Henderson-Hasselbalch Equation

In chemistry and biochemistry, the Henderson–Hasselbalch equation

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

can be used to estimate the pH of a buffer solution. The numerical value of the acid dissociation constant, K_a, of the acid is known or assumed. The pH is calculated for given values of the concentrations of the acid, HA and of a salt, MA, of its conjugate base, A^-; for example, the solution may contain acetic acid and sodium acetate.

Contents

• 1History
• 2Theory
• 3Application
• 4References

History[edit]

In 1908, Lawrence Joseph Henderson derived an equation to calculate the pH of a buffer solution.^[1] In 1917, Karl Albert Hasselbalch re-expressed that formula in logarithmicterms,^[2] resulting in the Henderson–Hasselbalch equation.

Theory[edit]

Main article: Acid dissociation constant

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.^[3] To derive the equation a number of simplifying assumptions have to be made. The mixture which has ability to resist change in pH when small amount of acid or base or water is added to it, is called Buffer solution.

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

${\displaystyle {\ce {HA}}\leftrightharpoons {\ce {H^+}}+{\ce {A^-}}}$

It is understood that the symbol H⁺ stands for the hydrated hydronium ion. The Henderson–Hasselbalch equation can be applied to a polybasic acid only if its consecutive pK values differ by at least 3. Phosphoric acid is such an acid.

Assumption 2. The self-ionization of water 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.

C_H = [H⁺] + K_a[H⁺][A⁻]- K_w[H⁺]⁻¹

C_A = [A⁻] + K_a[H⁺][A⁻]

and the pH will have to be found by solving the two mass-balance equations simultaneously for the two unknowns, [H⁺] and [A⁻].

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

Na(CH₃CO₂) → Na⁺ + CH₃CO₂^-

Assumption 4: The quotient of activity coefficients, ${\displaystyle \Gamma }$, is a constant under the experimental conditions covered by the calculations.

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

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

is a product of a quotient of concentrations ${\displaystyle {\frac {[{\ce {H+}}][{\ce {A^-}}]}{[{\ce {HA}}]}}}$ and a quotient, ${\displaystyle \Gamma }$, 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 can be assumed to be a constant which is independent of concentrations and pH, the dissociation constant, K_a can be 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[edit]

The Henderson–Hasselbalch 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

pK_a + pK_b = pK_w

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

If the "free acid" concentration, [HA], can be taken to be equal to the analytical concentration of the acid, T_AH (sometimes denoted as C_AH) an approximation is possible, which is widely used in biochemistry; it is valid for very dilute solutions.

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

The effect of this approximation is to introduce an error in the calculated pH, which becomes significant at low pH and high acid concentration. With bases the error becomes significant at high pH and high base concentration.^[4] (pdf)

Learning Objective

• Calculate the pH of a buffer system using the Henderson-Hasselbalch equation.

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Key Points

• The Henderson-Hasselbalch equation is useful for estimating the pH of a buffer solution and finding the equilibrium pH in an acid-base reaction.
• The formula for the Henderson–Hasselbalch equation is: [latex]pH=p{ K }_{ a }+log(\frac { { [A }^{ - }] }{ [HA] } )[/latex], where pH is the concentration of [H+], pK_a 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 solution of a certain pH.

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Term

• pKaA 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.

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The Henderson–Hasselbalch equation mathematically connects the measurable pH of a solution with the pK_a (which is equal to -log K_a) 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 pK_a for a weak acid or buffer. The balanced equation for an acid dissociation is:

[latex]HA\rightleftharpoons { H }^{ + }+{ A }^{ - }[/latex]

The acid dissociation constant is:

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

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

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

This equation can be rewritten as:

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

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

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

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 pK_a, the solution of the Henderson-Hasselbalch equation gives the logarithm of a ratio which can be solved by performing the antilogarithm of pH/pK­_a:

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

An example of how to use the Henderson-Hasselbalch equation to solve for the pH of a buffer solution is as follows:

What is the pH of a buffer solution consisting of 0.0350 M NH₃ and 0.0500 M NH₄⁺ (K_a for NH₄⁺ is 5.6 x 10^-10)? The equation for the reaction is:

