,

where you set the equilibrium constant . Practically, the different values for correspond to various operating temperatures of the adsorber. The mass ratios are defined by , the ratio of kg acetone to kg charcoal, and , the ratio of kg acetone to kg air.

Combining the material balance and equilibrium equations and rearranging gives the following Riccati equation:

, where , , and .

Here, , since clean charcoal enters the system, and .

The built-in Mathematica command RSolve makes it possible to solve algebraic difference equations similar to this Riccati equation and to estimate the number of theoretical equilibrium stages. For further details, see Analyzing a Rectifying Column Using the Calculus of Finite Differences.

The present Demonstration plots the McCabe–Thiele diagram and displays in red the theoretical number of equilibrium stages found by solving the Riccati equation given above. You can vary the values of the equilibrium constant, .