The Henderson-Hasselbalch equation

The key variable that determines [H30 ] is the concentration ratio of acid species to base species, so rearranging to isolate [H3O ] gives 

Taking the negative common logarithm (base 10) of both sides gives 

Generalizing the previous equation for any conjugate acid-base pair gives the Henderson-Hasselbalch equation  

This relationship is very useful for two reasons. First, it allows us to solve directly for pH instead of having to calculate [H30 j first. For instance, by applying the Henderson-Hasselbalch equation in part (b) of Sample Problem 19.1, we could have found the pH of the buffer after the addition of NaOH as follows  

Second, as we ll see shortly, it allows us to prepare a buffer of a desired pH just by mixing the appropriate amounts of A and HA. 

We can derive an equation that relates the pH of a buffer solution to the initial concentration of the buffer components, thus simpUlying the calculation of the pH of a buffer solution. Consider a buffer containing the generic weak acid HA and its conjugate base A . The acid ionizes as follows  

We derive an expression for the concentration of H30 from the acid ionization equilibrium expression by solving the expression for [H30 ]  

If we make the same x is small approximation that we make for weak acid or weak base equilibrium problems, we can consider the equilibrium concentrations of HA and A to be essentially identical to the initial concentrations of HA and A (see step 4 of Example 16.1). Therefore, to determine [H30 ] for any buffer solution, we multiply by the ratio of the 

Recall that the variable x in a weak acid equilihrium prohlem represents the change in the initial acid concentration. The x / sma// approximation is valid because so little of the weak acid ionizes compared to its Initial concentration. 

Note that, as expected, the pH of a buffer increases with an increase in the amount of base reiative to the amount of acid. 

For the ionization of a weak acid, we calculated above that the pH is given by the equation 

In both cases, we are making the assumption that the concentration of the ion (either AcO or NH4 ) is not signihcantly altered by the equilibrium and can, therefore, be considered to be equivalent to the molar concentration. 

Taking negative logarithms of each side, this becomes 

This is referred to as the Henderson-Hasselbalch eqnation, and it is sometimes written as 

Using this relationship, it is possible to determine the degree of ionization of an acid at a given pH. 

The central equation for buffers is the Henderson-Hasselbalch equation, which is merely a rearranged form of the equilibrium expression  

Useful logarithm rules logxy = logx + logy logx/y = logx- logy log x = y log X 

Henderson was a physician who wrote [H l = /CJacidJ/Isalt] in a physiology article in 1908, a year before the word buffer and the concept of pH were invented by the biochemist Sorensen. Henderson s contribution was the approximation of setting [acid] equal to the concentration of HA placed in solution and [salt] equal to the concentration of A placed in solution. In 1916, K. A. Hasselbalch wrote what we call the Henderson-Hasselbalch equation in a biochemical journal.  

The Henderson-Hasselbalch equation tells us the pH of a solution, provided we know the ratio of concentrations of conjugate acid and base, as well as p/ a for the acid. For the weak base B and its conjugate acid, the analogous equation is 

The Henderson-Hasselbalch equation tells us that a factor-of-10 change in the ratio [A ]/[HA] changes the pH by one unit (Table 9-1). As [A ] increases, pH goes up. As [HA] increases, the pH goes down. For any conjugate acid-base pair, you can say, for example, that, if pH = pK. - 1, there must be 10 times as much HA as A . HA is ten-elevenths of the mixture and A is one-eleventh. 

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This relationship is one form of the Henderson-Hasselbalch equation It is a useful relationship m chemistry and biochemistry One rarely needs to cal culate the pH of a solution—pH is more often mea sured than calculated It is much more common that one needs to know the degree of ionization of an acid at a particular pH and the Henderson-Hasselbalch equation gives that ratio... 

An alternative form of the Henderson-Hasselbalch equation for acetic acid is... 

As in Example 6.13, the Henderson-Hasselbalch equation provides a simple way to calculate the pH of a buffer and to determine the change in pH upon adding a strong acid or strong base. 

Substituting the new concentrations into the Henderson-Hasselbalch equation gives a pH of... 

Multiprotic weak acids can be used to prepare buffers at as many different pH s as there are acidic protons. For example, a diprotic weak acid can be used to prepare buffers at two pH s and a triprotic weak acid can be used to prepare three different buffers. The Henderson-Hasselbalch equation applies in each case. Thus, buffers of malonic acid (pKai = 2.85 and = 5.70) can be prepared for which... 

Although this treatment of buffers was based on acid-base chemistry, the idea of a buffer is general and can be extended to equilibria involving complexation or redox reactions. For example, the Nernst equation for a solution containing Fe + and Fe + is similar in form to the Henderson-Hasselbalch equation. 

Suppose you need to prepare a buffer with a pH of 9.36. Using the Henderson-Hasselbalch equation, you calculate the amounts of acetic acid and sodium acetate needed and prepare the buffer. When you measure the pH, however, you find that it is 9.25. If you have been careful in your calculations and measurements, what can account for the difference between the obtained and expected pHs In this section, we will examine an important limitation to our use of equilibrium constants and learn how this limitation can be corrected. 

Any solution containing comparable amounts of a weak acid, HA, and its conjugate weak base, A-, is a buffer. As we learned in Chapter 6, we can calculate the pH of a buffer using the Henderson-Hasselbalch equation. 

