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| In [[chemistry]], the '''Henderson–Hasselbalch equation''' describes the derivation of [[pH]] as a measure of acidity (using pK<sub>a</sub>, 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 [[Chemical equilibrium|equilibrium]] pH in [[acid-base reaction theories|acid-base reactions]] (it is widely used to calculate the [[isoelectric point]] of proteins).
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| The equation is given by:
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| :<math>\textrm{pH} = \textrm{pK}_{a}+ \log_{10} \left ( \frac{[\textrm{A}^-]}{[\textrm{HA}]} \right )</math>
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| Here, [HA] is the molar concentration of the undissociated weak acid, [A⁻] is the molar concentration of this acid's conjugate base and <math>\textrm{pK}_{a}</math> is <math>-\log (K_{a})</math> where <math>K_{a}</math> is the [[acid dissociation constant]], that is:
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| :<math>\textrm{pK}_{a} = - \log_{10} (K_{a}) = - \log_{10} \left ( \frac{[\mbox{H}_{3}\mbox{O}^+][\mbox{A}^-]}{[\mbox{HA}]} \right )</math> for the non-specific Brønsted acid-base reaction: <math>\mbox{HA} + \mbox{H}_{2}\mbox{O} \rightleftharpoons \mbox{A}^- + \mbox{H}_{3}\mbox{O}^+</math>
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| A second form of the equation, known as the Heylman Equation, expressed in terms of <math>K_{b}</math> where <math>K_{b}</math> is the [[acid dissociation constant#Bases|base dissociation constant]]:
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| <math>\textrm{pK}_{b} = - \log_{10} (K_{b}) = - \log_{10} \left ( \frac{[\mbox{O}\mbox{H}^-][\mbox{HA}]}{[\mbox{A}^-]} \right )</math>
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| In these equations, <math>\mbox{A}^-</math> denotes the ionic form of the relevant acid. Bracketed quantities such as [base] and [acid] denote the molar concentration of the quantity enclosed.
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| In analogy to the above equations, the following equation is valid:
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| :<math>\textrm{pOH} = \textrm{pK}_{b}+ \log_{10} \left ( \frac{[\textrm{BH}^+]}{[\textrm{B}]} \right )</math>
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| Where BH<sup>+</sup> denotes the conjugate acid of the corresponding base B.
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| ==Derivation==
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| The Henderson–Hasselbalch equation is derived from the [[acid dissociation constant]] equation by the following steps:<ref>[http://pharmaxchange.info/press/2012/10/henderson%E2%80%93hasselbalch-equation-derivation-of-pka-and-pkb/ Henderson Hasselbalch Equation: Derivation of pKa and pKb]</ref>
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| :<math>K_\textrm{a} = \frac{[\textrm{H}^+][\textrm{A}^-]} {[\textrm{HA}]}</math>
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| :<math>\log_{10}K_\textrm{a} = \log_{10} \left ( \frac{[\textrm{H}^+][\textrm{A}^-]}{[\textrm{HA}]} \right )</math>
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| :<math>\log_{10}K_\textrm{a} = \log_{10}[\textrm{H}^+] + \log_{10} \left ( \frac{[\textrm{A}^-]}{[\textrm{HA}]} \right )</math>
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| :<math>-\textrm{p}K_\textrm{a} = -\textrm{pH} + \log_{10} \left ( \frac{[\textrm{A}^-]}{[\textrm{HA}]} \right )</math>
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| :<math>\textrm{pH} = \textrm{p}K_\textrm{a} + \log_{10} \left ( \frac{[\textrm{A}^-]}{[\textrm{HA}]} \right )</math>
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| The ratio <math>[A^-]/[HA]</math> is unitless, and as such, other ratios with other units may be used. For example, the mole ratio of the components, <math>n_{A^-}/n_{HA}</math> or the fractional concentrations <math>\alpha_{A^-}/\alpha_{HA}</math> where <math>\alpha_{A^-}+\alpha_{HA}=1</math> will yield the same answer. Sometimes these other units are more convenient to use.
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| ==History==
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| [[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 [[logarithm]]ic terms, resulting in the Henderson–Hasselbalch equation [http://www.acid-base.com/history.php]. Hasselbalch was using the formula to study [[metabolic acidosis]].
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| ==Limitations==
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| 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 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).
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| 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.
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| ==Estimating blood pH==
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| The Henderson–Hasselbalch equation can be applied to relate the pH of [[blood]] to constituents of the [[bicarbonate buffering system]]:<ref name=Bray1999>[http://books.google.com.my/books?id=qyHu0Iu-XOUC&pg=PA556 page 556], section "Estimating plasma pH" in: {{Cite book | last1 = Bray | first1 = John J. | title = Lecture notes on human physiolog | year = 1999 | publisher = Blackwell Science | location = Malden, Mass. | isbn = 978-0-86542-775-4 | pages = }}</ref>
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| :<math> pH = pK_{a~H_2CO_3}+ \log_{10} \left ( \frac{[HCO_3^-]}{[H_2CO_3]} \right )</math>
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| , where:
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| *pK<sub>a H<sub>2</sub>CO<sub>3</sub></sub> is the [[cologarithm]] of the [[acid dissociation constant]] of [[carbonic acid]]. It is equal to 6.1.
