Henderson–Hasselbalch equation

In chemistry, the Henderson–Hasselbalch equation describes the derivation of pH as a measure of acidity (using pK_a, the acid dissociation constant) in biological and chemical systems. The equation is also useful for estimating the pH of a buffer solution and finding the equilibrium pH in acid-base reactions (it is widely used to calculate the isoelectric point of proteins).

Two equivalent forms of the equation are

$\textrm{pH} = \textrm{pK}_{a}+ \log \left ( \frac{[\textrm{A}^-]}{[\textrm{HA}]} \right )$

and

$\textrm{pH} = \textrm{pK}_{a}-\log \left ( \frac{[\textrm{HA}]}{[\textrm{A}^-]} \right ).$

Here, pK_a is − log(K_a) where K_a is the acid dissociation constant, that is:

$\textrm{pK}_{a} = - \log(K_{a}) = - \log \left ( \frac{[\mbox{H}_{3}\mbox{O}^+][\mbox{A}^-]}{[\mbox{HA}]} \right )$ for the non-specific Brønsted acid-base reaction: $\mbox{HA} + \mbox{H}_{2}\mbox{O} \rightleftharpoons \mbox{A}^- + \mbox{H}_{3}\mbox{O}^+$

In these equations, A ⁻ denotes the ionic form of the relevant acid. Bracketed quantities such as [base] and [acid] denote the molar concentration of the quantity enclosed.

In analogy to the above equations, the following equation is valid:

$\textrm{pOH} = \textrm{pK}_{b}+ \log \left ( \frac{[\textrm{BH}^+]}{[\textrm{B}]} \right )$

Where BH⁺ denotes the conjugate acid of the corresponding base B.

Derivation

The Henderson–Hasselbalch equation is derived from the acid dissociation constant equation by the following steps:

$K_\textrm{a} = \frac{[\textrm{H}^+][\textrm{A}^-]} {[\textrm{HA}]}$

$\log_{10}K_\textrm{a} = \log_{10} \left ( \frac{[\textrm{H}^+][\textrm{A}^-]}{[\textrm{HA}]} \right )$

$\log_{10}K_\textrm{a} = \log_{10}[\textrm{H}^+] + \log_{10} \left ( \frac{[\textrm{A}^-]}{[\textrm{HA}]} \right )$

$-\textrm{p}K_\textrm{a} = -\textrm{pH} + \log_{10} \left ( \frac{[\textrm{A}^-]}{[\textrm{HA}]} \right )$

$\textrm{pH} = \textrm{p}K_\textrm{a} + \log_{10} \left ( \frac{[\textrm{A}^-]}{[\textrm{HA}]} \right )$

The ratio [A ⁻ ] / [HA] is unitless, and as such, other ratios with other units may be used. For example, the mole ratio of the components, $n_{A^-}/n_{HA}$ or the fractional concentrations $\alpha_{A^-}/\alpha_{HA}$ where $\alpha_{A^-}+\alpha_{HA}=1$ will yield the same answer. Sometimes these other units are more convenient to use.

History

Lawrence Joseph Henderson wrote an equation, in 1908, describing the use of carbonic acid as a buffer solution. Karl Albert Hasselbalch later re-expressed that formula in logarithmic terms, resulting in the Henderson–Hasselbalch equation [1]. Hasselbalch was using the formula to study metabolic acidosis.

Limitations

There are some significant approximations implicit in the Henderson–Hasselbalch equation. The most significant is the assumption that the concentration of the acid and its conjugate base at equilibrium will remain the same as the formal concentration. This neglects the dissociation of the acid and the hydrolysis of the base. The dissociation of water itself is neglected as well. These approximations will fail when dealing with relatively strong acids or bases (pKa more than a couple units away from 7), dilute or very concentrated solutions (less than 1 mM or greater than 1M), or heavily skewed acid/base ratios (more than 100 to 1). Also, the equation does not take into effect the dilution factor of the acid and conjugate base in water. If the proportion of acid to base is 1, then the pH of the solution will be different if the amount of water changes from 1mL to 1L.

Estimating blood pH

A modified version of the Henderson–Hasselbalch equation can be used to relate the pH of blood to constituents of the bicarbonate buffering system:

$pH = pK_{a~H_2CO_3}+ \log \left ( \frac{[HCO_3^-]}{[H_2CO_3]} \right )$

, where:

• pK_{a H2CO3} is the acid dissociation constant of carbonic acid. It is equal to 6.1.
• [HCO₃^-] is the concentration of bicarbonate in the blood
• [H₂CO₃] is the concentration of carbonic acid in the blood

This is useful in arterial blood gas, but these usually state pCO₂, that is, the partial pressure of carbon dioxide, rather than H₂CO₃. However, these are related by the equation:

$[H_2CO_3] = k_{\rm H~CO_2}\, \times pCO_2$

, where:

• [H₂CO₃] is the concentration of carbonic acid in the blood
• k_{H CO2} is a constant including the solubility of carbon dioxide in blood. k_{H CO2} is approximately 0.03 mmol/mmHg
• pCO₂ is the partial pressure of carbon dioxide in the blood

Taken together, the following equation can be used to relate the pH of blood to the concentration of bicarbonate and the partial pressure of carbon dioxide:

$pH = 6.1 + \log \left ( \frac{[HCO_3^-]}{0.03 \times pCO_2} \right )$

, where:

• pH is the acidity in the blood
• [HCO₃^-] is the concentration of bicarbonate in the blood
• pCO₂ is the partial pressure of carbon dioxide in the blood

See also

• Acid
• Base
• Titration
• Acidosis
• Alkalosis

Further reading

• Lawrence J. Henderson (1 May 1908). "Concerning the relationship between the strength of acids and their capacity to preserve neutrality" (Abstract). Am. J. Physiol. 21 (4): 173–179. http://ajplegacy.physiology.org/cgi/content/abstract/21/4/465-s. 
• 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. 
• Po, Henry N.; Senozan, N. M. (2001). "Henderson–Hasselbalch Equation: Its History and Limitations". J. Chem. Educ. 78 (11): 1499–1503. doi:10.1021/ed078p1499. 
• de Levie, Robert. (2003). "The Henderson–Hasselbalch Equation: Its History and Limitations". J. Chem. Educ. 80 (2): 146. doi:10.1021/ed080p146. 
• de Levie, Robert (2002). "The Henderson Approximation and the Mass Action Law of Guldberg and Waage". The Chemical Educator 7 (3): 132–135. doi:10.1007/s00897020562a. 

External links

• Henderson–Hasselbalch Calculator
• Derivation and detailed discussion of Henderson–Hasselbalch equation
• True example of using Henderson–Hasselbalch equation for calculation net charge of proteins