Henderson–Hasselbalch equation explained

In chemistry and biochemistry, the Henderson–Hasselbalch equation$$\backslash ce\; =\; \backslash ceK\_\backslash ce\; +\; \backslash log\_\; \backslash left(\backslash frac\; \backslash right)$$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, $\backslash frac$ of the acid and its conjugate base in an equilibrium.^[1]

CH _{3CO} _{2H} \leftrightharpoons CH _{3CO} ^{-} ^{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,

RNH _{2} 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

$\backslash ce$

$\backslash mathrm$

$\backslash mathrm$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.

$\backslash mathrm$However, the term $\backslash mathrm$ can be omitted to a good approximation.

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

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

The thermodynamic equilibrium constant,

K ^{*} ,

$K^*\; =\; \backslash frac\; \backslash times\; \backslash frac$ is a product of a quotient of concentrations $\backslash frac$ and a quotient,

\Gamma , of activity coefficients $\backslash frac$.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 $\backslash gamma$ 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.

$$K\_a\; =\; \backslash frac\; =\; \backslash frac$$Rearrangement of this expression and taking logarithms provides the Henderson–Hasselbalch equation $$\backslash ce\; =\; \backslash ceK\_\backslash ce\; +\; \backslash log\_\; \backslash left(\backslash frac\; \backslash right)$$

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⁺.

$\backslash mathrm$The value of $\backslash mathrm$ 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

$\backslash ce\; +\; \backslash mathrm\; \backslash rightleftharpoons\; \backslash ce\; \backslash rightleftharpoons\; \backslash ce\; +\; \backslash ce$where $\backslash mathrm$ is the bicarbonate ion and $\backslash mathrm$ 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.

$\backslash mathrm\; =\; \backslash mathrm$$\backslash mathrm$ 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

$\backslash mathrm\; =\; 6.1\; +\; \backslash log\_\; \backslash left\; (\backslash frac\; \backslash right)$where is the molar concentration of bicarbonate in the blood plasma and 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:

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.

See also

• Davenport diagram

Further reading

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

Notes and References

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