158
W. B. INNES AND H. H. ROIVLEY
RELATIOSSHIPS BETKEEX T H E ADSORPTIOS ISOTHERM A S D T H E SPREADIXG FORCE
w. B. I”ES AND H. H. ROWLEI’ Division of Physical Chemistry, State University of Iowa, Iowa City, Iowa Received November 19, lQS9
Attempts to apply the Gibbs equation to liquid-vapor interfaces have been made by Micheli ( 5 ) and Cassel and Formstecher (3). In the case of solid-vapor systems, Palmer (6) and Armbruster and Austin (1) have utilized the following equation to calculate the spreading force, F , from adsorption data:
where f = dimensional constant for adjusting units, +a = potential of the surface at zero adsorption, s = slope log versus a curve, a = amount adsorbed, and I = constant of integration.
+
Equation 1 is derived by assuming the Gibbs equation, and a linear relationship between the log of the adsorption potential and the amount adsorbed. The latter relationship is approximately true for the isothermal data considered by Palmer, but certainly cannot be considered even a good approximation for most cases. Since the Gibbs adsorption equation still lacks adequate experimental verification and since the interrelationships between equilibrium pressure, the amount adsorbed, and the spreading force seem to need further elucidation, an attempt is made here to approach the problem from a slightly different view point. The process of the formation of a film on a surface can be made to take place by t x o methods: by an adsorption process and by spreading from one part of the surface to another part of the surface. Each of these processes can he carried on reversibly, and it should be possible to start with vapor at the same pressure and to end up with the same film in both cases. Therefore the work done in the two processes can be equated. In doing so, it should be kept in mind that the film finally obtained has the same area and the same number of moles whether formed by adsorption or by spreading, and that both processes are carried on isothermally. In carrying out the reversible spreading process the system pictured in figure 1 will be imagined. The spreading film is in equilibrium with its vapor a t the pressure p l . As the film spreads on the surface, more vapor
ADSORPTIOS ISOTHERM AND SPREADING FORCE
159
is adsorbed so as to keep the surface concentration of the film a t a constant value and therefore maintain a constant spreading force. The vapor phase is connected to a large reservoir of gas or vapor having a pressure of p l . The volume above the film is assumed so small that the pressure \.ohme work can be neglected in comparison with the work of spreading. The amount of reversible work done on the barrier by the film as it spreads so as to cover a new surface of area A will then be F.4, where F is the spreading force. T o form the same film by a reversible adsorption process, the starting point will again be vapor at a pressure of p l from a large reservoir. The set-up in figure 2 will be utilized. Again it will be assumed that the volume of the space above the adsorbent is so small that the magnitude of the pressure volume work of expansion can be neglected in comparison with the work of adsorption.' To carry out the reversible adsorption the following procedure could be utilized: the valve a would be opened, admitting rapor to the piston b ; the gas would then be allowed to expand,
FIG. 1. Formation of surface film by method 1
FIG. 2. Formation of surface film by method 2
doing work on the piston until the vapor was a t a pressure infinitesimally greater than the equilibrium pressure above the adsorbent; then valve c would be opened and this vapor allowed t o be adsorbed a t this pressure. The cycle would then be repeated until the reservoir pressure was equal to the equilibrium pressure above the adsorbent. If the piston is supposed to be infinitesimally small, the adsorption process would approach reversible conditions and the work done on the piston for the complete process would be:?
1 If the pressure volume work had not been neglected in the two cases, these work terms would have cancelled out when the work done in the two processes was equated. 2 Fugacities should be employed for exact results.
160
W . B. INNES AND H. H . ROWLEP
where p' = pressure on the piston, p = equilibrium pressure, p1 = reservoir pressure (limiting equilibrium pressure), R = gas constant, T = absolute temperature, n = moles adsorbed a t a pressure p , and n1 = moles adsorbed a t a pressure pl. Equating the work done in the formation of the film by both proresses: the following equation is obtained?
FA = Jn-"RTlnEdn
P
n-0
Equation 2 can be used directly to calculate the spreading force, if data are available relating the equilibrium pressure to the amount ad-
F
4,
FXG.3. Ethyl iodide on unreduced iron a t 20°C. (from the data of Armbruster and
Austin (1)). A, graphical integration of adsorption data;- - -, two-dimensional perfect gas law; 0, equation 1. F in dynes per centimeter and A , in square lingstroms per molecule.
sorbed, and the area of the surface is known. The integration can be carried out graphically or mathematically if p(n) ran be represented by a mathematical function, such as the Langmuir equation. To illustrate
* Since p 0 when n = 0, i t is necessary to show that this integral exists. This can be done if i t is assumed aa an experimental fact that the slope dpldn as n + 0 is finite. If this assumption is made, i t follows that, in the immediate neighborhood of zero adsorption, Henry's Law is obeyed (Lewis and Randall: Thermodynamics, p. 232. McGraw-Hill Book Company, Inc., New York (1923)). Hence p = kn. This is substituted in equation 2 and the integration carried out between the limits t h a t the law applies. 5
-I
ni
RT In PI - dn = n1RT In PI kn
RT In kn dn
= nr RT In PI
- R T [n In kn - kn];'
= nt RT In PI
- R T [nl In knl - knll
The expression above is finite and hence the integral in question haa a real exiatence if the postulate is true.
