## An ideal system, two components [15]

In the last equation, the macroscopic contribution P1 and the correction for the limited volume of the system (size contribution) ΔP1 are distinguished. For macrosystems with N1, NV → ∞, the size contribution is zero. An analogous equation for the second partial difference symmetric derivative with respect to molecules of the second kind can be written in the form

ln ( , , ) 1 ln ( , 1, ) ( , 1, ) , 2 2 2

P N N M P N N M P PN N

PN M δ∆ = + − − ∆

= +

1 ( ln / , 1)ln 2 ( )

P N P N N N

N Nβµ δ + +

+ =

= (32.2)

Equation (32.2) is written in a general form – it refers to any ideal system containing an arbitrary number of components (s-1). If the set of numbers of molecules of such a mixture is denoted by {Ni}, where 1 ≤ i ≤ s-1, and the number of free sites 11

− == − ∑ ,

then we can express in general form the conditions on the maximum

summand in the sum for ln ({ }, ): 0ii i

i P N M

N P dPΞ = + =∆

∆ , whence

follows

* 1 (ln / – 1)( ) ln . 2 ( 1)

N NN N N N

βµ + +

= (32.3)

The system of equations (32.3) describes the partial isotherms of the adsorption of a multicomponent mixture on a uniform surface that

is limited in area. In the particular case of adsorption of a singlecomponent substance, we pass to the equation (31.3). The solution of the system (32.3) determines the set of the most probable values of the numbers of the adsorbed molecules of the mixture {Ni

*}. This system is nonlinear with respect to the relationship between the given values of the chemical potentials of the mixture components {μi} and the numbers of adsorbed {Ni

*} molecules. With an increase in the surface area, the value of ΔPi decreases, and for macroscopic systems we have the well-known expressions βµ*i = ln(Ni /NV) (Langmuir partial isotherms).