Henderson–Hasselbalch equation

In chemistry, the Henderson–Hasselbalch equation describes the derivation of pH as a measure of acidity (using pK_a, the negative log of 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).

The equation is given by:

[math]\displaystyle{ \mathrm{pH} = \mathrm{p}K_\mathrm{a}+ \log_{10} \left ( \frac{[\mathrm{A}^-]}{[\mathrm{HA}]} \right ) }[/math]

Here, [HA] is the molar concentration of the undissociated weak acid, [A⁻] is the molar concentration (molarity, M) of this acid's conjugate base and pK_a is −log₁₀ K_a where K_a is the acid dissociation constant, that is:

[math]\displaystyle{ \mathrm{p}K_\mathrm{a} = - \log_{10} (K_\mathrm{a}) = - \log_{10} \left ( \frac{[\mathrm{H}_{3}\mathrm{O}^+][\mathrm{A}^-]}{[\mathrm{HA}]} \right ) }[/math] for the non-specific Brønsted acid-base reaction: [math]\displaystyle{ \mathrm{HA} + \mathrm{H}_{2}\mathrm{O} \rightleftharpoons \mathrm{A}^- + \mathrm{H}_{3}\mathrm{O}^+ }[/math]

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.

For bases

For the standard base equation:

[math]\displaystyle{ \mathrm{B} + \mathrm{H}^{+} \rightleftharpoons \mathrm{BH}^{+} }[/math]

A second form of the equation expressed in terms of [math]\displaystyle{ K_\mathrm{b} }[/math] where [math]\displaystyle{ K_\mathrm{b} }[/math] is the base dissociation constant:

[math]\displaystyle{ \mathrm{p}K_\mathrm{b} = - \log_{10} (K_\mathrm{b}) = - \log_{10} \left ( \frac{[\mathrm{O}\mathrm{H}^-][\mathrm{HA}]}{[\mathrm{A}^-]} \right ) }[/math]

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

[math]\displaystyle{ \mathrm{pOH} = \mathrm{p}K_\mathrm{b}+ \log_{10} \left ( \frac{[\mathrm{BH}^+]}{[\mathrm{B}]} \right ) }[/math],

where BH⁺ denotes the conjugate acid of the corresponding base B. Using the properties of these terms at 25 degrees Celsius, one can synthesise an equation for pH of basic solutions in terms of pK_a and pH:

[math]\displaystyle{ \mathrm{pH} = \mathrm{p}K_\mathrm{a} + \log_{10} \left(\frac{[\mathrm{B}]}{[\mathrm{BH}^{+}]}\right) }[/math]

Derivation

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

[math]\displaystyle{ K_\ce{a} = \frac{[\ce{H+}][\ce{A^-}]} {[\ce{HA}]} }[/math]

Taking the log, to base ten, of both sides gives:

[math]\displaystyle{ \log_{10}K_\ce{a} = \log_{10} \left ( \frac{[\ce{H+}][\ce{A^-}]}{[\ce{HA}]} \right ) }[/math]

Then, using the properties of logarithms:

[math]\displaystyle{ \log_{10}K_\ce{a} = \log_{10}[\ce{H+}] + \log_{10} \left ( \frac{[\ce{A^-}]}{[\ce{HA}]} \right ) }[/math]

Identifying the left-hand side of this equation as -pK_a and the [math]\displaystyle{ \log_{10} [\ce{H}^{+}] }[/math] as -pH:

[math]\displaystyle{ -\ce{p}K_\ce{a} = -\ce{pH} + \log_{10} \left ( \frac{[\ce{A^-}]}{[\ce{HA}]} \right ) }[/math]

Adding pH and pK_a to both sides:

[math]\displaystyle{ \ce{pH} = \ce{p}K_\ce{a} + \log_{10} \left ( \frac{[\ce{A^-}]}{[\ce{HA}]} \right ) }[/math]

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

History

In 1908, Lawrence Joseph Henderson wrote an equation describing the use of carbonic acid as a buffer solution. Later, Karl Albert Hasselbalch re-expressed that formula in logarithmic terms, resulting in the Henderson–Hasselbalch equation. 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 binding of H+ to the base. The dissociation of water and relative water concentration 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). In high buffer dilutions, where the concentration of protons arising from water become equally or more prevalent than the buffer species themselves (at pH 7, this means buffer component concentrations of <10⁻⁵ M formally, but practically much higher), the pKa of the "buffer" system will tend towards neutrality.

