Gina A. Tran

Ceng 402 Project for Dr. Davis

Spring 1999

Table 16.1-1, page 498

Notation for Concentrations in Binary Systems

> restart;

> rho:=array(1..2);

Setting up a vector for the mass density of each species.

> rho_total:=sum(rho[i],i=1..2);

Relation A , the mass density of solution (g/cm cubed).

> omega:=array(1..2);M:=array(1..2);c:=array(1..2);

Setting up vectors for mass fractions (unitless), molecular weights (g/mol), and molar concentrations (mol/cm cubed) for each species.

> for k from 1 to 2 do rho[k]:=c[k]*M[k] od;

Relation B , the mass concentration of each species (g/cm cubed of solution).

> for k from 1 to 2 do omega[k]:=rho[k]/rho_total od;

Relation C , the mass fraction of each species (unitless).

> c_total:=sum(c[i],i=1..2);

Relation D , the molar density of solution (mol/cm cubed).

> for k from 1 to 2 do c[k]:=rho[k]/M[k] od;

Relation E , molar concentration of each species (mol/cm cubed).

> x:=array(1..2);

Setting up a vector of the mole fractions of each species.

> for k from 1 to 2 do x[k]:=c[k]/c_total od;

Relation F , the mole fraction of each species (unitless).

> mean_M:=rho_total/c_total;

Relation G , the number-mean molecular weight of mixture (mol/g).

> relation_H:=sum(x[i],i=1..2);

> simplify(relation_H);

Relation H , summing the mole fractions of each species equals one.

> relation_I:=sum(omega[i],i=1..2);

> simplify(relation_I);

Relation I , summing the mass fractions of each species equals one.

> relation_J:=sum(x[i]*M[i],i=1..2);

> simplify(relation_J-mean_M);

Relation J , the mass fraction multiplied by the molecular weights of each respective species equals the mean molecular weight of the solution.

> relation_K:=sum(omega[i]/M[i],i=1..2);

> simplify(relation_K-1/mean_M);

Relation K , summing the mass fractions of each species divided by the respective molecular weight is equal to one divided by the total molecular weight.

> book_L:=(omega[1]/M[1])/sum(omega[i]/M[i],i=1..2);

> relation_L:=x[1]-book_L;

> simplify(relation_L);

Relation L is true because the mole fraction for each species is equivalent to the book's equation for L.

> relation_M:=(x[1]*M[1])/(x[1]*M[1]+x[2]*M[2]);

> simplify(omega[1]-relation_M);

Relation M , showing that the mass fraction is equal to the mole fraction multiplied by the molecular weight, then divided by the number-mean molecular weight of the mixture.

> x[1];

Showing the mole fraction for first species again. Note that x[1] depends on c[1] and c[2].

> omega[1];

Mass fraction for first species.

> d_x:=array(1..2);

Setting up a vector for the differential equations of x (mole fraction) with respect to omega (mass fraction).

> d_x[1]:=diff(x[1],c[1]);

Keeping c[2] constant.

> d_omega:=array(1..2);

Setting up a vector for the differential equations of omega (mass fraction) with respect to x (mole fraction).

> d_omega[1]:=diff(omega[1],c[1]);

Again, keeping c[2] constant.

> relation_N:=d_omega[1]/(M[1]*M[2]*(sum(omega[i]/M[i],i=1..2)^2));

Starting the proof of Relation N.

> simplify(d_x[1]-relation_N);

Relation N is true when compared to the differential equation of x (mole fraction).

> relation_O:=(d_x[1]*M[1]*M[2])/((x[1]*M[1]+x[2]*M[2])^2);

Starting the proof of Relation O.

> simplify(d_omega[1]-relation_O);

Relation O is true when compared to the differential equation of omega (mass fraction).