Easy Way to Figure Out Henderson Hasselbalch Equation

Equation used to estimate pH of a weak acid or base solution

In chemistry and biochemistry,
the Henderson–Hasselbalch equation

$${\displaystyle {\ce {pH}}={\ce {p}}K_{{\ce {a}}}+\log _{10}\left({\frac {[{\ce {Base}}]}{[{\ce {Acid}}]}}\right)}$$

relates the pH of a chemic solution of a weak acid to the numerical value of the acid dissociation constant, K _a, of acid and the ratio of the concentrations, ${\displaystyle {\frac {[{\ce {Base}}]}{[{\ce {Acid}}]}}}$ of the acid and its cohabit base in an equilibrium.

${\displaystyle \mathrm {{\underset {(acid)}{HA}}\leftrightharpoons {\underset {(base)}{A^{-}}}+H^{+}} }$

For example, the acid may be acetic acid

${\displaystyle \mathrm {CH_{3}CO_{2}H\leftrightharpoons CH_{3}CO_{2}^{-}+H^{+}} }$

The Henderson–Hasselbalch equation tin be used to approximate the pH of a buffer solution by approximating the actual concentration ratio equally the ratio of the analytical concentrations of the acid and of a table salt, MA.

The equation can too exist practical to bases by specifying the protonated form of the base as the acid. For case, with an amine, ${\displaystyle \mathrm {RNH_{2}} }$

${\displaystyle \mathrm {RNH_{3}^{+}\leftrightharpoons RNH_{2}+H^{+}} }$

Derivation, assumptions and limitations [edit]

A unproblematic buffer solution consists of a solution of an acid and a common salt of the cohabit base of the acid. For case, the acid may be acerb acid and the table 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.^[one]

Faux titration of an acidified solution of a weak acid (pThou _a = iv.seven) with alkali

To derive the equation a number of simplifying assumptions have to be made.^[2] (pdf)

Assumption 1: The acid, HA, is monobasic and dissociates according to the equations

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

${\displaystyle \mathrm {C_{A}=[A^{-}]+[H^{+}][A^{-}]/K_{a}} }$

${\displaystyle \mathrm {C_{H}=[H^{+}]+[H^{+}][A^{-}]/K_{a}} }$

C_A is the analytical concentration of the acrid 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 chemic substance Ten. It is understood that the symbol H⁺ stands for the hydrated hydronium ion. Grand_a is an acid dissociation constant.

The Henderson–Hasselbalch equation can be applied to a polybasic acid but if its consecutive pK values differ by at least three. Phosphoric acid is such an acrid.

Assumption 2. The self-ionization of h2o can be ignored. This assumption is not, strictly speaking, valid with pH values close to seven, half the value of pK_w, the constant for self-ionization of water. In this case the mass-balance equation for hydrogen should exist extended to take account of the self-ionization of h2o.

${\displaystyle \mathrm {C_{H}=[H^{+}]+[H^{+}][A^{-}]/K_{a}+K_{w}/[H^{+}]} }$

However, the term ${\displaystyle \mathrm {K_{w}/[H^{+}]} }$ can exist omitted to a proficient approximation.^[2]

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

${\displaystyle \mathrm {Na(CH_{3}CO_{ii})\rightarrow Na^{+}+CH_{three}CO_{2}^{-}} }$

the concentration of the sodium ion, [Na⁺] can be ignored. This is a expert approximation for i:i electrolytes, but not for salts of ions that have a higher charge such with magnesium sulphate, Mg(SO₄)₂, that form ion pairs.

Supposition 4: The caliber of activity coefficients, ${\displaystyle \Gamma }$ , is a constant under the experimental conditions covered past the calculations.

