The Henderson-Hasselbalch Equation

The Henderson–Hasselbalch equation connects the measurable value of the pH of a solution with the theoretical value pKa.

Key Points

• The formula for the Henderson–Hasselbalch equation is$pH=p{ K }_{ a }+log(\frac { { [A }^{ - }] }{ [HA] } )$, where pH is the concentration of [H+], pKa is the acid dissociation constant, and [A-] and [HA] are concentrations of the conjugate base and starting acid.

• The equation can be used to determine the amount of acid and conjugate base needed to make a buffer of a certain pH. With a given pH and known pKa, the solution of the equation gives the logarithm of a ratio which can be solved by performing the antilogarithm of pH/pKa.

• Titration can be used to determine the pKa of a solution. In the method, the pH of the solution is constantly monitored while a titrant is slowly added to create a titration curve. To calculate the pKa, one must find the volume at the half-equivalence point.

• The equation will fail with strong acids or bases, dilute or very concentration solutions, heavily skewed acid/base ratios, or when the buffer is diluted drastically.

Terms

• of an acid (or a base) that can donate (or accept) more than one proton; polybasic

Figures

1. Titration curve for ascorbic acid

During a titration, the pH of the solution is constantly monitored while the known acid or base (called the titrant) is slowly added to the unknown solution. The pH of the unknown solution will stay fairly constant until the moles of titrant added equals the moles of unknown acid or base. When the moles of acid and base are the same, further additions of titrant will cause a dramatic change in pH until the pH eventually stabilizes. A graph of pH versus the added titrant is called a titration curve, and the point at which the pH changes drastically is called the equivalence point and can be used to determine the pKa of the compound.

The Henderson-Hasselbalch Equation

The Henderson–Hasselbalch equation mathematically connects the measurable pH of a solution, the concentration of hydrogen ions, with the pKa, the negative log of the acid dissociation constant, of the acid. The equation is also useful for estimating the pH of a buffer solution and finding the equilibrium pH in an acid-base reaction. The equation can be derived from the formula of pKa for a weak acid or buffer. The balanced equation for an acid dissociation is:

$HA\rightleftharpoons { H }^{ + }+{ A }^{ - }$

The acid dissociation constant is:

${ K }_{ a }=\frac { [{ H }^{ + }][A^{ - }] }{ [HA] }$

After taking the log of the entire equation and rearranging it, the result is:

$log({ K }_{ a })=log[{ H }^{ + }]+log(\frac { { [A }^{ - }] }{ [HA] } )$

This equation can be rewritten as:

$-p{ K }_{ a }=-pH+log(\frac { [A^{ - }] }{ [HA] } )$

Distributing the negative sign gives the final version of the Henderson-Hasselbalch Equation:

$pH=p{ K }_{ a }+log(\frac { { [A }^{ - }] }{ [HA] } )$

In an alternate application, the equation can be used to determine the amount of acid and conjugate base needed to make a buffer of a certain pH. With a given pH and known pKa, the solution of the Henderson-Hasselbach equation gives the logarithm of a ratio which can be solved by performing the antilogarithm of pH/pK­a:

${ 10 }^{ pH-p{ K }_{ a } }=\frac { [base] }{ [acid] }$

FINDING THE pH OF A SOLUTION EXPERIMENTALLY

Titration is commonly used to determine the pKa of a solution. In this method, the pH of the solution is constantly monitored while a known acid or base (called the titrant) is slowly added. The pH of the unknown solution will stay fairly constant until the moles of titrant added equals the moles of unknown acid or base. When the moles of acid and base are the same, further additions of the titrant will cause a dramatic change in pH until the pH eventually stabilizes. A graph of pH versus added titrant is called a titration curve, and the point at which the pH changes drastically is called the equivalence point.

The equivalence point on the graph is where all of the starting solution (usually an acid) has been neutralized by the titrant (usually a base). It can be calculated precisely by finding the molarity of the titration curve and computing the points of inflection.  However, in most cases, simply looking at the graph is usually enough. To calculate the pKa, one must find the volume at the half-equivalence point (that is, where half the amount of titrant has been added to form the next compound).

The titration curve for a polyprotic acid will have more than one equivalence point. As the added base completely removes each proton from the acid, the pH will jump significantly. Figure 1 shows the titration curve for ascorbic acid, which is a polyprotic acid also known as Vitamin C.  Graphing the pH versus volume of base added during an acid-base titration shows the successive ionization steps taking place. To find the concentration of a polyprotic acid, the volume of base required to reach the first equivalence point is needed. The half-equivalence points on this graph can also be used to obtain the Ka value of each successive ionization.

LIMITATIONS OF THE EQUATION

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 of 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.

