5d408c81-3aed-4e36-810e-be01485ab52f 23 15 12 decimal

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Reference: Butler,Carbon Dioxide Equilibria, Prob. 2-11, pg 41 A solution containing 10^-3 M sodium bicarbonate [NaHCO3] is treated with 5*10^4 M sodium carbonate [Na2CO2] What is the alkalinity of the final solution ?

Coefficients pertaining to the "Equilibrium solution" k1 10 6.3 -^: 1rst dissociation constant carbonic acid ≡ k2 10 10.3 -^: 2nd dissociation constant carbonic acid ≡ kw 1014 -^: kh 10 1.5 -^: PCO2 10 3.5 -^: 61 line

3 appVersion 0.98.6179 ≡

Something else C 1.5 103 -^* : inorganic carbon ≡ Na 2103 -^* : inorganic sodium ≡ 21 line

balance H OH * kw ≡ CO2 kh PCO2 * ≡ H HCO3 * k1 CO2 * ≡ H CO3 * k2 HCO3 * ≡ H HCO3 2 CO3 * + OH + ≡ 51 mat :

ionisation of water: equilibrium expression Henry's law CO2: equilibrium gas/liquid equilibrium expression for bicarbonate HCO3 equilibrium expression for carbonate CO3 charge balance 51 mat

n 4 :

initial H 10 n -^≡ CO2 10 n -^≡ OH 10 n -^≡ HCO3 10 n -^≡ CO3 109 -^≡ 51 mat :

ultimate accuracy w/o ε ... SLOW !!

For this given example, FindRoot adequate. "roots" is 1/1 Levenberg-Marquardt Mathcad. "roots" preferred as suggested by Radovan.

ε 108 -^:

Optimiz => Numeric H CO2 OH HCO3 CO3 51 mat balance initial FindRoot :

H CO2 OH HCO3 CO3 51 mat 0.0000 0.0000 0.0000 0.0000 0.0000 51 mat

sanity H OH * kw - CO2 kh PCO2 * - H HCO3 * k1 CO2 * - H CO3 * k2 HCO3 * - 0 round H HCO3 2 CO3 * + OH + (- 51 mat : 0.00000 0.00001 - 0.00000 0.00000 0.00000 51 mat

Conjugate Gradient ... LM fails

H log10 - 5

sanity Mathcad  ... Butler problem Alkalinity HCO3 2 CO3 * + OH + H - : 0.000001 -

Alkalinity is a measure of a solution's ability to resist a change in pH and is often used by environmental Engineers. In this example, the alkalinity is zero because alkalinity is defined to begin at the CO2 equivalence point. Adding or removing CO2 by changing the PCO2 does not affect alkalinity, it remains zero

suite suggested by Radovan H CO2 OH HCO3 CO3 Clear 1

balance H OH * 1100000000000000 / ≡ CO2 1100000 / ≡ H HCO3 * CO2 1010 nthroot 63^/ ≡ H CO3 * HCO3 1010 nthroot 103^/ ≡ H HCO3 OH + 2 CO3 * + ≡ 51 mat

H CO2 OH HCO3 CO3 51 mat balance H CO2 OH HCO3 CO3 51 mat roots : 0000051 mat

sanity H OH * kw - CO2 kh PCO2 * - H HCO3 * k1 CO2 * - H CO3 * k2 HCO3 * - H HCO3 2 CO3 * + OH + (- 51 mat : 61022 -^* 51021 -^* 11021 -^* 71022 -^* 11015 -^* - 51 mat

H 0

H log10 - 5.6496

≠ sols roots   ...  solves Butler problem Alkalinity HCO3 2 CO3 * + OH + H - : 11015 -^*