Page 348 - Thermodynamics of Biochemical Reactions

348 Mathematica Solutions to Problems

8.1 (a) Calculate the the standard transformed Gibbs eneries for the Reaction C02tot = C02(g) + H20 at 283.15 K, 298.15
K, and 313.15 K at pHs 5, 6, 7, 8, and 9 at ionic strength 0.25 M. (b) Plot the Henry's law constants versus pH from pH 2 to
pH 12.

(a) Calculation of the standard transformed Gibbs energy of reaction:.

The following program can be used to calculate the function of temperature, pH, and ionic strength that represents the
standard transformed Gobbs energy of formation of a reactant.

ca~cdGHT[speciesmat-] :=
Module[{dGzero, dGzeroT,dHzero,zi, nH, gibbscoeff,pHterm,
istem,gpfnsp,dGfn,dHfn},(*This program produces the function of T (in Kelvin), pH and
ionic strength (is) that gives the standard transformed Gibbs energy of formation of a
reactant (sum of species) and the standard transformed enthalpy. The input speciesmat
is a matrix that gives the standard Gibbs energy of formation at 298.15 K, the
standard enthalpy of formation at 298.15 K, the electric charge, and the number of
hydrogen atoms in each species. There is a row in the matrix for each species of the
reactant. gpfnsp is a list of the functions for the transformed Gibbs energies of the
species. The output is in the form {dGfn,dHfn}, and energies are expressed in kJ
mol"-l. The values of the standard transformed Gibbs energy of formation and the
standard transformed enthalpy of formation can be calculated at any temperature in the
range 273.15 K to 313.15 K, any pH in the range 5 to 9, and any ionic strength in the
range 0 to 0.35 m by use of the assignment operator(/.).*)
{dGzero,dHzero,zi,nH}=Transpose[speciesmatl;
gibbscoeff=9.20483*1OA-3*t-l.284668*lOA-S*tA2+4.95l99*lOA-8*tA3~
dGzeroT=dGzero*t/298.15+dHzero*(l-t/298.15);
pHterm = nH*8.31451*(t/10O0)*Log[lOA-pH1;
istermG = gibbscoeff*((ziA2) - nH)*(is".5)/(1 + 1.6*isA.5);
gpfnsp=dGzeroT - pHterm - istermG;
dGfn=-8.31~51*(t/100~)*Log[App~y[P~us,Exp[-l*gpfnsp/(8.3~~5~*(t/~~~~~~~]~;
dHfn=-tA2*D[dGfn/t, tl ;
{dGfn,dHfn}l
The program calctrGerx is readily modified to take a list of temperatures.

calctrGerxT[eg_,pHlist_rislislist_l:},(*Calculates the standard
transformed Gibbs energy of reaction in kJ molA-l at specified pHs, ionic strengths,
and temperatures for a biochemical equation typed in the form atpt+h2ot+de==adpt+pit,
where the functions include temperature (in K). The names of the reactants call the
appropriate functions of pH, ionic strength, and temperature. pHlist, islist, and
tlist can be lists. This program can also be used to calculate the standard
transformed enthalpy of reaction.*)
energy=Solve [eq,del ;
energy[[l,l,2ll/.pH->pHliSt/.is->islist/.t->tlist]
co2gt=calcdGHT[co2gspl~~lll

(-120.272 (-393.5 (1 - 0.00335402 t) - 1.32269 t))/t
-0.00831451 t Log[E I
co2tott=calcdGHT[co2totspl~[lll
-0.00831451 t Log[Power[E, (-120.272 (-677.14 (1 - 0.00335402 t) - 1.77028 t -
0.5 2 -8 3 0.5
(4 is (0.00920483 t - 0.0000128467 t + 4.95199 10 t ))/(1 + 1.6 is )))
Power[E, (-120.272 (-699.63 (1 - 0.00335402 t) - 2.08992 t +
0.5 2 -8 3 0.5
(2 is (0.00920483 t - 0.0000128467 t + 4.95199 10 t ))/(1 + 1.6 is )
(-120.272 (-691.99 (1 - 0.00335402 t) - 1.96804 t - 0,00831451 t Log[lO-pHI))/t
E 1