Protein Adsorption Isotherms — Identify the appropriate correlation: Given Ymax (maximum adsorbed mass per mass of adsorbent), X (solute mass fraction in the diluent phase on a solute-free basis), KL (constant), and Y (equilibrium adsorbed mass per mass of adsorbent), which expression is commonly used?

Introduction:Adsorption of proteins onto solids is often described using a Langmuir-type isotherm that assumes a finite number of equivalent binding sites and monolayer coverage. Recognizing the correct algebraic form is essential for parameter estimation and scale-up of capture steps.

Given Data / Assumptions:

• Y is the mass of solute adsorbed per mass of adsorbent at equilibrium.
• Ymax is the saturation capacity (monolayer limit).
• X is the solute mass fraction in the liquid phase (solute-free basis).
• KL is an empirical constant linked to affinity.

Concept / Approach:

The classic Langmuir model in concentration terms is q = qmax * C / (KL + C). Replacing concentration with a proportional composition variable X yields Y = Ymax * X / (KL + X). This captures initial linear uptake at low X and asymptotic saturation as X increases, consistent with site-limited adsorption behavior often observed with proteins on hydrophobic or ion-exchange surfaces under noncompetitive conditions.

Step-by-Step Solution:

Verification / Alternative check:

Plotting 1/Y versus 1/X (Lineweaver–Burk linearization) yields a straight line if Langmuir holds, enabling estimates of Ymax and KL.

Why Other Options Are Wrong:

B omits X in the numerator and does not approach Ymax correctly; C lacks saturation behavior; D has incorrect subtraction form; E is not a standard isotherm representation and fails limiting checks.

Common Pitfalls:

Misusing X units or mixing mass fraction with concentration without appropriate conversion; ensure consistency when fitting data.

Final Answer:

Y = Ymax * X / (KL + X)