(%i1) kill(all)$
 find:general equations for strong acids H(n)A,n=1,2
 peter.vlasschaert@gmail.com, 25/09/2016
 calculus ,infinity limit of rational expression.
(%i1) p1:(5*x^3+x+9)/(3*x^3+5);
(p1)(5*x^3+x+9)/(3*x^3+5)
(%i2) limit(p1, x, infinity);
(%o2) 5/3
 general equation for (di)protic acid : (n=2,H(n)A)
(%i3) p2:h^4+k1*h^3+(k1*k2-kw-k1*ca)*h^2-h*(k1*kw+2*k1*k2*ca)-k1*k2*kw=0;
(p2)-h*(k1*kw+2*ca*k1*k2)-k1*k2*kw+h^2*(-kw+k1*k2-ca*k1)+h^3*k1+h^4=0
 monoprotic strong acid (n=1,H(n)A)
[H(+)] = h ,ka(1)=k1 = infinity , ka(2)=k2= 0
(%i4) p3:ev(lhs(p2),k2=0);
(p3)-h*k1*kw+h^2*(-kw-ca*k1)+h^3*k1+h^4
(%i5) p4:limit(lhs(p3/k1), k1, infinity);
(p4)-h*kw+h^3-ca*h^2
(%i6) p5:factor(lhs(p4))=0;
(p5)-h*(kw-h^2+ca*h)=0
(%i7) p6:part(p5,1,1,2);
(p6)kw-h^2+ca*h
 formula : monoprotic strong acid,HA
(%i8) p7:(-1)*lhs(p6)=0;
(p7)-kw+h^2-ca*h=0
 first approximation : monoprotic strong acid HA
(%i9) ap:ev(lhs(p7),kw=0);
(ap)h^2-ca*h
(%i10) ap1:factor(ap);
(ap1)h*(h-ca)
(%i11) ap2:part(ap1,2)=0;
(ap2)h-ca=0
(%i12) ap3:part(solve(ap2,h),1);
(ap3)h=ca
 pH = - log(ca)
 diprotic strong acid (n=2,H(n)A)
[H(+)] = h ,ka(1) = k1 = infinity , ka(2)=k2 = infinity,k1*k2 = infinity
 replace :  k1*k2 by k1*k2 = k, k -> infinity
(%i13) p8:ratsubst(k, k1*k2, lhs(p2))=0;
(p8)(-h*k1-k-h^2)*kw+(h^3-ca*h^2)*k1+h^2*k-2*ca*h*k+h^4=0
 first step : find coef,h^4,h^3...
(%i18) n1:ev(lhs(p2),h=0);
n2:coeff(lhs(p2),h^1);
n3:coeff(lhs(p2),h^2);
n4:coeff(lhs(p2),h^3);
n5:coeff(lhs(p2),h^4);
(n1)-k1*k2*kw
(n2)-k1*kw-2*ca*k1*k2
(n3)-kw+k1*k2-ca*k1
(n4)k1
(n5)1
(%i19) p9:n5*h^4+n4*h^3+n3*h^2+n2*h+n1=0;
(p9)h*(-k1*kw-2*ca*k1*k2)-k1*k2*kw+h^2*(-kw+k1*k2-ca*k1)+h^3*k1+h^4=0
 replace : coef's => k1*k2 = k divide every coef of h^i,i=0..4 through k
          n._k  -> * (k1*k2)  
          n._k1 -> *  k1     -> limit k2-> infinity
          n._k2 -> *  k2     -> limit k1-> infinity
          n._c  -> *  1      -> limit k -> infinity
. = 1,2,3,4,5
(%i24) p10:ratsubst(k,k1*k2,n1)$
