Download 134_2010_1897_MOESM1_ESM - Springer Static Content Server

Electronic Supplementary Material (ESM)
Additional aspects related to the Stewart approach
Werner Lang
In response to the article by Doberer D, Funk GC, Kirchner K, Schneeweiss B (2009) “A
critique of Stewart’s approach: the chemical mechanism of dilutional acidosis”. Intensive Care
Medicine DOI 10.1007/s00134-009-1528-y
Corresponding author:
Dr. Werner Lang, Institut für Physiologie und Pathophysiologie, Universitätsmedizin der
Johannes Gutenberg-Universität Mainz, Saarstraße 21, 55099 Mainz, Germany
Phone: ++49-6131-39-25761
Fax: ++49-6131-39-25560
e-mail: [email protected]
Stewart approach and standard physical chemistry
In Table 1 [1], the authors worked out apparent differences between the Stewart approach and
standard physical chemistry. However, Stewart is physicochemically well grounded on proton
dissociation equilibria, mass balance equations and electroneutrality condition. In particular, all
acids in plasma or other biological fluids are treated as proton donors, corresponding bases as
proton acceptors. Since plasma concentrations of the electrolytes are high, ion concentrations
must be corrected by individual activity coefficients. These depend on total ionic strength
(Debye-Hückel law), not on osmolality (0.290 osmol/ kgH2O), as mentioned by the authors.
Therefore, in Stewart’s master equation [2] apparent acidity constants are used at defined total
ionic strength (0.154 mol/L) for carbonic acid (K1, K2) or other weak acids, e.g., cholic acid (Ka).
SID  [H ] 
K 1  sCO2  pCO2
K  K  sCO2  pCO2
K
K A
2 1 2
 w  a tot  0
[H  ]
[ H  ]2
[H ] [H ]  K a
.
SID = [Na+] – [Cl-] strong ion difference (mol∙L-1); [H+] concentration of H+-ion (mol∙L-1),
which is set equal to the measured activity (aH+) in the solution with standardized glass electrode
system; K1, K2 apparent acidity constants of first and second dissociation of carbonic acid; sCO2
solubility of CO2 (mol∙L-1∙mmHg-1), which can be combined to Kc1 = K1∙sCO2; Ka apparent
acidity constant of weak acid (cholic acid); Atot total concentration of weak acid (Atot = [HA] +
[A-])
According to Stewart [3], alkalosis can be observed if the SID is increased, and acidosis if the
SID is decreased. However, one cannot argue that cations are bases and anions are acids on their
own. What counts is the difference in concentration between the sum of the strong cations and
the sum of the strong anions, i.e., the SID. Only the latter is associated with alkalinity (SID > 0)
or with acidity (SID < 0), whereas the pH is neutral if SID = 0, e.g., in pure water or in an
aqueous solution of 0.9 % NaCl.
This, however is nothing other than titrating, e.g., solution A (0.140 mol/l NaOH) by stepwise
addition of HCl until the neutral point is reached at pH = 6.81 by the equivalent amount of HCl
(0.140 mol/l). Further addition of HCl results in excess HCl, decreased pH and negative SID.
Phase rule: components and independent variables
When blood passes the lungs for gas exchange, it can be treated physicochemically as a twophase system, which is described at constant temperature (37°C) and pressure (760 mmHg) by
composition of the aqueous and gaseous phase. For demonstration purpose, solution B may be
used, which was equilibrated at constant pCO2 (40 mmHg) and final state can be achieved from
three different starting solutions with the same concentration of NaCl (0.105 mol/L), but
different concentrations of either NaOH (0.035 mol/L), or NaHCO3 (0.035 mol/L), or Na2CO3
(0.0175 mol/L).
According to the phase rule [4], the degrees of freedom (f) of a system depend on the number of
phases (p) and on the number of components (c): f = c + 2 – p. For solution B, the number of
components of the system can be derived from the total species in the aqueous phase, H2O, H+,
OH-, dissolved CO2, H2CO3, HCO3-, CO32-, Na+, Cl-, and in the gaseous phase, CO2, H2O, i.e., a
total of 11 species minus seven restraining conditions. These are the physical and chemical
equilibria of water vapour, water dissociation (Kw), CO2-solution (Kdiss), CO2-hydration (Kh),
first (K1) and second (K2) dissociation of carbonic acid and electroneutrality. Thus, the number
of degrees of freedom f = (11 – 7) + 2 – 2 is 4 and the whole system is completely determined by
choice of four independent variables. These are temperature (T), pCO2, and the analytical
concentrations of sodium and chloride, where the difference of the latter two concentrations is
the strong ion difference (SID).
In a two-phase system with additional non-bicarbonate buffer (solution C), the independent
variables are the SID, pCO2 and Atot, respectively.
Theoretical simulations
In simulation 1 (Table S1), an aqueous solution of 0.140 mol/L NaOH (solution A) was diluted
by pure water. In this case, the special conditions in Stewart’s equation are: pCO2 = 0, SID =
SID  [H ] 
0.140 mol/L and Atot= 0. Hence,
SID 
compared to SID, it follows that
Kw
0
[H ]
.
Kw
 [OH ]
[H ]
Furthermore, since [H+] is very small
[H ] 
or for calculation of the pH,
Kw
[SID]
.