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| *[HCO<sub>3</sub><sup>-</sup>] is the concentration of [[bicarbonate]] in the blood
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| *[H<sub>2</sub>CO<sub>3</sub>] is the concentration of carbonic acid in the blood
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| This is useful in [[arterial blood gas]], but these usually state ''pCO<sub>2</sub>'', that is, the [[partial pressure]] of [[carbon dioxide]], rather than H<sub>2</sub>CO<sub>3</sub>. However, these are related by the equation:<ref name=Bray1999/>
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| :<math> [H_2CO_3] = k_{\rm H~CO_2}\, \times pCO_2 </math>
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| , where:
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| *[H<sub>2</sub>CO<sub>3</sub>] is the concentration of carbonic acid in the blood
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| *''k<sub>H CO<sub>2</sub></sub>'' is the [[Henry's law]] constant for the [[solubility]] of carbon dioxide in blood. ''k<sub>H CO<sub>2</sub></sub>'' is approximately 0.03 [[millimole|mmol]]/[[mmHg]]
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| *''pCO<sub>2</sub>'' is the [[partial pressure]] of [[carbon dioxide]] in the blood
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| 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:<ref name=Bray1999/>
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| :<math> pH = 6.1 + \log_{10} \left ( \frac{[HCO_3^-]}{0.03 \times pCO_2} \right )</math>
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| , where:
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| *pH is the acidity in the blood
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| *[HCO<sub>3</sub><sup>-</sup>] is the concentration of bicarbonate in the blood
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| *''pCO<sub>2</sub>'' is the partial pressure of carbon dioxide in the blood
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| ==See also==
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| *[[Acid]]
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| *[[base (chemistry)|Base]]
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| *[[Titration]]
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| *[[Buffer solution]]
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| *[[Acidosis]]
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| *[[Alkalosis]]
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| * {{Cite journal| author = Lawrence J. Henderson | title = Concerning the relationship between the strength of acids and their capacity to preserve neutrality | journal = [[Am. J. Physiol.]] | date=1 May 1908| volume = 21 | pages = 173–179 | url = http://ajplegacy.physiology.org/cgi/content/abstract/21/4/465-s | format = Abstract | issue = 4 }}
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| * {{Cite journal| author = Hasselbalch, K. A. | title = Die Berechnung der Wasserstoffzahl des Blutes aus der freien und gebundenen Kohlensäure desselben, und die Sauerstoffbindung des Blutes als Funktion der Wasserstoffzahl | journal = [[Biochemische Zeitschrift]] | year = 1917 | volume = 78 | pages = 112–144}}
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| * {{Cite journal| author = Po, Henry N.; Senozan, N. M. | title = Henderson–Hasselbalch Equation: Its History and Limitations | journal = [[J. Chem. Educ.]] | year = 2001 | volume = 78 | pages = 1499–1503 | doi = 10.1021/ed078p1499| issue = 11|bibcode = 2001JChEd..78.1499P }}
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| * {{Cite journal| author = de Levie, Robert. | title = The Henderson–Hasselbalch Equation: Its History and Limitations | journal = [[J. Chem. Educ.]] | year = 2003 | volume = 80 | pages = 146 | doi = 10.1021/ed080p146| issue = 2|bibcode = 2003JChEd..80..146D }}
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| * {{Cite journal| author = de Levie, Robert | journal = [[The Chemical Educator]] | year = 2002 | volume = 7 | pages = 132–135 | doi = 10.1007/s00897020562a | title = The Henderson Approximation and the Mass Action Law of Guldberg and Waage| issue = 3}}
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| ==External links==
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| *[http://www.changbioscience.com/calculator/HendersonHasselbach.html Henderson–Hasselbalch Calculator]
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| *[http://www.chembuddy.com/?left=pH-calculation&right=pH-buffers-henderson-hasselbalch Derivation and detailed discussion of Henderson–Hasselbalch equation]
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| *[http://isoelectric.ovh.org True example of using Henderson–Hasselbalch equation for calculation net charge of proteins]
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| {{Use dmy dates|date=September 2010}}
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| {{DEFAULTSORT:Henderson-Hasselbalch Equation}}
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| [[Category:Acid-base chemistry]]
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| [[Category:Equilibrium chemistry]]
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| [[Category:Equations]]
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| [[Category:Mathematics in medicine]]
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| [[Category:Respiratory therapy]]
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Hello friend. Let me introduce ourselves. I am Merlin. One of his favorite hobbies is caving and he's been doing it for a while. After being out of his work for years he became a sale clerk but he's always wanted her own business. Nevada is where he's for ages been living. Go to my website unearth out more: http://m.scalarenergypendants.com/
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