ADSORPTION ISOTHERM AND SPREADING FORCE
161
the former method the data of Armbruster and Austin (1) for the adsorption of ethyl iodide on iron have been used to calculate the spreading force as a function of the area per molecule. These results are plotted in figure 3 and can he compared with those of Armbrustcr and Austin, who used equation 1. A plot according to the two-dimensional perfect gas law a t the same temperature is included for comparison. However, a more convenient way of handling this equation can be utilized. Differentiating both sides of the equation with respect t o nl, with T and A constant: dF LL R T l n E d n = A dn1 P dnl
i"
These relationships make it possible to calculate the adsorption equation if the two-dimensional equation of state is k n o w , or, conversely, the equation of state if the adsorption equation is known. Equation 4 is seen to be the Gibhs adsorption equation. To illustrate the application of these equations, two simple cases will be considered. If it is assumed that the film obeys the two-dimensional gas law, then :
FA = niRT A dF - = RT dnl
Substituting in equation 3:
p2
nl = / d l n p l
In n1 = In-pl
+ In K
n1 = Kpl
K = constant of integration
(5)
162
W. B. INNES AND H. H. ROWLEY
I t is concluded, therefore, that if a film Has a linear adsorption isotherm, it will obey the two-dimensional gas law. As the initial part of most adsorption isotherms is a straight line, it would appear that this is true of most adsorption films of low concentration. To illustrate further the application of these relationships, the twodimensional equation of state corresponding to the Langmuir adsorption equation will be determined. According to the Langmuir equation:
where
+ = moles adsorbed when the surface is completely covered, e=
- = fraction of surface covered,
n1
+
k = a constant a t constant temperature (the adsorption equilibrium constant). Substituting in equation 3:
RT
=
- + 1 n ( 4 -nl) + C
To evaluate C, the limiting case when n1 = 0 and F = 0 is considered:
C
= ++In+
Then :
Expanding by series :
Equation 10 is plotted in figure 4 so as to bring out the deviation from the perfect gas law as a function of the fraction of the surface covered.
ADSORPTION ISOTHERM AND SPREADING FORCE
163
To obtain the equation of state in terms of the two variables F and An, as is customary: From equation 8 FA RT=-41n-
4
- nl 4J
Let A/nl = A , = the area occupied per mole adsorbed when n1 moles are adsorbed.
e
FIG.4. Plot of equation 9
A / + = B = the area occupied per mo:e when the surface is completely covered.
F
-1 B
An-B An
= - In ___
Expanding by series :
Equation 12 is plotted in figure 5 , along with a corresponding plot for a perfect two-dimensional gas. This plot can be compared with the analo-
164
W. B. INNES AND H. H. ROWLEY
gous pressure-volume isothermals obtained for three-dimensional systems. The form of the curve is seen to correspond to that obtained for a real gas considerably above the critical temperature. Since the plot obtained, assuming the Langmuir equation, is much as would have been expected for a real two-dimensional gas, further light is thrown on the two-dimensional state of adsorbed substances which obey a Langmuir type of iso-
B
Fro. 5. Plot of equation 11; two-dimensional perfect gas law
FIQ.6. Nitrogen on iron-aluminum oxide catalyst 954 (50.4 g.) at -195.8'C. (from data of Brunauer, Emmett, and Teller (2)). Area assumed to be 13.2 square meters per gram. Values determined by the use of equation 2. F in dynes per square centimeter and A . in square Angstroms per molecule.
therm. This evidence, indicating that most adsorbed films are gaseous, serves further to corroborate the calculations of Devonshire (4), who estimates that the two-dimensional critical temperature is approximately one-half of the corresponding three-dimensional critical temperature when both are expressed in degrees Absolute. An interesting point may be noted in the foregoing derivation,-that is, the equation of state obtained (equation 11) is independent of the con-
ADSORPTION ISOTHERM AND SPREADING FORCE
165
stant k (equation 7). The latter constant is a measure of the attraction of the adsorbent for the adsorbate, as it is the equilibrium pressure when the surface is half covered. Hence the conclusion is drawn that the twodimensional equation of state of the adsorbate is independent of attractive forces between the adsorbent and adsorbate in cases where the Langmuir equation applies. A change in the two-dimensional state would be indicated by a constant value of F with a varying A,. This would correspond in the adsorption isotherm to a constant value of the equilibrium pressure when the amount adsorbed varied. An example of such a case is the adsorption of water on charcoal. The excellent agreement of adsorption data with the Emmett, Brunauer, and Teller (2) equation has lent credence to the hypothesis of multilayer adsorption when the pressure is near the saturated vapor pressure. The spreading force in such a case would be partially due to layers other than the first. To illustrate the type of force-area curve obtained when the data fit this type of adsorption isotherm, the data of Emmett and Brunauer and Teller (2) have been utilized and a plot is given in figure 6. Equation 2 was-employed and the integration carried out graphically. The dotted curve represents the behavior of an ideal two-dimensional gas a t the same temperature. SUMMARY
A thermodynamic relationship between spreading force and the data of the adsorption isotherm has been derived by considering two reversible processes whereby the surface film could be formed. The application of this relationship has been compared with the method of Palmer for calculating spreading force from adsorption data. A two-dimensional equation of state corresponding to the Langmuir adsorption isotherm has been derived which is similar to that which would be expected for a real twodimensional gas. A perfect two-dimensional gas was found to correspond to a linear adsorption isotherm. REFERENCES
(1) ARMBRUBTER, M. H.,AND AUSTIN,J. B . : J. Am. Chem. SOC. 61, 1117 (1939). (2) BRUNAUER, S.,EMMETT,P. H., AND TELLER,E . : J. Am. Chem. Sor. 60, 309 (1938). (3) CABSEL,H., AND FORMBTECRER, M.: Kolloid-2. 61, 18 (1932). (4) DEVONSEXIR~, A. F.: Proc.Roy. SOC.(London) 183, 132 (1937). (5) MICHELI,L. I. A,: Phil. Mag. 3, 895 (1927). (6)PALMER, W.G.:Proc.Roy. SOC.(London) A160, 254 (1937).
