Equilibrium Stages

Starting from the point corresponding to the desired raffinate concentration, draw a horizontal line to the operating line, then a vertical line down to the equilibrium curve. Repeating this process of horizontal and vertical lines represents the individual equilibrium stages. Continue this pattern until reaching the point corresponding to the initial feed concentration. Count the number of these 'steps' (vertical lines) which are needed – that is the estimated number of required equilibrium stages.

Find the Concentration in the Exit Extract Stream

The concentration of the peptide in the exit extract is given by the \(y\) value at the intersection of the last horizontal line with the operating line.

Analyze the Partitioning Trend

As the concentration of the peptide in the raffinate decreases, the slope of the equilibrium curve also decreases, implying more peptide is kept in the extract. Thus, the partitioning of the peptide into the extract is becoming more favorable.

Key concepts

Peptide Extraction

Peptide extraction is a critical process in biotechnology and pharmaceutical industries where peptides need to be separated from complex mixtures. The goal is to isolate peptides with high purity and yield. In the exercise, the focus is on extracting a specific peptide from an aqueous feed using an organic solvent. This is commonly executed in a liquid-liquid extraction system, where the peptide transfers from the aqueous phase (also known as the raffinate phase) into the organic solvent (the extract phase).

The exercise sets out to extract 95% of the peptide from a feed with a known concentration, a task which is not only pertinent in laboratory settings but also in large-scale industrial applications. The efficacy of this extraction process is pivotal, as it determines the product quality and the cost-effectiveness of the entire operation. Understanding the equilibrium partitioning behavior of the peptide between the two phases allows for the design of a suitable extraction process.

Equilibrium Partitioning

Equilibrium partitioning is a fundamental concept in the study of extraction processes. It describes the distribution of a substance between two immiscible solvents at equilibrium. This distribution can be depicted mathematically, as shown by the logarithmic relationship in the provided equation where different concentrations of the peptide in the extract phase correspond to the raffinate phase values.

The nonlinear nature of the equation suggests that the efficiency of the extraction varies with the peptide concentration in the raffinate, implying that it may become easier or harder to extract the peptide as the process proceeds. Understanding this behavior is essential for predicting how the substance will separate under various conditions and designing extraction systems that are both efficient and economical.

Countercurrent Flow

Countercurrent flow is a mechanism employed in many separation processes, where two phases flow in opposite directions to each other. In the context of this exercise, the aqueous feed and the organic solvent flow counter to each other, allowing for efficient contact and transfer of the peptide between them. This mechanism enhances the efficiency of the extraction process by continuously providing fresh solvent to the feed, which helps maximize the driving force for mass transfer. It is a widely used technique in industrial processes where high-quality separation is required.

The graphical analysis of equilibrium stages, often depicted through the McCabe-Thiele method as shown in the solution steps, is a practical approach to determine the number of stages required for the desired separation. The intersection points between the equilibrium curve and the operating line represent the changes in concentration for each stage, and the steps constructed graphically mirror the stages in a real countercurrent extraction column.

Extract and Raffinate Phases

In the extraction process, two crucial phases are involved: the extract and the raffinate phases. The extract phase is typically an organic solvent that has a high affinity for the substance being extracted—in this case, the peptide. The raffinate phase is the remaining aqueous phase that initially contains the peptide. After the extraction, this phase has a reduced concentration of the peptide as it has been transferred into the extract phase.

The performance of an extraction process can be evaluated based on the concentrations of the extracted substance in both phases. A successful extraction leads to a high concentration in the extract phase and a low concentration in the raffinate phase. The question in the exercise on how the partitioning changes with decreasing peptide concentration in the raffinate phase assesses the understanding of this balance between the extract and raffinate throughout the extraction process. The process becomes more favorable for extract phase as the concentration in raffinate drops, enabling a better separation.