n1_k:ratsubst(k1*k2,k,(coeff(p10,k)/(k)))*(k1*k2);
n1_k1:limit(ratsubst(k1*k2,k,(coeff(p10,k1)/(k)))*k1,k2,infinity);
n1_k2:limit(ratsubst(k1*k2,k,(coeff(p10,k2)/(k)))*k2,k1,infinity);
n1_c:limit(ev(p10,k1=0,k2=0,k=0)/(k),k,infinity);
(n1_k)-kw
(n1_k1)0
(n1_k2)0
(n1_c)0
(%i29) p11:ratsubst(k,k1*k2,n2)$
n2_k:ratsubst(k1*k2,k,(coeff(p11,k)/(k)))*(k1*k2);
n2_k1:limit(ratsubst(k1*k2,k,(coeff(p11,k1)/(k)))*k1,k2,infinity);
n2_k2:limit(ratsubst(k1*k2,k,(coeff(p11,k2)/(k)))*k2,k1,infinity);
n2_c:limit(ev(p11,k1=0,k2=0,k=0)/(k),k,infinity);
(n2_k)-2*ca
(n2_k1)0
(n2_k2)0
(n2_c)0
(%i34) p12:ratsubst(k,k1*k2,n3)$
n3_k:ratsubst(k1*k2,k,(coeff(p12,k)/(k)))*(k1*k2);
n3_k1:limit(ratsubst(k1*k2,k,(coeff(p12,k1)/(k)))*k1,k2,infinity);
n3_k2:limit(ratsubst(k1*k2,k,(coeff(p12,k2)/(k)))*k2,k1,infinity);
n3_c:limit(ev(p12,k1=0,k2=0,k=0)/(k),k,infinity);
(n3_k)1
(n3_k1)0
(n3_k2)0
(n3_c)0
(%i39) p13:ratsubst(k,k1*k2,n4)$
n4_k:ratsubst(k1*k2,k,(coeff(p13,k)/(k)))*(k1*k2);
n4_k1:limit(ratsubst(k1*k2,k,(coeff(p13,k1)/(k)))*k1,k2,infinity);
n4_k2:limit(ratsubst(k1*k2,k,(coeff(p13,k2)/(k)))*k2,k1,infinity);
n4_c:limit(ev(p13,k1=0,k2=0,k=0)/(k),k,infinity);
(n4_k)0
(n4_k1)0
(n4_k2)0
(n4_c)0
(%i44) p14:ratsubst(k,k1*k2,n5)$
n5_k:ratsubst(k1*k2,k,(coeff(p14,k)/(k)))*(k1*k2);
n5_k1:limit(ratsubst(k1*k2,k,(coeff(p14,k1)/(k)))*k1,k2,infinity);
n5_k2:limit(ratsubst(k1*k2,k,(coeff(p14,k2)/(k)))*k2,k1,infinity);
n5_c:limit(ev(p14,k1=0,k2=0,k=0)/(k),k,infinity);
(n5_k)0
(n5_k1)0
(n5_k2)0
(n5_c)0
 formula : diprotic strong acid,H2A
(%i50) t1:n1_k+n1_k1+n1_k2+n1_c$
t2:n2_k+n2_k1+n2_k2+n2_c$
t3:n3_k+n3_k1+n3_k2+n3_c$
t4:n4_k+n4_k1+n4_k2+n4_c$
t5:n5_k+n5_k1+n5_k2+n5_c$
p15:t5*h^4+t4*h^3+t3*h^2+t2*h+t1=0;
(p15)-kw+h^2-2*ca*h=0
 first approximation : diprotic strong acid H2A
(%i51) aq:ev(lhs(p15),kw=0)=0;
(aq)h^2-2*ca*h=0
(%i52) aq1:factor(lhs(aq))=0;
(aq1)h*(h-2*ca)=0
(%i53) aq2:part(lhs(aq1),2)=0;
(aq2)h-2*ca=0
(%i54) aq3:solve(aq2,h);
(aq3)[h=2*ca]
(%i55) aq4:part(aq3,1);
(aq4)h=2*ca
 pH = - log(2*ca)