Edwin F. Meyer De Paul Unlverslty Chicago. IL 60614
On Thermodynamics of Adsorption Using Gas-Solid Chromatography
The gas chromatographic experiment provides both a unique research tool, dealing directly with the infinitely dilute state, and a valuable pedagogic device for illustrating the difficult thermodynamic concepts of standard state (and the role played by concentration units in establishing it), reference state, and activity coefficient in a real and accessible situation. A previous paper in this Journal deals with these matters as they pertain to the solution process in gas-liquid chromatography ( I ) . This article extends that discussion to consider the reference and standard states for the surface phase, or adsorbate, in the study of thermodynamics of adsorption using gas-solid chromatography (gsc). It should he noted a t the outset that the present discussion assumes that the adsorption actually occurring in the gsc column does so in the linear oortion of the isotherm charac-~~~~~~~~~ terizing the particular adsorbate-adsorbent system under studv. The ooerational test for this restriction is the symmetry of tGe elution peak observed; thus, we assume t h a t t h e data under discussion were obtained from chromatograms displaying symmetric peaks. In nddition, we will neglect adsorbate-carrier eas interactions, and assume the g;~*-phase adsorbate behaves ideally. An attempt will he made to include the following topics, with an emphasis on clarity rather than rigor. The concept of two-dimensional pressure and its relationship to surface concentration and to the three-dimensional pressure of the equilibrium vapor ahwe the curtire; the re~ati&lshi~ betwem the limi~ineslooe uf the adsorption isotherm {which can he thought ofas the inverse of t h e ~ e n r y ' sLaw constant for the svstem) and the specific retention volume, VF, observed in the gsc experiment; the choice of standard and reference states for the vapor and the adsorbed species, and the corresponding equations relating VT to AGO, AH",and ASo of adsorption; an example using literature data; and finally the effect of the choice of standard states on the statistical mechanical calculation of entropies of adsorption for comparison with the experimental values. ~~~
~
~
The Two-Dimensional Pressure The standard state of a gas is traditionally (though arhitrarilv) .. chosen to he devoid of intermolecular interactions, a t 1atm pressure and the temperature of interest. The standard state of an adsorbed speiies in the asc experiment is similar, in that adsorbate-adsoibate interactions are considered absent (the interactions of the adsorhed molecules with the surface are of course present in the standard state) and the temperature is likewise whatever value we are interested in. Instead of 1atm pressure, however, we must recognize that the surface state is essentially two-dimensional, and substitute some "surface nressure" for the usual easeous uressure. Furthermore, the'same arbitrariness characterizesthe choice of some oarticular value of surface pressure in the standard state, and there is by no means the universal agreement for this value that exists for the three-dimensional 1atm pressure state. The idea of a two-dimensional, or surface, pressure is discussed in detail bv N. K. Adam in his classical hook (2). It stems from cnrly e;periment* dealing with film-covered watvr surfi~ccs:n which the o u t w r d furre of the lilni against a harrier was measured directly. Whereas in three dimensions, pressure has units of force per unit area (e.g. dyne/cm2), the force exerted by an adsorhed layer in two dimensions is along a harrier of some length, rather than against a piston of some 120 1 Journal of Chemical Education
with
the infinitely law coverage state as reterence
area. so the two-dimensional oressure has units of force oer unit length (e.g., dynelcm). In like manner, the three-dimensional volume of " eas is replaced bv the two-dimensional area of the adsorbed layer. An important consequence of chaneine both auantities is that their product maintains the samcun& as thk P V product; i.e., units of energy. Plots of th_e product of two-dimensional pressure, n, and molar area, A, versus n were observed to behave remarkably like plots of PV versusP for three-dimensional gases; hence the model of the "two-dimensional gas" for the adsorhed layer. The analogy includes an equation of state for the surface layer. rA =n 8 T
(1)
where n, is the number of moles contained therein. As in the three-dimensional case, it is obeyed exactly only in the limit of zero surface pressure hut is approximately true for low values of a.If we define thesurface concentration, u, as moles per unit area, we have u = A-' and a = uRT
