Henderson–Hasselbalch equation

In chemistry and biochemistry, the Henderson–Hasselbalch equation $${\ce {pH}}={\ce {p}}K_{{\ce {a}}}+\log _{10}\left({\frac {[{\ce {Base}}]}{[{\ce {Acid}}]}}\right)$$ 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, ${\frac {[{\ce {Base}}]}{[{\ce {Acid}}]}}$ of the acid and its conjugate base in an equilibrium.

$\mathrm {{\underset {(acid)}{HA}}\leftrightharpoons {\underset {(base)}{A^{-}}}+H^{+}}$

For example, the acid may be acetic acid

$\mathrm {CH_{3}CO_{2}H\leftrightharpoons CH_{3}CO_{2}^{-}+H^{+}}$

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, $\mathrm {RNH_{2}}$

$\mathrm {RNH_{3}^{+}\leftrightharpoons RNH_{2}+H^{+}}$

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.

Simulated titration of an acidified solution of a weak acid (pK_a = 4.7) 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

${\ce {HA <=> H^+ + A^-}}$

$\mathrm {C_{A}=[A^{-}]+[H^{+}][A^{-}]/K_{a}}$

$\mathrm {C_{H}=[H^{+}]+[H^{+}][A^{-}]/K_{a}}$

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.

$\mathrm {C_{H}=[H^{+}]+[H^{+}][A^{-}]/K_{a}+K_{w}/[H^{+}]}$

However, the term $\mathrm {K_{w}/[H^{+}]}$ can be omitted to a good approximation.

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

$\mathrm {Na(CH_{3}CO_{2})\rightarrow Na^{+}+CH_{3}CO_{2}^{-}}$

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, $\Gamma$, is a constant under the experimental conditions covered by the calculations.

The thermodynamic equilibrium constant, $K^{*}$,

$K^{*}={\frac {[{\ce {H+}}][{\ce {A^-}}]}{[{\ce {HA}}]}}\times {\frac {\gamma _{{\ce {H+}}}\gamma _{{\ce {A^-}}}}{\gamma _{HA}}}$

is a product of a quotient of concentrations ${\frac {[{\ce {H+}}][{\ce {A^-}}]}{[{\ce {HA}}]}}$ and a quotient, $\Gamma$, of activity coefficients ${\frac {\gamma _{{\ce {H+}}}\gamma _{{\ce {A^-}}}}{\gamma _{HA}}}$. 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 $\gamma$ 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.

$$K_{a}={\frac {K^{*}}{\Gamma }}={\frac {[{\ce {H+}}][{\ce {A^-}}]}{[{\ce {HA}}]}}$$

Rearrangement of this expression and taking logarithms provides the Henderson–Hasselbalch equation $${\ce {pH}}={\ce {p}}K_{{\ce {a}}}+\log _{10}\left({\frac {[{\ce {A^-}}]}{[{\ce {HA}}]}}\right)$$

Application to bases

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

B(base) + H⁺ ⇌ BH⁺(acid)

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

$\mathrm {pK_{a}=\mathrm {pK_{w}} -\mathrm {pK_{b}} }$

The value of $\mathrm {pK_{w}}$ 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

${\ce {HCO3-}}+\mathrm {H^{+}} \rightleftharpoons {\ce {H2CO3}}\rightleftharpoons {\ce {CO2}}+{\ce {H2O}}$

where $\mathrm {HCO_{3}^{-}}$ is the bicarbonate ion and $\mathrm {H_{2}CO_{3}}$ 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.

$\mathrm {[H^{+}][HCO_{3}^{-}]} =\mathrm {K^{m}[CO_{2}(g)]}$

$\mathrm {CO_{2}(g)}$ represents the carbon dioxide liberated as gas. In this equation, which is widely used in biochemistry, $K^{m}$ is a mixed equilibrium constant relating to both chemical and solubility equilibria. It can be expressed as

$\mathrm {pH} =6.1+\log _{10}\left({\frac {[\mathrm {HCO} _{3}^{-}]}{0.0307\times P_{\mathrm {CO} _{2}}}}\right)$

where [HCO⁻
₃] is the molar concentration of bicarbonate in the blood plasma and P_CO2 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:

[H⁺] [HCO₃^–] = K [CO₂] [H₂O]

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
• Gastric tonometry

Further reading

Davenport, Horace W. (1974). The ABC of Acid-Base Chemistry: The Elements of Physiological Blood-Gas Chemistry for Medical Students and Physicians (Sixth ed.). Chicago: The University of Chicago Press.