Critical-point universality in adsorption: The effect of charcoal on a mixture of isobutyric acid and water near the consolute point

The mixture of isobutyric acid and water has a consolute point at a temperature of 25.75 degrees C and mole fraction 0.1148 isobutyric acid. When charcoal is added to this mixture, the concentration of isobutyric acid is reduced by adsorption. We have measured the action of charcoal on solutions of isobutyric acid and water as a function of isobutyric acid mole fraction at temperatures of 25.85 and 32.50 degrees C. At the higher temperature, the specific adsorption density (y(2)(alpha)/m) satisfies the Freundlich equation (y(2)(alpha)/m) = KX21/n, where y(2)(alpha) is the mass of isobutyric acid adsorbed, m is the mass of charcoal, X-2 is the equilibrium mole fraction of isobutyric acid, n is the Freundlich index, and K = K(T) is an amplitude that depends upon the temperature T. At 25.85 degrees C, a critical endpoint is located at an isobutyric acid mole fraction X-2(ce) = 0.09. When compared with the Freundlich equation at this temperature, a plot of the specific adsorption density as a function of X-2 in the vicinity of the critical-endpoint composition assumes a shape which is reminiscent of the derivative of a Dirac delta function. Using critical-point scaling theory, we show that this divergent pattern is consistent with the principle of critical point universality.