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.^[1]

\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.^[2]

To derive the equation a number of simplifying assumptions have to be made.^[3]

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.^[3]

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,

+ H⁺ ⇌

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 $P CO 2$ is the partial pressure of carbon dioxide in the supernatant gas.

History

In 1908, Lawrence Joseph Henderson^[4] 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,^[5] resulting in the Henderson–Hasselbalch equation.

References

^ Petrucci, Ralph H.; Harwood, William S.; Herring, F. Geoffrey (2002). General Chemistry (8th ed.). Prentice Hall. p. 718. ISBN 0-13-014329-4.
^ For details and worked examples see, for instance, Skoog, Douglas A.; West, Donald M.; Holler, F. James; Crouch, Stanley R. (2004). Fundamentals of Analytical Chemistry (8th ed.). Belmont, Ca (USA): Brooks/Cole. pp. 251–263. ISBN 0-03035523-0.
^ ^a ^b Po, Henry N.; Senozan, N. M. (2001). "Henderson–Hasselbalch Equation: Its History and Limitations". J. Chem. Educ. 78 (11): 1499–1503. Bibcode:2001JChEd..78.1499P. doi:10.1021/ed078p1499.
^ Lawrence J. Henderson (1908). "Concerning the relationship between the strength of acids and their capacity to preserve neutrality". Am. J. Physiol. 21 (2): 173–179. doi:10.1152/ajplegacy.1908.21.2.173.
^ Hasselbalch, K. A. (1917). "Die Berechnung der Wasserstoffzahl des Blutes aus der freien und gebundenen Kohlensäure desselben, und die Sauerstoffbindung des Blutes als Funktion der Wasserstoffzahl". Biochemische Zeitschrift. 78: 112–144.

[1] Petrucci, Ralph H.; Harwood, William S.; Herring, F. Geoffrey (2002). General Chemistry (8th ed.). Prentice Hall. p. 718. ISBN 0-13-014329-4.

[2] For details and worked examples see, for instance, Skoog, Douglas A.; West, Donald M.; Holler, F. James; Crouch, Stanley R. (2004). Fundamentals of Analytical Chemistry (8th ed.). Belmont, Ca (USA): Brooks/Cole. pp. 251–263. ISBN 0-03035523-0.

[:0-3] Po, Henry N.; Senozan, N. M. (2001). "Henderson–Hasselbalch Equation: Its History and Limitations". J. Chem. Educ. 78 (11): 1499–1503. Bibcode:2001JChEd..78.1499P. doi:10.1021/ed078p1499.

[4] Lawrence J. Henderson (1908). "Concerning the relationship between the strength of acids and their capacity to preserve neutrality". Am. J. Physiol. 21 (2): 173–179. doi:10.1152/ajplegacy.1908.21.2.173.

[5] Hasselbalch, K. A. (1917). "Die Berechnung der Wasserstoffzahl des Blutes aus der freien und gebundenen Kohlensäure desselben, und die Sauerstoffbindung des Blutes als Funktion der Wasserstoffzahl". Biochemische Zeitschrift. 78: 112–144.

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