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 carbonic acid

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

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, $\mathrm {RNH_{2}}$

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

The Henderson–Hasselbach buffer system also has many natural and biological applications.

History

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

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.

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, 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.

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

Derivation

Source:

Following these assumptions, the Henderson–Hasselbalch equation is derived in a few logarithmic steps.$$K_{a}={[H^{+}][A^{-}] \over [HA]}$$

Solve for $[H^{+}]$:$$[H^{+}]=K_{a}{[HA] \over [A^{-}]}$$

On both sides, take the negative logarithm:$$-\log[H^{+}]=-\log K_{a}-\log {[HA] \over [A^{-}]}$$

Based on previous assumptions, $pH=-\log[H^{+}]$ and $pK_{a}=-\log K_{a}$$$pH=pK_{a}-\log {[HA] \over [A^{-}]}$$

Inversion of $-\log {[HA] \over [A^{-}]}$ by changing its sign, provides the Henderson–Hasselbalch equation$$pH=pK_{a}+\log {[A^{-}] \over [HA]}$$

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. 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.

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

$\mathrm {CO_{2}(g)}$ 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. The concentration of $\mathrm {H_{2}CO_{3}}$ is dependent on the $[\mathrm {CO_{2}(aq)} ]$which is also dependent on P_CO2.

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 $\mathrm {H_{2}CO_{3}}$, carbonic acid, working in conjunction with [HCO⁻
₃], bicarbonate, to form the bicarbonate system. This is effective near physiological pH of 7.4 as carboxylic acid is in equilibrium with $\mathrm {CO_{2}(g)}$ in the lungs. As blood travels through the body, it gains and loses H+ from different processes including lactic acid fermentation and by NH3 protonation from protein catabolism. Because of this the $[\mathrm {H_{2}CO_{3}} ]$, changes in the blood as it passes through tissues. This correlates to a change in the partial pressure of $\mathrm {CO_{2}(g)}$ in the lungs causing a change in the rate of respiration if more or less $\mathrm {CO_{2}(g)}$ is necessary. For example, a decreased blood pH will trigger the brain stem to perform more frequent respiration. The Henderson–Hasselbalch equation can be used to model these equilibria. It is important to maintain this pH of 7.4 to ensure enzymes are able to work optimally.

Life threatening Acidosis (a low blood pH resulting in nausea, headaches, and even coma, and convulsions) is due to a lack of functioning of enzymes at a low pH. As modelled by the Henderson–Hasselbalch equation, in severe cases this can be reversed by administering intravenous bicarbonate solution. If the partial pressure of $\mathrm {CO_{2}(g)}$ does not change, this addition of bicarbonate solution will raise the blood pH.

Natural buffers

The ocean contains a natural buffer system to maintain a pH between 8.1 and 8.3. The oceans 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 $\mathrm {CO_{2}}$ into bicarbonate. The carbonate buffer reaction helps maintain a constant H+ concentration in the ocean because it consumes hydrogen ions, and thereby maintains a constant pH. The ocean has been experiencing ocean acidification due to humans increasing $\mathrm {CO_{2}}$ in the atmosphere. About 30% of the $\mathrm {CO_{2}}$ that is released in the atmosphere is absorbed by the ocean, and the increase in $\mathrm {CO_{2}}$ absorption results in an increase in H+ ion production. The increase in atmospheric $\mathrm {CO_{2}}$ increases H+ ion production because in the ocean $\mathrm {CO_{2}}$ reacts with water and produces carbonic acid, and carbonic acid releases H+ ions and bicarbonate ions. Overall, since the Industrial Revolution the ocean has experienced a pH decrease by about 0.1 pH units due to the increase in $\mathrm {CO_{2}}$ production.

Ocean acidification affects marine life that have shells that are made up of carbonate. In a more acidic environment it is harder organisms to grow and maintain the carbonate shells. The increase of ocean acidity can cause carbonate shell organisms to experience reduced growth and reproduction.

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.