(2)
for the ideal two-dimensional gas. Let us consider a conventional adsorption isotherm and its relation to the two-dimensional pressure of the adsorbate. In Figure 1,surface concentration is plotted versus equilibrium pressure of adsorbate in the gas phase over the surface. The Gibbs isotherm, which provides the general relationship between the directly measured quantities, o and p, and the conceptual two-dimensional pressure, n, may he written (3)
In rreneral. the two-dimensional uressure would be evaluated from the expression s = RTJudlnp
(4)
For the purposes of the gsc experiment, however, which is carried out in the linear portion of the isotherm (dashed line
in Fig. 11, this expression is readily shown to reduce to eqn. V : and the Limiting Slope of the Adsorption Isotherm
The relationship between VT and the limiting slope in Figure 1 is readily established. While it is not strictly essential to the present discussion, i t is worth considering, since the connection between the time for adsorbate elution from the column and an equilibrium slope is not immediately obvious to the non-specialist. Let the limiting slope in Figure 1he k = iim
(-1
dm
aP
,
A T
= ,
'
(r
p
(5)
The primed quantities are characteristic of an imaginary state in which the molecular interactions on the surface are the same as a t infinitely low coverage (the reference state for the adsorhed layer). If n, is moles adsorhed, ng is moles of adsorbate in the gas phase and V,,, is the dead space volume in the column, we have
- V-I n8 (6) ~,RTIV,,I ART n, so that k is nronortional to the ratio of moles adsorhed to . moles in the vapor phase a t equilibrium. Consider now the time i t takes for the adsorbate to oass through the column, t ~ Let . t~ he the time required for a non-adsorbed s ~ e c i e (usuallv s air). I t is intuitivelv acceptable to claim that t i is simply t~ divided by the fraction ofsolute molecules in the mobile, or gas phase as they are swept through the column. If a molecule spends only half its time in the mobile phase, for example, it takes twice as long to elute as a molecule which spends all of its time there. Thus =
=
.
p'
and n, - t~ - t~ - t k -
(RI \-,
n, t ta t k is called the "adjusted retention time" by chromatographers; ironically, this time is the essential link with thermodynamics. Substituting into eqn. (6), we have
V-l f k f t, (9) ARTLA ART where we have substituted f , the flow rate of carrier gas, for Vcollt~. The specific retention volume a t temperature T is defined ( 4 ) as
k-
v,'='tkP
(10)
whereg is the mass of adsorbent in the column. (In gas-liquid chromatography, VT is corrected to 0°C by multiplying by 273.15lT to give V;). The final step combines eqns. (9) and (10) to yield
where it must he remembered that this a is characteristic of the infinitelv low nressure eanilihrium. whatever value we choose forp'. This kquation theconnection between the two- and three-dimensional Dressures that allows us to calculate numerical values for standard Gibbs energy and entropy of adsorption. (We will see later that, as long as the infinitely dilute state serves as reference, the standard enthalpy of adsorption is independent of our choice of a for the standard surface state.) Standard State for the Adsorbate; Evaluation of and A 9
AGO, Ano,
We are now in a position to discuss the choice of standard state for the adsorhed molecules. Since we do not live in a two-dimensional world, there is no "natural" choice for the surface pressure akin to the 1 atm three-dimensional standard state. There are two frequently used states which serve to illustrate the arbitrariness of our choice: the Kemhall-Rideal, (K-R), ( 5 )and the deBoer, (dB), ( 6 )standard surface states. Both strive to provide conditions analogous to the 1 atrn three-dimensional gas state. The approach of Kemball and Rideal was to consider the thickness of the adsorhed layer as a third dimension, giving it a volume of sorts. If the arhitrarv thickness is 6. the laver has a volume A6. Using the ideal law, P(A6) = ~ , R T , and substitutina the standard values 1 atm for P and 273.15 K for T, we can calculate a surface concentration 6
(13) 273.15 which mav be considered to exert a two-dimensional nressure equivalent in a sense to the three-dimensional pressure of 1 atm. Using 6 A for 6, we calculate = 2.68 X 10-l2 moleslcm2. Equation (1) yields for the two-dimensional pressure 0.0608 dvneslcm, surorisinalv small compared with the three-dimensional pressure