Key Term Glossary

acid
an electron pair acceptor; generally capable of donating hydrogen ions
Appears in these related concepts:
acid dissociation constant
quantitative measure of the strength of an acid in solution; typically written as a ratio of the equilibrium concentrations
Appears in these related concepts:
balanced equation
The law of conservation of mass dictates the quantity of each element does not change in a chemical reaction. Thus, each side of the chemical equation must represent the same quantity of any particular element. Similarly, the charge is conserved in a chemical reaction. Therefore, the same charge must be present on both sides of the balanced equation.
Appears in these related concepts:
base
A proton acceptor, or an electron pair donor.
Appears in these related concepts:
buffer
a solution used to stabilize the pH (acidity) of a liquid
Appears in these related concepts:
buffers
a weak acid or base used to maintain the acidity (pH) of a solution near a chosen value, prevent a rapid change in pH when acids or bases are added to the solution
Appears in these related concepts:
compound
A substance made from any combination elements.
Appears in these related concepts:
concentration
the proportion of a substance in a mixture
Appears in these related concepts:
conjugate base
The species that is created after the donation of a proton.
Appears in these related concepts:
constant
Consistently recurring over time; persistent
Appears in these related concepts:
dilution
A solution that has had additional solvent, such as water, added to it into order to make it less concentrated.
Appears in these related concepts:
dissociation
Referring to the process by which compounds split into smaller constituent molecules, usually in a reversible manner.
Appears in these related concepts:
equilibrium
the state of a reaction in which the rates of the forward and reverse reactions are the same
Appears in these related concepts:
equivalence point
the point at which an added titrant is stoichiometrically equal to the number of moles of substance present in the sample: the smallest amount of titrant that is sufficient to fully neutralize or react with the analyte.
Appears in these related concepts:
hydrolysis
A chemical process of decomposition involving the splitting of a bond and the addition of the hydrogen cation and the hydroxide anion of water.
Appears in these related concepts:
ion
An atom or group of atoms bearing an electrical charge, such as the sodium and chlorine atoms in a salt solution.
Appears in these related concepts:
ionization
Any process that leads to the dissociation of a neutral atom or molecule into charged particles (ions).
Appears in these related concepts:
logarithm
For a number $x$, the power to which a given base number must be raised in order to obtain x. Written logbx. For example, log216 = 4 because 24 = 16.
Appears in these related concepts:
molarity
The concentration of a substance in solution, expressed as the number moles of solute per litre of solution.
Appears in these related concepts:
mole
In the International System of Units, the base unit of the amount of substance; the amount of substance of a system that contains as many elementary entities as there are atoms in 0.012 kg of carbon-12. Symbol: mol. The number of atoms in a mole is known as Avogadro’s number.
Appears in these related concepts:
pH
the negative of the logarithm to base 10 of the concentration of hydrogen ions, measured in moles per liter; a measure of acidity or alkalinity of a substance, which takes numerical values from 0 (maximum acidity) through 7 (neutral) to 14 (maximum alkalinity)
Appears in these related concepts:
pKa
A quantitative measure of the strength of an acid in solution; a weak acid has a pKa value in the approximate range −2 to 12 in water and a strong acid has a pKa value of less than about −2.
Appears in these related concepts:
polyprotic
of an acid (or a base) that can donate (or accept) more than one proton; polybasic
Appears in these related concepts:
polyprotic acids
Referring to an acid with multiple protons capable of dissociating from the compound.
Appears in these related concepts:
proton
A positively charged subatomic particle forming part of the nucleus of an atom and determining the atomic number of an element; the nucleus of the most common isotope of hydrogen; composed of two up quarks and a down quark
Appears in these related concepts:
ratio
The relative magnitudes of two quantities (usually expressed as a quotient).
Appears in these related concepts:
solution
A homogeneous mixture, which may be liquid, gas or solid, formed by dissolving one or more substances.
Appears in these related concepts:
Solution
A homogeneous mixture, which may be liquid, gas or solid, formed by dissolving one or more substances.
Appears in these related concepts:
strong acid
A strong acid is one that completely ionizes (dissociates) in water; in other words, one mole of a strong acid HA dissolves in water yielding one mole of H+ and one mole of the conjugate base, A−.
Appears in these related concepts:
titrant
the reagent of known concentration and volume used in titrations
Appears in these related concepts:
titration
the determination of the concentration of some substance in a solution by slowly adding measured amounts of some other substance (normally using a burette) until a reaction is shown to be complete -- for instance, by the color change of an indicator
Appears in these related concepts:
volume
A unit of three-dimensional measure of space that comprises a length, a width, and a height. It is measured in units of cubic centimeters in metric, or cubic inches or cubic feet in English measurement.
Appears in these related concepts:
weak acid
one that dissociates incompletely, releasing only some of its hydrogen atoms into solution