[latex]{NH_4^+}\rightleftharpoons { H }^{ + }+{ NH_3}[/latex]

Assuming that the change in concentrations is negligible in order for the system to reach equilibrium, the Henderson-Hasselbalch equation will be:

[latex]pH=p{ K }_{ a }+log(\frac { { [NH_3}] }{ [NH_4^+] } )[/latex]

[latex]pH=9.25+log(\frac{0.0350}{0.0500} )[/latex]

pH = 9.095

The pH is a measure of the concentration of hydrogen ions in an aqueous solution. pKa (acid dissociation constant) and pH are related, but pKa is more specific in that it helps you predict what a molecule will do at a specific pH. Essentially, pKa tells you what the pH needs to be in order for a chemical species to donate or accept a proton.

The relationship between pH and pKa is described by the Henderson-Hasselbalch equation.

pH, pKa, and Henderson-Hasselbalch Equation

• The pKa is the pH value at which a chemical species will accept or donate a proton.
• The lower the pKa, the stronger the acid and the greater the ability to donate a proton in aqueous solution.
• The Henderson-Hasselbalch equation relates pKa and pH. However, it is only an approximation and should not be used for concentrated solutions or for extremely low pH acids or high pH bases.

pH and pKa

Once you have pH or pKa values, you know certain things about a solution and how it compares with other solutions:

• The lower the pH, the higher the concentration of hydrogen ions, [H+].
• The lower the pKa, the stronger the acid and the greater its ability to donate protons.
• pH depends on the concentration of the solution. This is important because it means a weak acid could actually have a lower pH than a diluted strong acid. For example, concentrated vinegar (acetic acid, which is a weak acid) could have a lower pH than a dilute solution of hydrochloric acid (a strong acid).
• On the other hand, the pKa value is a constant for each type of molecule. It is unaffected by concentration.
• Even a chemical ordinarily considered a base can have a pKa value because the terms "acids" and "bases" simply refer to whether a species will give up protons (acid) or remove them (base). For example, if you have a base Y with a pKa of 13, it will accept protons and form YH, but when the pH exceeds 13, YH will be deprotonated and become Y. Because Y removes protons at a pH greater than the pH of neutral water (7), it is considered a base.

Relating pH and pKa With the Henderson-Hasselbalch Equation

If you know either pH or pKa, you can solve for the other value using an approximation called the Henderson-Hasselbalch equation:

pH = pKa + log ([conjugate base]/[weak acid])
pH = pka+log ([A-]/[HA])

pH is the sum of the pKa value and the log of the concentration of the conjugate base divided by the concentration of the weak acid.

At half the equivalence point:

pH = pKa

It's worth noting sometimes this equation is written for the Ka value rather than pKa, so you should know the relationship: 

pKa = -logKa

Assumptions for the Henderson-Hasselbalch Equation

The reason the Henderson-Hasselbalch equation is an approximation is because it takes water chemistry out of the equation. This works when water is the solvent and is present in a very large proportion to the [H+] and acid/conjugate base. You shouldn't try to apply the approximation for concentrated solutions. Use the approximation only when the following conditions are met:

• −1 < log ([A−]/[HA]) < 1
• Molarity of buffers should be 100x greater than that of the acid ionization constant Ka.
• Only use strong acids or strong bases if the pKa values fall between 5 and 9.

Example pKa and pH Problem

Find [H+] for a solution of 0.225 M NaNO2 and 1.0 M HNO2. The Ka value (from a table) of HNO2 is 5.6 x 10-4.

pKa = −log Ka = −log(7.4×10−4) = 3.14

pH = pka + log ([A-]/[HA])

pH = pKa + log([NO2-]/[HNO2])

pH = 3.14 + log(1/0.225)

pH = 3.14 + 0.648 = 3.788

[H+] = 10−pH = 10−3.788 = 1.6×10−4

If you're working with acids and bases, two familiar values are pH and pKa. Here is the definition of pKa and a look at how it relates to acid strength.

pKa Definition

pKa is the negative base-10 logarithm of the acid dissociation constant (Ka) of a solution.
pKa = -log10Ka
The lower the pKa value, the stronger the acid. For example, the pKa of acetic acid is 4.8, while the pKa of lactic acid is 3.8. Using the pKa values, one can see lactic acid is a stronger acid than acetic acid.