The thermodynamic equilibrium constant, ${\displaystyle M^{*}}$ ,

${\displaystyle Thou^{*}={\frac {[{\ce {H+}}][{\ce {A^-}}]}{[{\ce {HA}}]}}\times {\frac {\gamma _{{\ce {H+}}}\gamma _{{\ce {A^-}}}}{\gamma _{HA}}}}$

is a product of a quotient of concentrations ${\displaystyle {\frac {[{\ce {H+}}][{\ce {A^-}}]}{[{\ce {HA}}]}}}$ and a caliber, ${\displaystyle \Gamma }$ , of activity coefficients ${\displaystyle {\frac {\gamma _{{\ce {H+}}}\gamma _{{\ce {A^-}}}}{\gamma _{HA}}}}$ . 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 ${\displaystyle \gamma }$ are the corresponding activity coefficients. If the quotient of activity coefficients tin can be assumed to be a abiding which is contained of concentrations and pH, the dissociation constant, K _a can exist expressed as a quotient of concentrations.

$${\displaystyle K_{a}=K^{*}/\Gamma ={\frac {[{\ce {H+}}][{\ce {A^-}}]}{[{\ce {HA}}]}}}$$

Rearrangement of this expression and taking logarithms provides the Henderson–Hasselbalch equation

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

Application to bases [edit]

The equilibrium constant for the protonation of a base, B,

B (base) + H⁺ ⇌ BH⁺ (acid)

is an association constant, K _b, which is simply related to the dissociation constant of the conjugate acid, BH⁺.

${\displaystyle \mathrm {pK_{a}=\mathrm {pK_{w}} -\mathrm {pK_{b}} } }$

The value of ${\displaystyle \mathrm {pK_{w}} }$ 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 [edit]

With homeostasis the pH of a biological solution is maintained at a abiding value by adjusting the position of the equilibria

${\displaystyle \mathrm {HCO_{iii}^{-}} +\mathrm {H^{+}} \rightleftharpoons \mathrm {H_{2}CO_{3}} \rightleftharpoons CO_{2}+H_{2}O}$

where ${\displaystyle \mathrm {HCO_{3}^{-}} }$ is the bicarbonate ion and ${\displaystyle \mathrm {H_{2}CO_{iii}} }$ is carbonic acid. However, the solubility of carbonic acid in water may be exceeded. When this happens carbon dioxide gas is liberated and the post-obit equation may be used instead.

${\displaystyle \mathrm {[H^{+}][HCO_{three}^{-}]} =\mathrm {Chiliad^{m}[CO_{2}(g)]} }$

${\displaystyle \mathrm {CO_{2}(m)} }$ represents the carbon dioxide liberated equally gas. In this equation, which is widely used in biochemistry, ${\displaystyle K^{m}}$ is a mixed equilibrium constant relating to both chemic and solubility equilibria. It tin can be expressed as

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

where [HCO ⁻
_three ] is the molar concentration of bicarbonate in the claret plasma and P _CO2 is the partial pressure of carbon dioxide in the supernatant gas.

History [edit]

In 1908, Lawrence Joseph Henderson^[3] derived an equation to calculate the hydrogen ion concentration of a bicarbonate buffer solution, which rearranged looks like this:

[H⁺] [HCO₃ ^–] = K [CO₂] [H₂O]

In 1909 Søren Peter Lauritz Sørensen introduced the pH terminology, which allowed Karl Albert Hasselbalch to re-express Henderson's equation in logarithmic terms,^[4] resulting in the Henderson–Hasselbalch equation.

Meet also [edit]

Davenport diagram

Further reading [edit]

Davenport, Horace Westward. (1974). The ABC of Acid-Base Chemistry: The Elements of Physiological Claret-Gas Chemistry for Medical Students and Physicians (Sixth ed.). Chicago: The University of Chicago Printing.

References [edit]

1. ^ For details and worked examples see, for instance, Skoog, Douglas A.; West, Donald M.; Holler, F. James; Crouch, Stanley R. (2004). Fundamentals of Belittling Chemistry (eighth ed.). Belmont, Ca (USA): Brooks/Cole. pp. 251–263. ISBN0-03035523-0.
2. ^ ^{a b} 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.
3. ^ Lawrence J. Henderson (1908). "Apropos the relationship between the strength of acids and their chapters to preserve neutrality". Am. J. Physiol. 21 (2): 173–179. doi:10.1152/ajplegacy.1908.21.two.173.
4. ^ Hasselbalch, Grand. 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.

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Source: https://en.wikipedia.org/wiki/Henderson%E2%80%93Hasselbalch_equation

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