A DERIVATION qj' LANGMUIR'S ADSORPTION ISOTHERM PAUL G. BIRD Iowa State College, Ames, Iowa
I
RVING Langmuirl has submitted the equation:
P
I
P
- = z f z ?I
for the simple adsorption of a gas on a surface, where P is the pressure, q the amount of gas adsorbed by a unit of surface, and a and b are constants. The same expression may be obtained by a different development based on the adsorption of an electrolyte in solution. Let it be assumed that a t equilibrium the tendency of the ions to adsorb, or the adsorption force, is equal to the tendency of the ions to desorb, or the desorptiou force. At the interface, the electrolyte concentration will be greater than in the body of the solution. For this reason both the adsorbed cation and anion will tend to ditfuse back into the body of the solution. This tendency to diffuse away from the surface constitutes the desorption force. The adsorption force may be assumed to be proportional to the concentration of the cation in .the body of the solution, the available or unoccupied spaces on the adsorbing surface, and a constant (K) depending upon the electrolyte and the surface. If S is the saturation value of a given surface, and X the number of spaces occupied, S minus X is the number of unoccupied spaces. The adsorption force then becomes equal to KC(S - X), where C is the concentration of the cation in the body of the solution. 1 LANGMUIR, I., I.Am. Chm. Soc., 40, 1361 (1918).
-
The tendency of the cation to diffuse back into the solution can be given as V I ( X A C), and the desorption force of the anion as Vt(XA - C). A is the conversion factor which permits the product X A to be expressed in the same terms as C. The tenns VI and Vx represent the diffusion pressures across the interface of the cation and anion per unit of concentration gradient. The term ( X A - C) represents the concentration gradient between the surface of the solid adsorbent and the body of the solution. The expression obtained upon equating the adsorption force to the sum of the desorption forces is:
-
KC(S - X ) = V l ( X A - C )
+ V 2 ( X A- C )
(1)
Expanding and collecting terms,
-
KCS - KCX V t X A - VIC f V X A - VsC -KCX = XA( VI ' I Vz) C( Vr Vs K S )
-
Dividing by X and transposing, C(Vl VZ K S ) / X = KC A(Vl
-+
+
+
+ +
+ V S )or
(2) (3)
(4)
+ Vs + K S ) + A(V, + V M Vt + Vr + K S ) For a given set of conditions let K / A ( V l + VZ) = the constant a, and (Vl + V S f K S ) / K = the conC/X
KC/( VI
stant b. By substitution the equation becomes, C/X = C/b
+l/nb
(5)
This is the same equation as the one given by Langmuir for the adsorption of a gas.