The dissociation constant is most accurately estimated from kinetic data when all of the data points are used in the evaluation. There are several ways to do this. The Henderson-Hasselbalch equation... 

This relationship is known as the Henderson-Hasselbalch equation. Thus, the pH of a solution can be calculated, provided and the concentrations of the weak acid HA and its conjugate base A are known. Note particularly that when [HA] = [A ], pH = pAl,. For example, if equal volumes of 0.1 MHAc and 0.1 M sodium acetate are mixed, then... 

The Henderson-Hasselbalch equation provides a general solution to the quantitative treatment of acid-base equilibria in biological systems. Table 2.4 gives the acid dissociation constants and values for some weak electrolytes of biochemical interest. 

If the pfC, value of a given acid and the pH of the medium are knowrn, the percentages of dissociated and undissociated forms can be calculated using what is called the Henderson-Hasselbalch equation. 

As an example of how to use the Henderson-Hasselbalch equation, let s find out what species are present in a 0.0010 M solution of acetic acid at pH = 7.3. According to Table 20.3, the pKa of acetic acid is 4.76. From the Henderson-Hasselbalch equation, we have... 

We saw in Section 20.3 that the extent of dissociation of a carboxylic acid HA in an aqueous solution buffered to a given pH can be calculated with the Henderson-Hasselbalch equation. Furthermore, we concluded that at the physiological... 

Amino Acids, the Henderson-Hasselbalch Equation, and Isoelectric Points... 

To apply the Henderson-Hasselbalch equation to an amino acid, let s find out what species are present in a 1.00 M solution of alanine at pH = 9.00. According to Table 26.1, protonated alanine [ NCHfCH CC H] has p/Cal =2.34, and neutral zwitteTionlc alanine [+ll3NCH(CH3)C02-] has pK52 = 9.69 ... 

Figure 26.1 A titration curve for alanine, plotted using the Henderson-Hasselbalch equation. Each of the two legs is plotted separately. At pH < 1, alanine is entirely protonated at pH = 2.34, alanine is a 50 50 mix of protonated and neutral forms at pH 6.01, alanine is entirely neutral at pH = 9.69, alanine is a 50 50 mix of neutral and deprotonated forms at pH > 11.5, alanine is entirely deprotonated.

Chapter 20, Carboxylic Aciils and Nitriles—A new Section 20.3 discusses biological carboxylic acids and the Henderson-Hasselbalch equation. 

This relation, known as the Henderson-Hasselbalch equation, is often used in biology and biochemistry to calculate the pH of buffers. Historically, it was Henderson who discovered Equation 14.1 in 1908. Hasselbalch put it in logarithmic form eight years later. 

The values of [HA] and [A ] in this expression are the equilibrium concentrations of acid and base in the solution, not the concentrations added initially. However, a weak acid HA typically loses only a tiny fraction of its protons, and so [HA] is negligibly different from the concentration of the acid used to prepare the buffer, [HA]initia. Likewise, only a tiny fraction of the weakly basic anions A- accept protons, and so [A-] is negligibly different from the initial concentration of the base used to prepare the buffer. With the approximations A ] [base]initia and [HA] [acid]initia, we obtain the Henderson-Hasselbalch equation ... 

A note on good practice Keep in mind the approximations required for the use of the Henderson-Hasselbalch equation (that the concentrations of both the weak acid and its conjugate base are much greater than the hydronium ion concentration). Because the equation uses molar concentration instead of activities, it also ignores the interactions between ions. 

In practice, the Henderson-Hasselbalch equation is used to make rapid estimates of the pH of a mixed solution intended to be used as a buffer, and then the pH is adjusted to the precise value required by adding more acid or base and monitoring the solution with a pH meter. 

We can use these numbers to express the range of buffer action in terms of the pH of the solution. The Henderson-Hasselbalch equation shows us that,... 

Step 5 Use an equilibrium table to find the H.O concentration in a weak acid or the OH concentration in a weak base. Alternatively, if the concentrations of conjugate acid and base calculated in step 4 are both large relative to the concentration of hydronium ions, use them in the expression for /<, or the Henderson—Hasselbalch equation to determine the pH. In each case, if the pH is less than 6 or greater than 8, assume that the autoprotolysis of water does not significantly affect the pH. If necessary, convert between Ka and Kh by using Kw = KA X Kb. 

The Henderson-Hasselbalch Equation Describes the Behavior of Weak Acids Buffers... 

The Henderson-Hasselbalch equation is derived below. A weak acid, HA, ionizes as follows ... 

The Henderson-Hasselbalch equation has great predictive value in protonic equilibria. For example,... 

The buffer equation, which is often called the Henderson-Hasselbalch equation, is used to calculate the equilibrium pH of a buffer solution directly from initial concentrations. The approximation is valid as long as the difference between initial concentrations and equilibrium concentrations is negligibly small. As a rule of thumb, the buffer equation can be applied when initial concentrations of H j4 and A differ by less than a factor of 10. Example provides an illustration of the use of the buffer equation. 

Generally it is only the non-dissociated or unionised drug that is lipid-soluble and a drug s degree of ionisation depends on its dissociation constant (pA) and the pH of the environment in which it finds itself. For an acidic drug this is represented by the Henderson Hasselbalch equation as... 

Absorption, the Henderson-Hasselbalch Equation and the pH-partition Hypothesis... 

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