If solution A is diluted (1:2), excess OH-- ions from solution A (0.140 mol/L) mop up excess
H+-ions from added water (1.55E-07 mol/L). Hence, in the fictive moment before the chemical
reaction in the expanded volume (2 L), there is an equivalent net consumption of H+-ions
(-1.55E-07 mol/L) and OH-- ions (-1.55E-07 mol/L) – incorrect sign, used in Table S1 – to form
1.55E-07 mol/L H2O. Therefore, at initial state before chemical reaction, the concentrations of
0.140 mol/L NaOH and 1.55E-07 mol/L H+-ion are halved: 0.070 mol/L and 7.75E-08 mol/L,
respectively. This, however, is not consistent with the ion product of water and [H+] must be
doubled with respect to the initial concentration. Hence, the dilution process can be described by
formation of H2O (neutralization reaction) and increased H2O-dissociation, caused by decreased
SID or [OH-] due to dilution. This contradicts the author’s interpretation.
Similarly, in simulation 3 (Table S3), the special conditions in Stewart’s equation are: pCO2 = 40
mmHg, SID = 0.035 mol/l and Atot= 0. Since in the pH range 7 to 8 the second dissociation
[H ]  [OH ] 
reaction of the carbonic acid can be neglected and since in addition,
SID 
same order, one obtains,
Kw
[H ]
are of the
K1
 sCO2  pCO2
[H ]
.
This is the non-logarithmic form of the Henderson-Hasselbalch equation, if SID is replaced by
[HCO3-]. In the Siggaard-Andersen nomogram - log pCO2 versus pH – the pH of supranormal
NaHCO3 (35 mmol/L)/NaCl (105 mmol/L) solution at normal pCO2 = 40 mmHg is on the
alkaline side at 7.55 and is a straight line with slope -1, which is shifted to the acid side on the
left upon dilution with 0.9 % NaCl. At half concentration (17.5 mmol/L), the calculated pH and
[H+] are 7.252 and 5.60E-08 mol/L.
In simulation 5 (Table S5), the special conditions are, pCO2 = 40 mmHg, SID = 0.035 mol/L and
Atot= 9.66 mmol/L. With the same approximations as in simulation 3, one obtains the following
quadratic equation for [H+]: ax2 + bx +c = 0, where
x  [H ] 
 b  b 2  4 ac
2a
a = SID; b = SID∙Ka – Kc1∙pCO2 - Ka∙Atot; c = - Kc1∙pCO2∙Ka .
and
Chemical equilibrium and driving force
If the equilibrium state in a chemical system is disturbed by changes in composition, e.g., by
dilution, the driving force is reestablishment of all chemical acid-base equilibria in the system
(water, carbonic acid, weak acids). The quantity, related to chemical equilibrium at constant
temperature and pressure is the Gibbs free energy [4], ∆G = ∆G° + 2.303∙RT log K, where the
first term is the free energy change at equilibrium conditions and the second term is caused by
displacement from equilibrium. If ∆G = 0, net reaction is zero and the system exists in a
stationary state, which is characterized by the equilibrium constant (Keq). If ∆G < 0, the reaction
runs spontaneously, and, if ∆G > 0, it does not.
In simulation 1, the only reaction is, OH- + H+ = H2O, which is the reverse reaction of water
dissociation (Kw). The gain in free energy, after reaction of the disturbed chemical system in the
mixture, can be calculated as follows, ∆G = 2.303∙RT (log Kw – log [H+]∙[OH-]). With the
constants R = 1.987 cal∙K-1∙mol-1, T = 310.15 K and the values from Table 1, [H+] = 7.75E-08
mol/L, [OH-] = 0.07 mol/L and Kw = 10-13.62, this yields -7.6 kcal/mol and the reaction proceeds
spontaneously.
In the more general case simulation 5, formation of bicarbonate by several reactions with excess
carbonic acid have to be considered: (1) A- + H2CO3 = HA + HCO3-; (2) CO32- + H2CO3 = 2
HCO3-; (3) OH- + H2CO3 = H2O + HCO3-. The extent of these competing reactions depends on
the initial concentrations of the reactants (law of mass action) and on the gain in free energy.
From initial concentrations before reaction (Table 1) and from known acidity constants
according to Watson [3], ∆G can be calculated for each reaction.
For reaction (1), ∆G(1) = 2.303∙RT (log{[HA]∙[HCO3-]}/{[A-]∙pCO2}– log K(1)) = -437 cal/mol,
and composed constant K(1) = 4.359E-05 (Kc1/Ka). For reaction (2), ∆G(2) = 2.303∙RT (log
[HCO3-]2 /{[CO32-]∙pCO2}– log K(2)) = -427 cal/mol, and composed constant K(2) = 0.425
(Kc1/K2). For reaction (3), ∆G(3) = 2.303∙RT (log [HCO3-]/{[OH-]∙pCO2}– log K(3)) = + 2
cal/mol, and composed constant K(3) = 1.021E+03 (Kc1/Kw). Hence, only reaction (1) and
reaction (2) proceed spontaneously (∆G(1), ∆G(2) < 0), whereas reaction (3) does not (∆G(3) >0).
Furthermore, since initial [CO32-] is small (1.97E-05 mol/L) in comparison to [A-] (4.53E-03
mol/L) by a ratio of 1:230, reaction (2) is also negligible. This explains why in simulation 5, the
approximate quadratic equation yields the same results as the complete quartic equation by
Stewart.
References
1. Doberer D, Funk GC, Kirchner K, Schneeweiss B (2009) A critique of Stewart’s
approach: the chemical mechanism of dilutional acidosis. Intensive Care Medicine
35:2173-2180
2. Watson PD (1999) Modeling the effects of proteins on pH in plasma. J Appl Physiol
86:1421-1427.
3. Jones NL (1991) Acid-Base Physiology, in The Lung: Scientific Foundations, Crystal
RG, West JB et al. (Eds.), Raven Press, New York
4. Moore WJ (1962) Physical Chemistry. Longmans, London