Johner

About me

I am Jean Johner.

I am a researcher in Thermonuclear Fusion at the CEA/Cadarache (France). I am also interested in nutrition, biology (and of course organic chemistry) which are the subjects of my contributions to Wikipedia.

Work in progress

pH and composition of a carbonic acid solution

At a given temperature, the composition of a pure carbonic acid solution (or of a pure CO₂ solution) is completely determined by the partial pressure $\scriptstyle p_{CO_{2}}$ of carbon dioxide above the solution. To calculate this composition, account must be taken of the above equilibria between the three different carbonate forms (H₂CO₃, HCO₃⁻ and CO₃²⁻) as well as of the hydratation equilibrium between dissolved CO₂ and H₂CO₃ with constant $\scriptstyle K_{h}={\frac {[H_{2}CO_{3}]}{[CO_{2}]}}$ (see above) and of the following equilibrium between the dissolved CO₂ and the gaseous CO₂ above the solution:

CO₂(gas) ↔ CO₂(dissolved) with $\scriptstyle {\frac {[CO_{2}]}{p_{CO_{2}}}}={\frac {1}{k_{\mathrm {H} }}}$ where k_H=29.76 atm/(mol/L) at 25°C (Henry constant)

The corresponding equilibrium equations together with the $\scriptstyle [H^{+}][OH^{-}]=10^{-14}$ relation and the neutrality condition $\scriptstyle [H^{+}]=[OH^{-}]+[HCO_{3}^{-}]+2[CO_{3}^{2-}]$ result in six equations for the six unknowns [CO₂], [H₂CO₃], [H⁺], [OH⁻], [HCO₃⁻] and [CO₃²⁻], showing that the composition of the solution is fully determined by $\scriptstyle p_{CO_{2}}$. The equation obtained for [H⁺] is a cubic whose numerical solution yields the following values for the pH and the different species concentrations:

$\scriptstyle P_{CO_{2}}$ (atm)	pH	[CO2] (mol/L)	[H2CO3] (mol/L)	[HCO3−] (mol/L)	[CO32−] (mol/L)
10−8	7.00	3.36 × 10-10	5.71 × 10−13	1.42 × 10−9	7.90 × 10−13
10−6	6.81	3.36 × 10−8	5.71 × 10−11	9.16 × 10−8	3.30 × 10−11
10−4	5.92	3.36 × 10−6	5.71 × 10−9	1.19 × 10−6	5.57 × 10−11
3.5 x 10−4	5.65	1.18 × 10−5	2.00 × 10−8	2.23 × 10−6	5.60 × 10−11
10−3	5.42	3.36 × 10−5	5.71 × 10−8	3.78 × 10−6	5.61 × 10−11
10−2	4.92	3.36 × 10−4	5.71 × 10−7	1.19 × 10−5	5.61 × 10−11
10−1	4.42	3.36 × 10−3	5.71 × 10−6	3.78 × 10−5	5.61 × 10−11
1	3.92	3.36 × 10−2	5.71 × 10−5	1.20 × 10−4	5.61 × 10−11
2.5	3.72	8.40 × 10−2	1.43 × 10−4	1.89 × 10−4	5.61 × 10−11
10	3.42	0.336	5.71 × 10−4	3.78 × 10−4	5.61 × 10−11

• We see that in the total range of pressure, the pH is always largely lower than pKa₂ so that the CO₃²⁻ concentration is always negligible with respect to HCO₃⁻ concentration. In fact CO₃²⁻ play no quantitive role in the present calculation (see remark below).

• For vanishing $\scriptstyle p_{CO_{2}}$, the pH is close to the one of pure water (pH = 7) and the dissolved carbon is essentially in the HCO₃⁻ form.

• For normal atmospherics conditions ($\scriptstyle P_{CO_{2}}=3.5\times 10^{-4}$ atm), we get a slightly acid solution (pH = 5.7) and the dissolved carbon is now essentially in the CO₂ form. From this pressure on, [OH⁻] becomes also negligible so that the ionized part of the solution is now an equimolar mixture of H⁺ and HCO₃⁻.

• For a CO₂ pressure typical of the one in soda drinks bottles ($\scriptstyle P_{CO_{2}}$ ~ 2.5 atm), we get a relatively acid medium (pH = 3.7) with a high concentration of dissolved CO₂. These features are responsible for the sour and sparkling taste of these drinks.

• Between 2.5 and 10 atm, the pH crosses the pKa₁ value (3.60) giving a dominant H₂CO₃ concentration (with respect to HCO₃⁻) at high pressures.