The reason pKa is used is because it describes acid dissociation using small decimal numbers. The same type of information may be obtained from Ka values, but they are typically extremely small numbers given in scientific notation that are hard for most people to understand.

Key Takeaways: pKa Definition

• The pKa value is one method used to indicate the strength of an acid.
• pKa is the negative log of the acid dissociation constant or Ka value.
• A lower pKa value indicates a stronger acid. That is, the lower value indicates the acid more fully dissociates in water.

pKa and Buffer Capacity

In addition to using pKa to gauge the strength of an acid, it may be used to select buffers. This is possible because of the relationship between pKa and pH:

pH = pKa + log10([A-]/[AH])

Where the square brackets are used to indicate the concentrations of the acid and its conjugate base.

The equation may be rewritten as:

Ka/[H+] = [A-]/[AH]

This shows that pKa and pH are equal when half of the acid has dissociated. The buffering capacity of a species or its ability to maintain pH of a solution is highest when the pKa and pH values are close. So, when selecting a buffer, the best choice is the one that has a pKa value close to the target pH of the chemical solution.

The strength of an acid is measured by both its pH and its pKa, and the two are related by the Henderson-Hasslebalch equation. This equation is: pH = pKa + log[A-]/[AH], where [AH] is the concentration of the acid and [A-] is the concentration of its conjugate base after dissociation. pH is a variable that depends on concentration, so if you want to derive its value from this relationship, you need to know the concentrations of the acid and its conjugate base.

What Are pH and pKa?

The acronym pH stands for "power of hydrogen," and it's a measure of the concentration of hydrogen ions in an aqueous solution. The following equation expresses this relationship:

pH = -log [H+]

The value of pKa, on the other hand, depends on the concentrations of acid and conjugate base in solution after the acid dissociation has achieved equilibrium. The ratio of the concentrations of conjugate base and conjugate acid to the acid in question, in an aqueous solution, is called the dissociation constant, Ka. The value for pKa is given by:

pKa = -log (Ka)

Although pH varies by solution, pKa is a constant for each acid.

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch formula comes directly from the definition of the dissociation constant Ka. For an acid HA that dissociates into H+ and A-in water, the dissociation constant is given by:

Ka = [H+][A-]/[HA]

We can take the logarithm of both sides:

log (Ka) = log ([H+][A-]/[HA]), or log Ka = log (H+) + log [A-]/[HA]

Referring to the definitions of pH and pKa, this becomes:

-pKa = -pH + log [A-]/[HA]

Finally, after adding pH and pKa to both sides:

pH = pKa + log [A-]/[HA].

This equation allows you to calculate pH if the dissociation constant, pKa, and the concentrations of the acid and conjugate base are known.

Some chemical reactions are known as reversible reactions because they can go in two directions: forward and reverse. These reactions happen simultaneously and never stop, so they are also called dynamic reactions. A reaction is at equilibrium when the rate of both reactions is the same. However, while the concentrations of the reactants and products are constant, they are not necessarily equal. Equilibrium constants are sometimes called keq values. If you are experimenting with an acid-base reaction, the keq value is Ka, also known as acidity constant, which measures the strength of an acid in solution.

The pKa Value

When an acid dissociates in water, it releases a proton to make the solution acidic. However, only weak acids, which only partially dissociate in water, have both a dissociated state (A-) and undissociated state (AH). They exist together according to the equilibrium equation AH ⇌ A- + H+. The concentration ratio of both sides is constant provided analytical conditions are fixed. This is the Ka, which is defined by the equation Ka = [A-] [H+] ÷ [AH], where the square brackets indicate the concentration of the relative components. Because the Ka constants for acids can be long numbers (for example, the Ka for acetic acid is 0.000018), it is inconvenient to express acidity using the Ka constant alone. The pKa value was introduced as an index to describe the acidity of weak acids, defined as pKa = -log Ka.

