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)

can be used to estimate the pH of a buffer solution. The numerical value of the acid dissociation constant, K_a, of the acid is known or assumed. The pH is calculated for given values of the concentrations of the acid, HA and of a salt, MA, of its conjugate base, A^–; for example, the solution may contain acetic acid and sodium acetate.

History

In 1908, Lawrence Joseph Henderson^[1] derived an equation to calculate the hydrogen ion concentration of a buffer solution which, rearranged looks like this:

[H⁺] [HCO₃^–] = K [CO₂] [H₂O]

This can be simplified: [H₂O] remains constant and physicians are much more familiar with: P_CO₂:

[H⁺] [HCO₃^–] = K

P CO 2

Clinically, this simple equation provides all the information required. It is often easy to anticipate how changes in one variable will affect another: When the PCO₂ is constant, then an increase in [H⁺] must be associated with a fall in [HCO₃^–], and an increase in the $P CO 2$ will normally increase both [H⁺] and [HCO₃^–].

Sørensen and Hasselbalch

In 1909 Søren Peter Lauritz Sørensen introduced the pH terminology which allowed Karl Albert Hasselbalch to re-express that equation in logarithmic terms,^[2] resulting in the Henderson–Hasselbalch equation (see Acid-Base History):

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

where:

$pH$ is the negative logarithm of molar concentration of hydrogen ions in the extracellular fluid (ECF), as before.
$[HCO - 3]$ is the molar concentration of bicarbonate in the blood plasma
$P CO 2$ is the partial pressure of carbon dioxide in the blood plasma.

Theory

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, and the concentrations of the species in solution.^[3] To derive the equation a number of simplifying assumptions have to be made. The mixture has the ability to resist changes in pH when a small amount of acid or base is added, which is the defining property of a buffer solution.

Assumption 1: The acid is monobasic and dissociates according to the equation

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

It is understood that the symbol H⁺ stands for the hydrated hydronium ion. 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 valid with pH values near pK_w/2. Normally the pK_w is taken as 14 at 25 °C. For such instances the mass-balance equation for hydrogen must be extended to take account of the self-ionization of water.

C_H = [H⁺] + [H⁺][A⁻]/K_a − K_w/[H⁺]

C_A = [A⁻] + [H⁺][A⁻]/K_a

and the pH will have to be found by solving the two mass-balance equations simultaneously for the two unknowns, [H⁺] and [A⁻].

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

Na(CH₃CO₂) → Na⁺ + CH₃CO₂⁻

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}=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

The Henderson–Hasselbalch equation can be used to calculate the pH of a solution containing the acid and one of its salts, that is, of a buffer solution. With bases, if the value of an equilibrium constant is known in the form of a base association constant, K_b the dissociation constant of the conjugate acid may be calculated from

pK_a + pK_b = pK_w

where K_w is the self-dissociation constant of water. pK_w has a value of approximately 14 at 25 °C.

If the "free acid" concentration, [HA], can be taken to be equal to the analytical concentration of the acid, T_AH (sometimes denoted as C_AH) an approximation is possible, which is widely used in biochemistry; it is valid for very dilute solutions.

{\ce {pH}}={\ce {p}}K_{{\ce {a}}}+\log _{10}\left({\frac {[{\ce {A^-}}]}{T_{AH}}}\right)

The effect of this approximation is to introduce an error in the calculated pH, which becomes significant at low pH and high acid concentration. With bases the error becomes significant at high pH and high base concentration.^[4] (pdf)

References

↑ 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.
↑ 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.
↑ 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.

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[1] 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.

[2] 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.

[3] 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.

[4] 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.

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