Remark: As noted above, [CO₃²⁻] may be neglected for this specific problem, resulting in the following very precise analytical expression for [H⁺]:

$\scriptstyle [H^{+}]\simeq \left(10^{-14}+{\frac {K_{h}K_{a1}}{k_{\mathrm {H} }}}p_{CO_{2}}\right)^{1/2}$

Overall oxidation reactions of Pyruvate and Glucose after the citric acid cycle

Combining the reactions occuring during the pyruvate oxydation with those occuring during the citric acid cycle, we get the following overall pyruvate oxydation reaction before the respiratory chain:

Pyruvic acid + 4 NAD⁺ + FAD + GDP + P_i + 2 H₂O → 4 NADH + 4 H⁺ + FADH₂ + GTP + 3 CO₂

Combining the above reaction with the ones occuring in the course of glycolysis, we get the following overall glucose oxydation reaction before the respiratory chain:

Glucose + 10 NAD⁺ + 2 FAD + 2 ADP + 2 GDP + 4 P_i + 2 H₂O → 10 NADH + 10 H⁺ + 2 FADH₂ + 2 ATP + 2 GTP + 6 C0₂

(the above reactions are equilibrated if P_i represents the H₂PO₄^- ion, ADP and GDP the ADP^2- and GDP^2- ions respectively, ATP and GTP the ATP^3- and GTP^3- ions respectively).

Considering the future conversion of GTP to ATP and the maximum 26 ATP produced by the 10 NADH and the 2 FADH₂ in the oxidative phosphorylation, we see that each glucose molecule is able to produce a maximum of 30 ATP.

Synthesis of glutathione

Glutathione is not an essential nutrient since it can be synthesized from the amino acids L-cysteine, L-glutamate and glycine.

It is synthesized in two ATP-dependent steps:

• first, gamma-glutamylcysteine is synthesized from L-glutamate and cysteine via the enzyme gamma-glutamylcysteine synthetase. This is the rate limiting step.
• second, glycine is added to the C-terminal of gamma-glutamylcysteine via the enzyme glutathione synthetase.

The liver is the principal site of glutathione synthesis. In healthy tissue, more than 90% of the total glutathione pool is in the reduced form and less than 10% exists in the disulfide form.

Solubility of calcium carbonate in water

Solubility in pure water with varying CO₂ pressure

Calcium carbonate is poorly soluble in pure water. The equilibrium of its solution is given by the equation (with dissolved calcium carbonate on the right):

CaCO3 ⇋ Ca2+ + CO32–

Ksp = 3.7×10–9 to 8.7×10–9 at 25 °C

where the solubility product for [Ca²⁺][CO₃^2–] is given as anywhere from K_sp = 3.7×10^–9 to K_sp = 8.7×10^–9 at 25 °C, depending upon the data source. What the equation means is that the product of molar concentration of calcium ions (moles of dissolved Ca²⁺ per liter of solution) with the molar concentration of dissolved CO₃^2– cannot exceed the value of K_sp. This seemingly simple solubility equation, however, must be taken along with the more complicated equilibrium of carbon dioxide with water (see carbonic acid). Some of the CO₃^2– combines with H⁺ in the solution according to:

HCO3– ⇋ H+ + CO32–

Ka2 = 5.61×10–11 at 25 °C

HCO₃^– is known as the bicarbonate ion. Calcium bicarbonate is many times more soluble in water than calcium carbonate -- indeed it exists only in solution.

Some of the HCO₃^– combines with H⁺ in solution according to:

H2CO3 ⇋ H+ + HCO3–

Ka1 = 2.5×10–4 at 25 °C

Some of the H₂CO₃ breaks up into water and dissolved carbon dioxide according to:

H2O + CO2(dissolved) ⇋ H2CO3

Kh = 1.70×10–3 at 25 °C

And dissolved carbon dioxide is in equilibrium with atmospheric carbon dioxide according to:

${\frac {P_{\mathrm {CO} _{2}}}{[\mathrm {CO} _{2}]}}\ =\ k_{\mathrm {H} }$

where kH = 29.76 atm/(mol/L) at 25°C (Henry constant), $\scriptstyle P_{\mathrm {CO} _{2}}$ being the CO2 partial pressure.

