,

where , , , is the reflux ratio that measures the amount of liquid returned to the column, and is the specified distillate composition leaving the condenser. This nonlinear Ricatti equation can be solved using Mathematica's RSolve function. The solution then has two unknowns: and an arbitrary constant , which can be determined by specifying at and at . Hence the solutions of the Riccati equation for values of in the range are bounded by mole fractions that lie in the range .

Integer values of in the range define the liquid composition that leaves an equilibrium stage such that the vapor leaving that stage has composition . The largest integer value in the interval determines the number of theoretical equilibrium stages () for a set value of (i.e., mole fraction at the bottom of the rectifying section).

This Demonstration plots the McCabe and Thiele diagram and displays (in red) found by solving the Riccati equation. You can change the values of the constant-relative volatility , the reflux ratio , the distillate mole fraction , and the mole fraction at the bottom of the rectifying section . The blue line in the plot defines the operating line for the column, the requirement that mass is conserved over an equilibrium stage.

Similar treatment can be performed for a stripping column.