“Henderson–Hasselbalch equation”的意思、由来-开放百科全书

In chemistry and biochemistry, the Henderson–Hasselbalch equation can be used to estimate the pH of a buffer solution containing given concentrations of an acid and its conjugate base (or a base and its conjugate acid). The numerical value of the acid dissociation constant of the acid must also be known.

History

Theory and application

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, and the concentrations of the species in solution.^[4] To derive the equation a number of simplifying assumptions have to be made.

Assumption 1: The acid is monobasic and dissociates according to the equation

It is understood that the symbol H⁺ stands for the hydrated hydronium ion. The Henderson–Hasselbalch equation can only reasonably be applied to a polybasic acid if its consecutive pK values differ by at least 3, and so all but one dissociation can be ignored.

Assumption 2: The dissociation constant, K_a can be expressed as a quotient of concentrations.

In this expression, the quantities in square brackets signify the concentration of the undissociated acid, HA, of the hydrogen ion H⁺, and of the anion A^-. It is implicit in the use of this expression that the quotient,

, of activity coefficients,

, is a constant which is independent of concentrations and pH.

With this approximation,

is proportional to the thermodynamic dissociation constant,

Assumption 3: The analytical concentrations of the acid, C_H, and of a salt of its conjugate base, C_A, are known quantities. At equilibrium the concentrations of the three species are related by the law of mass action, which can be represented in this case by the two mass-balance equations

For any given value for K_a, these are two equations in two unknown quantities, [H⁺], the concentration of hydrogen ions deriving from the acid, and [A^-], the concentration of anions deriving from the acid, HA, and its salt, MA (M = Na⁺, K⁺ (R₄N)⁺, etc.) Note that these equations can be reduced to a single quadratic equation that can be solved without further approximation.

can be ignored. This assumption is not valid with pH values more than about 10. For such instances the mass-balance equation for hydrogen must be extended to take account of the self-ionization of water.

With this extension, the pH will have to be found by solving the two mass-balance equations simultaneously for the two unknowns, [H⁺] and [A^-], or by reducing the two equations to a single cubic equation and solving that equation.

Assumption 5. In dilute solutions the concentration of undissociated acid, [HA] can be taken as equal to the total concentration of the acid, C_A.

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

This equation can be used to calculate the pH of a solution containing the acid and one of its salts, that is, of a buffer solution.

With bases, if the value of an equilibrium constant is known in the form of a base association constant, K_b the dissociation constant of the conjugate acid may be calculated from

where K_w is the self-dissociation constant of water. pK_w has a value of approximately 14 at 25C.

References

1. ^{{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. |year=1908| volume = 21 | pages = 173–179}}
2. ^{{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}}
3. ^{{Cite journal|author1=Po, Henry N. |author2=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 }}
4. ^For details and worked examples see, for instance,{{cite book |last1=Skoog |first1=Douglas A. |last2=West |first2=Donald M. |last3=Holler |first3=F. James |last4=Crouch |first4=Stanley R. |title=Fundamentals of Analytical Chemistry |date=2004 |publisher=Brooks/Cole |location=Belmont, Ca (USA) |isbn=0-03035523-0 |pages=251-263 |edition=8th}}