Calcium Ion Solubility as a function of CO2 partial pressure at 25 °C
$\scriptstyle P_{\mathrm {CO} _{2}}$ (atm)	pH	[Ca2+] (mol/L)
10−12	12.0	5.19 × 10−3
10−10	11.3	1.12 × 10−3
10−8	10.7	2.55 × 10−4
10−6	9.83	1.20 × 10−4
10−4	8.62	3.16 × 10−4
3.5 × 10−4	8.27	4.70 × 10−4
10−3	7.96	6.62 × 10−4
10−2	7.30	1.42 × 10−3
10−1	6.63	3.05 × 10−3
1	5.96	6.58 × 10−3
10	5.30	1.42 × 10−2

For ambient air, $\scriptstyle P_{\mathrm {CO} _{2}}$ is around 3.5×10^–4 atmospheres (or equivalently 35 Pa). The last equation above fixes the concentration of dissolved CO₂ as a function of $\scriptstyle P_{\mathrm {CO} _{2}}$, independent of the concentration of dissolved CaCO₃. At atmospheric partial pressure of CO₂, dissolved CO₂ concentration is 1.2×10^{–5 moles/liter. The equation before that fixes the concentration of H₂CO₃ as a function of [CO₂]. For [CO₂]=1.2×10–5, it results in [H₂CO₃]=2.0×10–8 moles per liter. When [H₂CO₃] is known, the remaining three equations together with}

H2O ⇋ H+ + OH–

K = 10–14 at 25 °C

(which is true for all aqueous solutions), and the fact that the solution must be electrically neutral,

2[Ca²⁺] + [H⁺] = [HCO₃^–] + 2[CO₃^2–] + [OH^–]

make it possible to solve simultaneously for the remaining five unknown concentrations (note that the above form of the neutrality equation is valid only if calcium carbonate has been put in contact with pure water or with a neutral pH solution; in the case where the origin water solvent pH is not neutral, the equation is modified).

The table on the right shows the result for [Ca²⁺] and [H⁺] (in the form of pH) as a function of ambient partial pressure of CO₂ (K_sp = 4.47×10⁻⁹ has been taken for the calculation). At atmospheric levels of ambient CO₂ the table indicates the solution will be slightly alkaline. The trends the table shows are

1) As ambient CO₂ partial pressure is reduced below atmospheric levels, the solution becomes more and more alkaline. At extremely low $\scriptstyle P_{\mathrm {CO} _{2}}$, dissolved CO₂, bicarbonate ion, and carbonate ion largely evaporate from the solution, leaving a highly alkaline solution of calcium hydroxide, which is more soluble than CaCO₃.

2) As ambient CO₂ partial pressure increases to levels above atmospheric, pH drops, and much of the carbonate ion is converted to bicarbonate ion, which results in higher solubility of Ca²⁺.

The effect of the latter is especially evident in day to day life of people who have hard water. Water in aquifers underground can be exposed to levels of CO₂ much higher than atmospheric. As such water percolates through calcium carbonate rock, the CaCO₃ dissolves according to the second trend. When that same water then emerges from the tap, in time it comes into equilibrium with CO₂ levels in the air by outgassing its excess CO₂. The calcium carbonate becomes less soluble as a result and the excess precipitates as lime scale. This same process is responsible for the formation of stalactites and stalagmites in limestone caves.

Two hydrated phases of calcium carbonate, monohydrocalcite, CaCO₃.H₂O, and ikaite, CaCO₃.6H₂O, may precipitate from water at ambient conditions and persist as metastable phases.

Solubility at atmospheric CO₂ pressure with varying pH

We now consider the problem of the maximum solubility of calcium carbonate in normal atmospheric conditions ($\scriptstyle P_{\mathrm {CO} _{2}}$ = 3.5 × 10⁻⁴ atm) when the pH of the solution is adjusted. This is for example the case in a swimming pool where the pH is maintained between 7 and 8 (by addition of NaHSO₄ to decrease the pH or of NaHCO₃ to increase it). From the above equations for the solubility product, the hydrolysis reaction and the two acid reactions, the following expression for the maximum [Ca²⁺] can be easily deduced:

$[{\text{Ca}}^{2+}]_{\mathrm {max} }={\frac {K_{\mathrm {sp} }k_{\mathrm {H} }}{K_{\mathrm {h} }K_{\mathrm {a1} }K_{\mathrm {a2} }}}{\frac {[\mathrm {H} ^{+}]^{2}}{P_{\mathrm {CO} _{2}}}}$

showing a quadratic dependence in [H⁺]. The numerical application with the above values of the constants gives

pH		7.0	7.2	7.4	7.6	7.8	8.0	8.2	8.27	8.4
[Ca2+]max (10-4mol/L or °f)		1590	635	253	101	40.0	15.9	6.35	4.70	2.53
[Ca2+]max (mg/L)		6390	2540	1010	403	160	63.9	25.4	18.9	10.1

Comments:

• decreasing the pH from 8 to 7 increases the maximum Ca²⁺ concentration by a factor 100
• note that the Ca²⁺ concentration of the previous table is recovered for pH = 8.27
• keeping the pH to 7.4 (which gives optimum HClO/OCl^- ratio in the case of "chlorine" maintenance) results in a maximum Ca²⁺ concentration of 1010 mg/L. This means that successive cycles of water evaporation and partial renewing may result in a very hard water before CaCO₃ precipitates. Addition of a calcium sequestrant or complete renewing of the water will solve the problem.

