Table 16.1-1 Notation for concentrations in Binary Systems
> restart;
> rho:=array(1...2); One of many vectors of two elements: the mass density of each species
> rhotot:=sum(rho[i],i=1..2); Definition A for the total mass density
Here are the vectors of mass fractions, molecular weights and molar densities
> omega:=array(1...2); MW:=array(1...2);c:=array(1...2);
> for k from 1 to 2 do rho[k]:=c[k]*MW[k] od; Definition B: mass density is the molar density times the molecular weight
> for k from 1 to 2 do omega[k]:=rho[k]/rhotot od; Definition C of mass fractions
> ctot:=sum(c[i],i=1..2); The total molar density
> x:=array(1...2); The mol fraction vector
> for k from 1 to 2 do x[k]:=c[k]/ctot od; Definition F of mol fractions
> M:=rhotot/ctot; The mean molecular weight of the mixture
> sum(x[i],i=1...2); Showing H: the sum of the mol frations is one
> simplify(%);
> sum(x[i]*MW[i],i=1..2)-M; Showing J: another way to get M
> simplify(%);
> sum(omega[i],i=1..2); Showinh I: the mass fractions also add up to 1
> simplify(%);
> x[1]-(omega[1]/MW[1])/sum(omega[i]/MW[i],i=1...2); Showing L: one relation between mass fractions and molfractions
> simplify(%);
> x[1]; x[1] depends on c[1] and c[2]
> omega[1]; So does the mol fraction as long as the mol weights are fixed
> dx:=array(1...2); This will be used to hold differentials of x
> dx[1]:=diff(x[1],c[1]); Holding c[2] constant
> domeg:=array(1...2); To hold differentials of the mass fractions
> domeg[1]:=diff(omega[1],c[1]); Holding c[2] constant
> dxodomeg:=simplify(dx[1]/domeg[1]); Should give the ratio of dx/domega and can be used to verify N
> %-1/(MW[1]*MW[2]*(sum(omega[i]/MW[i],i=1..2)^2));
> simplify(%); This proves N
>