User: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 $$\scriptstyle K_h=\frac{[H_2CO_3]}{[CO_2]}$$ (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 kH=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&minus;], [HCO3&minus;] and [CO32&minus;], 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:


 * 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&minus;] 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&minus;] may be neglected for this specific problem, resulting in the following very precise analytical expression for [H+]:


 * $$\scriptstyle[H^+] \simeq \left( 10^{-14}+\frac {K_hK_{a1}}{k_\mathrm{H}} p_{CO_2}\right)^{1/2}$$

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 + Pi + 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 Pi + 2 H2O → 10 NADH + 10 H+ + 2 FADH2 + 2 ATP + 2 GTP + 6 C02

(the above reactions are equilibrated if Pi represents the H2PO4- ion, ADP and GDP the ADP2- and GDP2- ions respectively, ATP and GTP the ATP3- and GTP3- 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 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):
 * {| width="450"


 * width="50%" height="30"| CaCO3 ⇋ Ca2+ + CO32–
 * Ksp = 3.7×10–9 to 8.7×10–9 at 25 °C
 * }

where the solubility product for [Ca2+][CO32–] is given as anywhere from Ksp = 3.7×10–9 to Ksp = 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 CO32– cannot exceed the value of Ksp. 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 CO32– combines with H+ in the solution according to:


 * {| width="450"


 * width="50%" height="25"| 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:


 * {| width="450"


 * width="50%" height="25"|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:


 * {| width="450"


 * width="50%" height="25"| 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:


 * {| width="450"


 * width="50%" |$$\frac{P_{\mathrm{CO}_2}}{[\mathrm{CO}_2]}\ =\ k_\mathrm{H}$$
 * where kH = 29.76 atm/(mol/L) at 25°C (Henry constant), $$\scriptstyle P_{\mathrm{CO}_2}$$ being the CO2 partial pressure.
 * }

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


 * {| width="450"


 * width="50%" height="25"| 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[CO32–] + [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 (Ksp = 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&minus;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

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&minus;], we obtain (with CaCO3 molar mass = 100 g):

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&minus; so that the neutrality equation reduces approximately to 2[Ca2+] = [A&minus;] yielding $$\scriptstyle[\mathrm{Ca}^{2+}] \simeq \frac{[\mathrm{A}^-]}{2}$$. When the concentration decreases, [HCO3&minus;] 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 pKA = 4.76) with decreasing concentration [A] = [A&minus;]+[AH], we obtain:

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 pKA, 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&minus;], [CO32&minus;], [Ca2+], [H+] and [OH&minus;]. The system may be reduced to a seventh degree equation for [H+] the numerical solution of which gives

where [A] = [H3PO4] + [H2PO4&minus;] + [HPO42&minus;] + [PO43&minus;]. We see that phosphoric acid is more efficient than a monoacid since at the final almost neutral pH, the second dissociated state concentration [HPO42&minus;] 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-] + [HPO42-] + [PO43-] ([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:

For large acid concentrations, the solution is mainly composed of H3PO4. For [A] = 10-2, the pH is closed to pKa1, giving an equimolar mixture of H3PO4 and H2PO4-. For [A] below 10-3, the solution is mainly composed of H2PO4- with [HPO42-] becoming non negligible for very dilute solutions. [PO43-] is always negligible.

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

&alpha; α &beta; β &gamma; γ &epsilon; ε&delta; δ &nu; 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

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

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

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

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

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