Solubility in a strong or weak acid solution

Solutions of strong (HCl) or weak (acetic, phosphoric) acids are commercially available. They are commonly used to remove limescale deposits. The maximum amount of CaCO₃ that can be "dissolved" by one liter of an acid solution can be calculated using the above equilibrium equations.

• In the case of a strong monoacid with decreasing concentration [A] = [A⁻], we obtain (with CaCO₃ molar mass = 100 g):

[A] (mol/L)		1	10-1	10-2	10-3	10-4	10-5	10-6	10-7	10-10
Initial pH		0.00	1.00	2.00	3.00	4.00	5.00	6.00	6.79	7.00
Final pH		6.75	7.25	7.75	8.14	8.25	8.26	8.26	8.26	8.27
Dissolved CaCO3 (g per liter of acid)		50.0	5.00	0.514	0.0849	0.0504	0.0474	0.0471	0.0470	0.0470

where the initial state is the acid solution with no Ca²⁺ (not taking into account possible CO₂ dissolution) and the final state is the solution with saturated Ca²⁺. For strong acid concentrations, all species have a negligible concentration in the final state with respect to Ca²⁺ and A⁻ so that the neutrality equation reduces approximately to 2[Ca²⁺] = [A⁻] yielding $\scriptstyle [\mathrm {Ca} ^{2+}]\simeq {\frac {[\mathrm {A} ^{-}]}{2}}$. When the concentration decreases, [HCO₃⁻] becomes non negligible so that the preceding expression is no longer valid. For vanishing acid concentrations, we recover the final pH and the solubility of CaCO₃ in pure water.

• In the case of a weak monoacid (here we take acetic acid with pK_A = 4.76) with decreasing concentration [A] = [A⁻]+[AH], we obtain:

[A] (mol/L)		1	10-1	10-2	10-3	10-4	10-5	10-6	10-7	10-10
Initial pH		2.38	2.88	3.39	3.91	4.47	5.15	6.02	6.79	7.00
Final pH		6.75	7.25	7.75	8.14	8.25	8.26	8.26	8.26	8.27
Dissolved CaCO3 (g per liter of acid)		49.5	4.99	0.513	0.0848	0.0504	0.0474	0.0471	0.0470	0.0470

We see that for the same total acid concentration, the initial pH of the weak acid is less acid than the one of the strong acid; however, the maximum amount of CaCO₃ which can be dissolved is approximately the same. This is because in the final state, the pH is larger that the pK_A, so that the weak acid is almost completely dissociated, yielding in the end as many H⁺ ions as the strong acid to "dissolve" the calcium carbonate.

• The calculation in the case of phosphoric acid (which is the most widely used for domestic applications) is more complicated since the concentrations of the four dissociation states corresponding to this acid must be calculated together with [HCO₃⁻], [CO₃²⁻], [Ca²⁺], [H⁺] and [OH⁻]. The system may be reduced to a seventh degree equation for [H⁺] the numerical solution of which gives

[A] (mol/L)		1	10-1	10-2	10-3	10-4	10-5	10-6	10-7	10-10
Initial pH		1.08	1.62	2.25	3.05	4.01	5.00	5.97	6.74	7.00
Final pH		6.71	7.17	7.63	8.06	8.24	8.26	8.26	8.26	8.27
Dissolved CaCO3 (g per liter of acid)		62.0	7.39	0.874	0.123	0.0536	0.0477	0.0471	0.0471	0.0470

where [A] = [H₃PO₄] + [H₂PO₄⁻] + [HPO₄²⁻] + [PO₄³⁻]. We see that phosphoric acid is more efficient than a monoacid since at the final almost neutral pH, the second dissociated state concentration [HPO₄²⁻] is not negligible (see phosphoric acid ).

