Английская Википедия:Henderson–Hasselbalch equation

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Шаблон:Short description In chemistry and biochemistry, the Henderson–Hasselbalch equation <math chem display="block">\ce{pH} = \ce{p}K_\ce{a} + \log_{10} \left( \frac{[\ce{Base}]}{[\ce{Acid}]} \right)</math> relates the pH of a chemical solution of a weak acid to the numerical value of the acid dissociation constant, K_a, of acid and the ratio of the concentrations, <math chem > \frac{[\ce{Base}]}{[\ce{Acid}]}</math> of the acid and its conjugate base in an equilibrium.

<math> \mathrm{\underset {(acid)} {HA} \leftrightharpoons \underset {(base)} {A^-} + H^+}</math>

For example, the acid may be acetic acid

<math>\mathrm{CH_3CO_2H \leftrightharpoons CH_3CO_2^- + H^+}</math>

The Henderson–Hasselbalch equation can be used to estimate the pH of a buffer solution by approximating the actual concentration ratio as the ratio of the analytical concentrations of the acid and of a salt, MA.

The equation can also be applied to bases by specifying the protonated form of the base as the acid. For example, with an amine, <math>\mathrm{RNH_2}</math>

<math>\mathrm{RNH_3^+ \leftrightharpoons RNH_2 + H^+}</math>

Derivation, assumptions and limitations

A simple buffer solution consists of a solution of an acid and a salt of the conjugate base of the acid. For example, the acid may be acetic acid and the salt may be sodium acetate. The Henderson–Hasselbalch equation relates the pH of a solution containing a mixture of the two components to the acid dissociation constant, K_a of the acid, and the concentrations of the species in solution.

Файл:Buffer titration graph.svg

Simulated titration of an acidified solution of a weak acid (Шаблон:Math) with alkali

To derive the equation a number of simplifying assumptions have to be made.

Assumption 1: The acid, HA, is monobasic and dissociates according to the equations

<math chem> \ce{HA <=> H^+ + A^-} </math>

<math chem> \mathrm{C_A = [A^-] + [H^+][A^-]/K_a} </math>

<math chem> \mathrm{C_H = [H^+] + [H^+][A^-]/K_a} </math>

C_A is the analytical concentration of the acid and C_H is the concentration the hydrogen ion that has been added to the solution. The self-dissociation of water is ignored. A quantity in square brackets, [X], represents the concentration of the chemical substance X. It is understood that the symbol H⁺ stands for the hydrated hydronium ion. K_a is an acid dissociation constant.

The Henderson–Hasselbalch equation can be applied to a polybasic acid only if its consecutive pK values differ by at least 3. Phosphoric acid is such an acid.

Assumption 2. The self-ionization of water can be ignored. This assumption is not, strictly speaking, valid with pH values close to 7, half the value of pK_w, the constant for self-ionization of water. In this case the mass-balance equation for hydrogen should be extended to take account of the self-ionization of water.

<math chem> \mathrm{C_H = [H^+] + [H^+][A^-]/K_a + K_w/[H^+]}</math>

However, the term <math chem> \mathrm{K_w/[H^+]}</math> can be omitted to a good approximation.

Assumption 3: The salt MA is completely dissociated in solution. For example, with sodium acetate

<math chem>\mathrm{Na(CH_3CO_2) \rightarrow Na^ + + CH_3CO_2^-} </math>

the concentration of the sodium ion, [Na⁺] can be ignored. This is a good approximation for 1:1 electrolytes, but not for salts of ions that have a higher charge such as magnesium sulphate, MgSO₄, that form ion pairs.

Assumption 4: The quotient of activity coefficients, <math chem>\Gamma</math>, is a constant under the experimental conditions covered by the calculations.

The thermodynamic equilibrium constant, <math>K^*</math>,

<math chem>K^* = \frac{ [\ce{H+}][\ce{A^-}]} { [\ce{HA}] } \times \frac{ \gamma_{\ce{H+}} \gamma _{\ce{A^-}}} {\gamma _{HA} }</math>

is a product of a quotient of concentrations <math chem>\frac{ [\ce{H+}][\ce{A^-}]} { [\ce{HA}] } </math> and a quotient, <math> \Gamma </math>, of activity coefficients <math chem> \frac{ \gamma_{\ce{H+}} \gamma _{\ce{A^-}}} {\gamma _{HA} }</math>. In these expressions, the quantities in square brackets signify the concentration of the undissociated acid, HA, of the hydrogen ion H⁺, and of the anion A⁻; the quantities <math chem>\gamma</math> are the corresponding activity coefficients. If the quotient of activity coefficients can be assumed to be a constant which is independent of concentrations and pH, the dissociation constant, K_a can be expressed as a quotient of concentrations.

<math chem display="block">K_a = K^* / \Gamma = \frac{[\ce{H+}][\ce{A^-}]} { [\ce{HA}] }</math>

Rearrangement of this expression and taking logarithms provides the Henderson–Hasselbalch equation <math chem display="block">\ce{pH} = \ce{p}K_\ce{a} + \log_{10} \left( \frac{[\ce{A^-}]}{[\ce{HA}]} \right)</math>

Application to bases

The equilibrium constant for the protonation of a base, B,

Шаблон:Underset + H⁺ Шаблон:Eqm Шаблон:Underset

is an association constant, K_b, which is simply related to the dissociation constant of the conjugate acid, BH⁺.

<math chem>\mathrm{pK_a = \mathrm{pK_w} - \mathrm{pK_b}}</math>

The value of <math chem>\mathrm{pK_w}</math> is ca. 14 at 25°C. This approximation can be used when the correct value is not known. Thus, the Henderson–Hasselbalch equation can be used, without modification, for bases.

Biological applications

With homeostasis the pH of a biological solution is maintained at a constant value by adjusting the position of the equilibria

<math chem=""> \ce{HCO3-} + \mathrm{H^+} \rightleftharpoons \ce{H2CO3} \rightleftharpoons \ce{CO2} + \ce{H2O}</math>

where <math chem> \mathrm{HCO_3^-}</math> is the bicarbonate ion and <math chem>\mathrm{H_2CO_3} </math> is carbonic acid. However, the solubility of carbonic acid in water may be exceeded. When this happens carbon dioxide gas is liberated and the following equation may be used instead.

<math chem>\mathrm{[H^+] [HCO_3^-]} = \mathrm{K^m [CO_2(g)]} </math>

<math chem>\mathrm{CO_2(g)} </math> represents the carbon dioxide liberated as gas. In this equation, which is widely used in biochemistry, <math chem>K^m</math> is a mixed equilibrium constant relating to both chemical and solubility equilibria. It can be expressed as

<math chem> \mathrm{pH} = 6.1 + \log_{10} \left ( \frac{[\mathrm{HCO}_3^-]}{0.0307 \times P_{\mathrm{CO}_2}} \right )</math>

where Шаблон:Math is the molar concentration of bicarbonate in the blood plasma and Шаблон:Math is the partial pressure of carbon dioxide in the supernatant gas.

History

In 1908, Lawrence Joseph Henderson derived an equation to calculate the hydrogen ion concentration of a bicarbonate buffer solution, which rearranged looks like this:

In 1909 Søren Peter Lauritz Sørensen introduced the pH terminology, which allowed Karl Albert Hasselbalch to re-express Henderson's equation in logarithmic terms, resulting in the Henderson–Hasselbalch equation.

See also

• Davenport diagram