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The Clapeyron equation, derived in Sec 64 for the latent heat of phase transition of pure chemical species, is also applicable to pure-gas adsorption equilibrium Here, however, the two-phase equilibrium pressure depends not only on temperature, but on surface coverage or the amount adsorbed Thus the analogous equation for adsorption is written

where subscript n signifies that the derivative is taken at constant amount adsorbed Superscript av denotes a property change of desorption, ie, the difference between the vapor-phase and the adsorbed-phase property The quantity AH"" H U- H a is defined as the isosteric heat ofadsorption, and is usually a positive quantity28 The heat of adsorption is a useful indication of the strength of the forces binding adsorbed molecules to the surface of the adsorbent, and its magnitude can therefore often be used to distinguish between physical adsorption and chemisorption The dependence of heats of adsorption on surface coverage has its basis in the energetic heterogeneity of most solid surfaces The first sites on a surface to be occupied are those which attract adsorbate molecules most strongly and with the greatest release of energy Thus the heat of adsorption decreases with surface coverage Once all sites are occupied and multilayer adsorption begins, the dominant forces become those between adsorbate molecules, and for subcritical species the decreasing heat of adsorption approaches the heat of vaporization Assumed in the derivation of the Langmuir isotherm is the energetic equivalence of all adsorption sites, implying that the heat of adsorption is independent of surface coverage This explains in part the inability of the Langmuir isotherm to provide a close fit to most experimental data over a wide range of surface coverage The Freundlich isotherm, Eq (1411 I), implies a logarithmic decrease in the heat of adsorption with surface coverage As in the development of the Clausius/Clapeyron equation (Example 65), if for low pressures one assumes that the gas phase is ideal and that the adsorbate is of negligible volume

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W H AT I T TA K E S T O B E # 1

An unsaturated solution has less than the maximum amount of solute dissolved Sometimes, if the temperature, purity of the solute and solvent, and other factors are just right, you might be able to dissolve more than the maximum amount of solute, resulting in a supersaturated solution Supersaturated solutions are unstable, and sooner or later separation of the excess solute will occur, until a saturated solution and separated solute remain The formation of a solution depends on many factors, such as the nature of the solvent, the nature of the solute, the temperature, and the pressure Some of these factors were addressed in the Reactions and Periodicity chapter In general, the solubility of a solid or liquid will increase with temperature and be unaffected by pressure changes The solubility of a gas will decrease with increasing temperature and will increase with increasing partial pressure of the gas (Henry s law)

2 7 ~ Smith, Chemical Kinetics,3d ed, sec 8-1, McGraw-Hill, New York, 1981 M "other heats of adsorption, defined differently, are also in use However, the isosteric heat is the most common, and is the one needed for energy balances on adsorption columns

compared with the gas-phase volume, Eq (14112) becomes:

Application of this equation requires the measurement of isotherms, such as the one at 32315 K (50 C) in Fig 1423, at several temperatures Cross plotting yields sets of P vs T relations at constant n, from which values for the partial derivative of Eq (14113) can be obtained For chemisorption, AH aU values usually range from 60 to 170 kJ mol-' For physical adsorption, they are smaller For example, measured values at very low coverage for the physical adsorption of nitrogen and n-butane on 5A zeolite are 180 and 431 k mol-l, respectively29 J

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