Conventions

For bases, it is conventional to report [math]\displaystyle{ \mathrm{p}\!\,K_{\text{a}} }[/math] of the conjugate acid; conversely, for acids, it is conventional to report [math]\displaystyle{ \mathrm{p}\!\,K_{\text{b}} }[/math] of the conjugate base. Consequently, a base with a low [math]\displaystyle{ \mathrm{p}\!\,K_{\text{a}} }[/math] has a conjugate acid that readily dissociates and therefore that the basic compound is likely to be neutral at physiological pH. This can be contrasted with a low pK_a for an acid, which means that the acid readily dissociates and the molecule is therefore likely to exist as its conjugate base at physiological pH. To interpret literature [math]\displaystyle{ \mathrm{p}\!\,K_{\text{a}} }[/math] and [math]\displaystyle{ \mathrm{p}\!\,K_{\text{b}} }[/math] values an inspection of the molecule to determine whether it contains acid or base functionality (or both) is important.

The Henderson–Hasselbalch equation can also be rearranged to calculate the net charge of an amino acid at a certain [math]\displaystyle{ \mathrm{pH} }[/math]. Written in terms of the degree of dissociation [math]\displaystyle{ \alpha }[/math], the Henderson–Hasselbalch equation is:

[math]\displaystyle{ \mathrm{pH}=\mathrm{p}\!\,K_{\text{a}} + \log_{10}\left(\frac{\alpha}{1-\alpha}\right) }[/math];

solving for [math]\displaystyle{ \alpha }[/math] yields

[math]\displaystyle{ \alpha = \frac{1}{10^{\mathrm{p}\!\,K_{\text{a}}-\mathrm{pH}}+1} }[/math]

This equation above can be used to calculate the degree of dissociation for each ionizable group at the desired [math]\displaystyle{ \mathrm{pH} }[/math]. For ionizable groups that are able to deprotonate to a charge of -1 (e.g., carboxyl or sulfhydryl groups), multiply the calculated dissociation constant by -1. For ionizable groups that are able to deprotonate to a charge of 0, take the complement of the dissociation constant [math]\displaystyle{ (1-\alpha) }[/math] and multiply the constant by +1. The net charge of the amino acid will be the sum of the charges of all of the ionizable groups.

Estimating blood pH

The Henderson–Hasselbalch equation can be applied to relate the pH of blood to constituents of the bicarbonate buffering system:

[math]\displaystyle{ \mathrm{pH} = \mathrm{p}K_{\mathrm{a}~\mathrm{H}_2\mathrm{CO}_3} + \log_{10} \left ( \frac{[\mathrm{HCO}_3^-]}{[\mathrm{H}_2\mathrm{CO}_3]} \right ), }[/math]

where:

• pK_{a H2CO3} is the cologarithm of 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 p_CO2, that is, the partial pressure of carbon dioxide, rather than H₂CO₃. However, these are related by the equation:

[math]\displaystyle{ [\mathrm{H}_2\mathrm{CO}_3] = k_{\rm H~CO_2}\, \times p_{\mathrm{CO}_2}, }[/math]

where:

• [H₂CO₃] is the concentration of carbonic acid in the blood
• k_{H CO2} is the Henry's law constant for the solubility of carbon dioxide in blood. k_{H CO2} is approximately 0.0307 mmol/(L-torr)
• p_CO2 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:

[math]\displaystyle{ \mathrm{pH} = 6.1 + \log_{10} \left ( \frac{[\mathrm{HCO}_3^-]}{0.0307 \times p_{\mathrm{CO}_2}} \right ), }[/math]

where:

• pH is the acidity in the blood
• [HCO₃⁻] is the concentration of bicarbonate in the blood
• p_CO2 is the partial pressure of carbon dioxide in the arterial blood

See also

• Acid
• Acid base homeostasis
• Acidosis
• Alkalosis
• Base
• Buffer solution
• Titration

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. Bibcode: 2001JChEd..78.1499P. 
• de Levie, Robert. (2003). "The Henderson–Hasselbalch Equation: Its History and Limitations". J. Chem. Educ. 80 (2): 146. doi:10.1021/ed080p146. Bibcode: 2003JChEd..80..146D. 
• 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

• Applications and Example Problems Using Henderson-Hasselbalch Equation
• Derivation and detailed discussion of Henderson–Hasselbalch equation