Trigonometric integrals

We have:

${\rm {ci}}(x)={\rm {Ci}}(x)$

${\rm {Cin}}(x)=\gamma +\ln x-{\rm {Ci}}(x)$

pH and composition of a phosphoric acid solution

For a given total acid concentration [A] = [H₃PO₄] + [H₂PO₄^-] + [HPO₄^2-] + [PO₄^3-] ([A] is the total number of moles of pure H₃PO₄ which have been used to prepare 1 liter of solution) , the composition of an aqueous solution of phosphoric acid can be calculated using the equilibrium equations associated with the three reactions described above together with the [H⁺][0H^-] = 10^-14 relation and the electrical neutrality equation. The system may be reduced to a fifth degree equation for [H⁺] which can be solved numerically, yielding:

[A] (mol/L)	pH	[H3PO4]/[A] (%)	[HPO42-]/[A] (%)	[H2PO4-]/[A] (%)	[PO43-]/[A] (%)
1	1.08	91.7	8.29	6.20×10-6	1.60×10-17
10-1	1.62	76.1	23.9	6.20×10-5	5.55×10-16
10-2	2.25	43.1	56.9	6.20×10-4	2.33×10-14
10-3	3.05	10.6	89.3	6.20×10-3	1.48×10-12
10-4	4.01	1.30	98.6	6.19×10-2	1.34×10-10
10-5	5.00	0.133	99.3	0.612	1.30×10-8
10-6	5.97	1.34×10-2	94.5	5.50	1.11×10-6
10-7	6.74	1.80×10-3	74.5	25.5	3.02×10-5
10-10	7.00	8.24×10-4	61.7	38.3	8.18×10-5

For large acid concentrations, the solution is mainly composed of H₃PO₄. For [A] = 10^-2, the pH is closed to pK_a1, giving an equimolar mixture of H₃PO₄ and H₂PO₄^-. For [A] below 10^-3, the solution is mainly composed of H₂PO₄^- with [HPO₄^2-] becoming non negligible for very dilute solutions. [PO₄^3-] is always negligible.

Table test

[A] (mol/L)		pH	[H3PO4]/[A] (%)	[HPO42-]/[A] (%)	[H2PO4-]/[A] (%)	[PO43-]/[A] (%)
1		1.08	91.7	8.29	6.20×10-6	1.60×10-17
10-1		1.62	76.1	23.9	6.20×10-5	5.55×10-16
10-2		2.25	43.1	56.9	6.20×10-4	2.33×10-14
10-3		3.05	10.6	89.3	6.20×10-3	1.48×10-12
10-4		4.01	1.30	98.6	6.19×10-2	1.34×10-10
10-5		5.00	0.133	99.3	0.612	1.30×10-8
10-6		5.97	1.34×10-2	94.5	5.50	1.11×10-6
10-7		6.74	1.80×10-3	74.5	25.5	3.02×10-5
10-10		7.00	8.24×10-4	61.7	38.3	8.18×10-5

Proanthocyanidin

Proanthocyanidins have been sold as nutritional and therapeutic supplements in Europe since the 1980s, but their introduction to the United States market has been relatively recent.

Images of proanthocyanidin molecules.

Equations tests

Test greek letters

α α β β γ γ ε εδ δ ν ν

This is an equation which is in the text $E\left(k\right)=\int _{0}^{\frac {\pi }{2}}{\sqrt {1-k^{2}\sin ^{2}\theta }}\,d\theta$ and the rest of the text.

This is an equation in scriptstyle which is in the text $\scriptstyle E\left(k\right)=\int _{0}^{\frac {\pi }{2}}{\sqrt {1-k^{2}\sin ^{2}\theta }}\,d\theta$ and the rest of the text.

This is an equation which is in a new paragraph

$E\left(k\right)=\int _{0}^{\frac {\pi }{2}}{{\sqrt {1-k^{2}\sin ^{2}\theta }}\ }{\rm {{d}\theta \,0\leq k\leq 1}}$

$\mathbf {E} \left(k\right)=\int _{0}^{\frac {\pi }{2}}{{\sqrt {1-k^{2}\sin ^{2}\theta }}\ }d\theta \;0\leq k\leq 1$

$E\left(k\right)=\int _{0}^{\frac {\pi }{2}}{{\sqrt {1-k^{2}\sin ^{2}\theta }}\ }d\theta \ 0\leq k\leq 1$

$E\left(k\right)=\int _{0}^{\frac {\pi }{2}}{{\sqrt {1-k^{2}\sin ^{2}\theta }}\ }d\theta \quad 0\leq k\leq 1$

$E\left(k\right)=\int _{0}^{\frac {\pi }{2}}{{\sqrt {1-k^{2}\sin ^{2}\theta }}~}d\theta \qquad 0\leq k\leq 1$