The function ALFA[nco,eeta,con,ratio] calculates the "alpha" parameter for a mixture where:
->eeta=packing fraction
->con=list with the molar fractions of each species,e.g, {x_1,x_2,x_3}, 
        x_1 being the corresponding to the species with the largest diameter
->ratio=list with the diameter ratios, e.g., {1,ratio_2,ratio_3} where ratio_n=sigma_n/sigma_1=sigma_n

Off[General::spell];Off[General::spell1]

ALFA[nco_, eeta_, con_, ratio_] := 
ALFA[nco,eeta,con,ratio]=
  Module[{PI, c, sig, rho, zet, rhot, eta, lam, lamp, g, L, A, 
    delA, invdelA, der0, mder0, mH0, H, mder1, mH1, htil, invih, 
    pri, bpr,pto,solu},
     PI = N[Pi]; 
     Do[c[n] = con[[n]], {n, 1, nco}]; 
    Do[sig[n, n] = ratio[[n]], {n, 1, nco}]; 
    rhototal = (6*eeta)/(PI*Sum[c[i]*sig[i, i]^3, {i, 1, nco}]); 
    rho[n_] = rhototal*c[n]; 
    sig[i_, j_] := 1/2*(sig[i, i] + sig[j, j]) /; i != j; 
    zeta[n_, ncomp_] := Sum[rho[i]*sig[i, i]^n, {i, 1, ncomp}]; 
    rhot[ncomp_] := zeta[0, ncomp]; 
    eta[ncomp_] := 1/6*PI*zeta[3, ncomp]; 
    lam[ncomp_] := (2*PI)/(1 - eta[ncomp]); 
    lamp[ncomp_] := (PI^2*zeta[2, ncomp])/(1 - eta[ncomp])^2; 
    g[i_, j_, nc_] := 
     (lam[nc] + (lamp[nc]*sig[i, i]*sig[j, j])/(2*sig[i, j]) + 
        (lamp[nc]^2*((sig[i, i]*sig[j, j])/sig[i, j])^2)/
         (18*lam[nc]))/(2*PI); 
    L[i_, j_, 2, nc_] :=2*PI*alfa*sig[i, j]*g[i, j, nc]; 
    L[i_, j_, 0, nc_] := 
      lam[nc] + lamp[nc]*sig[j, j] + 2*lamp[nc]*alfa - 
       lam[nc]*Sum[rho[k]*sig[k, k]*L[k, j, 2, nc], {k, 1, nc}]; 
    L[i_, j_, 1, nc_] := 
      lam[nc]*sig[i, j] + 1/2*lamp[nc]*sig[i, i]*sig[j, j] + 
       (lam[nc] + lamp[nc]*sig[i, i])*alfa - 
       1/2*lam[nc]*sig[i, i]*
        Sum[rho[k]*sig[k, k]*L[k, j, 2, nc], {k, 1, nc}]; 
    A[i_, j_, n_, nc_] := 
          Simplify[(-1)^n*((L[i, j, 2, nc]*rho[i]*sig[i, i]^(n + 1))/
           (n + 1)! - (L[i, j, 1, nc]*rho[i]*sig[i, i]^(n + 2))/
           (n + 2)! + (L[i, j, 0, nc]*rho[i]*sig[i, i]^(n + 3))/
           (n + 3)!)]; 
   delA[nc_] :=IdentityMatrix[nc] - Table[A[k, j, 0, nc], {k, nc}, {j, nc}]; 
    invdelA[nc_] :=  Inverse[delA[nc]]; 
    der0[i_, j_, nc_] := 
     L[i, j, 2, nc]/(2*PI) + Sum[A[k, j, 2, nc], {k, 1, nc}] - 
      alfa*sig[i, j] + Sum[sig[i, k]*A[k, j, 1, nc], {k, 1, nc}] - 
      1/2*sig[i, j]^2 + Sum[1/2*sig[i, k]^2*A[k, j, 0, nc], 
       {k, 1, nc}]; mder0[nc_] := 
     Table[der0[i, j, nc], {i, nc}, {j, nc}]; 
    mH0[nc_,eet_,co_,rati_] :=mH0[nc,eet,co,rati] = 
     Simplify[mder0[nc] . invdelA[nc]]; 
    H[i_, j_, 0, nc_] := mH0[nc,eeta,con,ratio][[i,j]]; 
    der1[i_, j_, nc_] := 
     Sum[A[k, j, 3, nc], {k, 1, nc}] + 
      Sum[sig[i, k]*A[k, j, 2, nc], {k, 1, nc}] - 
      (1/2*sig[i, j]^2 + H[i, j, 0, nc])*alfa + 
      Sum[(1/2*sig[i, k]^2 + H[i, k, 0, nc])*A[k, j, 1, nc], 
       {k, 1, nc}] - (1/6*sig[i, j]^3 + sig[i, j]*H[i, j, 0, nc]) + 
      Sum[(1/6*sig[i, k]^3 + sig[i, k]*H[i, k, 0, nc])*
        A[k, j, 0, nc], {k, 1, nc}]; 
    mder1[nc_] := Table[der1[i, j, nc], {i, nc}, {j, nc}]; 
    mH1[nc_,eet_,co_,rati_] :=mH1[nc,eet,co,rati] =
     Simplify[mder1[nc] . invdelA[nc]]; 
    H[i_, j_, 1, nc_] := mH1[nc,eeta,con,ratio][[i,j]]; 
    htil[i_, j_, nc_] := -4*PI*rho[i]*H[i, j, 1, nc]; 
    inv1h[nc_,eet_,co_,rati_] :=inv1h[nc,eet,co,rati] = 
      Inverse[IdentityMatrix[nc] + 
        Table[htil[i, j, nc], {i, nc}, {j, nc}]]; 
    pri[i_, nc_] := pri[i, nc] =
     Sum[inv1h[nc,eeta,con,ratio][[j,i]], {j, 1, nc}]; 
    pto[nc_,eet_,co_,rati_] :=pto[nc,eet,co,rati] =
     Together[Sum[rho[i]*pri[i, nc], {i, 1, nc}]]; 
    bpr[nc_] := 
     rhot[nc]/(1 - eta[nc])^2 + 
      (PI*zeta[1, nc]*zeta[2, nc])/(1 - eta[nc])^3 + 
      (PI^2*zeta[2, nc]^3*(9 - 4*eta[nc] + eta[nc]^2))/
       (36*(1 - eta[nc])^4);
    solu = N[Solve[bpr[nco] == pto[nco,eeta,con,ratio], alfa]]; 
    solu = Table[solu[[n,1,2]], {n, 1, Length[solu]}]; 
    First[Sort[Cases[Cases[solu, _Real], x_ /; x > 0]]]]

EXAMPLES

An example for a binary mixture:

ncoa=2;   
eetaa=49/100;
cona={1/2,1/2};
ratioa={1,3/10};

alfa2a=ALFA[ncoa,eetaa,cona,ratioa]

An example for a binary mixture:

nco2b=2;   
eeta2b=49/100;
con2b={1/16,15/16};
ratio2b={1,3/10};

alfa2b=ALFA[nco2b,eeta2b,con2b,ratio2b]

An example for a ternary mixture:

nco3=3;  
eeta3=49/100;
con3={1/102,1/102,100/102};
ratio3={1,3/10,1/10};

ClearAll[ALFA]

alfa3=ALFA[nco3,eeta3,con3,ratio3]