Henderson–Hasselbalch equation

In chemistry and biochemistry, the pH of weakly acidic chemical solutions

can be estimated using 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>

The equation relates the pH of the weak acid to the numerical value of the acid dissociation constant, Ka, of the acid, and the ratio of the concentrations of the acid and its conjugate base.

Acid-base Equilibrium Reaction

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

The Henderson-Hasselbalch equation is often used for estimating the pH of buffer solutions by approximating the actual concentration ratio as the ratio of the analytical concentrations of the acid and of a salt, MA. It is also useful for determining the volumes of the reagents needed before preparing buffer solutions, which prevents unnecessary waste of chemical reagents that may need to be further neutralized by even more reagents before they are safe to expose.

For example, the acid may be carbonic acid

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

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>

The Henderson–Hasselbalch buffer system also has many natural and biological applications, from physiological processes (e.g., metabolic acidosis) to geological phenomena.

History

The Henderson–Hasselbalch equation was developed by Lawrence Joseph Henderson and Karl Albert Hasselbalch. Henderson was a biological chemist and Hasselbalch was a physiologist who studied pH.

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

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

Assumptions, limitations, and derivation

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, Ka of the acid, and the concentrations of the species in solution.

thumb|200px|right|Simulated [[titration of an acidified solution of a weak acid () 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>

CA is the analytical concentration of the acid and CH 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. Ka 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 pKw, 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.

Following these assumptions, the Henderson–Hasselbalch equation is derived in a few logarithmic steps.<math display="block">K_a = {[H^{+}][A^{-}] \over [HA] }</math>

Solve for <math>[H^{+}]

</math>:<math display="block">[H^{+}] = K_a {[HA] \over [A^{-}] }

</math>

On both sides, take the negative logarithm:<math display="block">-\log [H^{+}] = -\log K_a -\log {[HA] \over [A^{-}] }</math>

Based on previous assumptions, <math>pH = - \log[H^{+}]</math> and <math>pK_a = -\log K_a</math><math display="block">pH = pK_a -\log {[HA] \over [A^{-}] }</math>

Inversion of <math>-\log {[HA] \over [A^{-}] } </math> by changing its sign, provides the Henderson–Hasselbalch equation<math chem="" display="block">pH = pK_a + \log {[A^{-}] \over [HA] }</math>

Application to bases

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

: + H+

is an association constant, Kb, 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. Carbonic acid is formed reversibly from carbon dioxide and water. 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 is the molar concentration of bicarbonate in the blood plasma and is the partial pressure of carbon dioxide in the supernatant gas. The concentration of <math chem="">\mathrm{H_2CO_3} </math> is dependent on the <math>[\mathrm{CO_2(aq)}]</math>which is also dependent on .

thumb|440x440px|Carbon dioxide, a by-product of [[cellular respiration, is dissolved in the blood. From the blood it is taken up by red blood cells and converted to carbonic acid by the carbonate buffer system. Most carbonic acid then dissociates to bicarbonate and hydrogen ions.]]

One of the buffer systems present in the body is the blood plasma buffering system. This is formed from <math chem="">\mathrm{H_2CO_3} </math>, carbonic acid, working in conjunction with , bicarbonate, to form the bicarbonate system. This is effective near physiological pH of 7.4 as carboxylic acid is in equilibrium with <math chem="">\mathrm{CO_2(g)} </math> in the lungs. The ocean buffer system is known as the carbonate buffer system. The carbonate buffer system is a series of reactions that uses carbonate as a buffer to convert <math chem="">\mathrm{CO_2} </math> into bicarbonate. and thereby maintains a constant pH. About 30% of the <math chem="">\mathrm{CO_2} </math> that is released in the atmosphere is absorbed by the ocean, The increase in atmospheric <math chem="">\mathrm{CO_2} </math> increases H+ ion production because in the ocean <math chem="">\mathrm{CO_2} </math> reacts with water and produces carbonic acid, and carbonic acid releases H+ ions and bicarbonate ions.