Finding Keq From pKa

If you already have the pKa value of a compound, you can work out its Ka. For example, the pKa value of lactic acid is 3.86. The first thing you do is multiply the pKa value by negative one to invert its sign. In the case of lactic acid, this is 3.86 x (-1) = -3.86. Then use a calculator to raise 10 to the power of the negative pKa. In math, this is known as the antilog, and the key is normally marked 10x on scientific calculators. This means the Ka of lactic acid is 10(-3.86), which is 1.38 x 10-4 or 0.000138. The smaller the pKa value, the stronger the acid. This means lactic acid, with a pKa value of 3.86, is a stronger acid than acetic acid, which has a pKa value of 4.75.

pKa = -log(Ka) and so we get an equation relating pH and pKa:

pH = -log(Ka) + log([HA]/[A-])

So, the only way to relate the two is if you know the concentrations of the acid and its conjugate base. If these values are known, then you can just put the values into this equation. If not, then there is no way to find the pKa from the pH.

However, if you know the Ka value of the substance, you can pKa by simply taking the -log value of the Ka. This represents the pH of an acid at its half titration point, the point at which the concentrations of the acid and its conjugate base are equal.

Buffer solutions are used as a means of keeping pH at a nearly constant value in a wide variety of chemical applications. In nature, there are many systems that use buffering for pH regulation. For example, the bicarbonate buffering system is used to regulate the pH of blood.

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aintenance of life

Most biochemical processes work within a relatively small pH range.

The body uses buffers solution to maintain a constant pH.

For example, blood contains a carbonate/bicarbonate buffer that keeps the pH close to 7.4.

Biochemical Assays

Enzyme activity depends on pH, so the pH during an enzyme assay must stay constant.

In shampoos.

Many shampoos use a citric acid/sodium citrate shampoo to maintain a slightly acidic "pH balance".

This counteracts the basicity of the detergents present in the shampoo.

In baby lotions.

Baby lotions are buffered to a pH of about 6.

This hinders the growth of bacteria within the diaper and helps prevent diaper rash.

In the brewing Industry

Buffer solutions are added before fermentation begins.

This prevents the solutions becoming too acidic and spoiling the product.

In the textile Industry.

Many dyeing processes use buffers to maintain the correct pH for various dyes.

In laundry detergents.

Many laundry detergents use buffers to prevent their natural ingredients from breaking down.

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A strong acid and a strong base will react together to produce a neutral salt. E.g., HCl (strong acid) and NaOH (strong base) will react together to form H20 and NaCl (salt). The salt is neutral (if you dump table salt into water, the solution will be neutral) this is because the Na+ and Cl- are perfectly happy being charged atoms.

If you have something that doesn't really like to be ionized, which is a weak acid or base (for example acetic acid, (vinegar) which is only 1.1% ionized (charged) in a water solution) will only be ionized if something forces it to be ionized, i.e., a strong acid or base. When there is a mixture of a weak acid and its conjugate salt (or weak base and its conjugate salt) a buffer is formed. This is due to the fact that if you add some strong acid it will simply react with the conjugate salt, and if you add some strong base it will react with the weak acid. This is how they "buffer solutions" by keeping things pretty balanced.

So to answer your question, a buffer must contain something that is only weakly reactive, and can react further when the need is present. A strong acid/base will totally react, so there is nothing left over to do any buffering.

How good it is to change the pH of a solution. To an acid, it is related to its ability to dissociate to release H+ ions into the solution. In the case of the base, it is its ability to join to H+ ions and form its conjugate acid

The strength of an acid or base is measured with the pka and the pkb values (Note: the highest the value, the weaker the acid or base), which is the negative of log of the constant of equilibrium of the dissociation reaction.

When do we use Henderson-Hasselbalch Equation and when do we solve for the pH the regular way (using the ICE box)?

Only use the Henderson-Hasselbalch Equation for buffers. It is an approximation that is used for buffers and only at equilibrium.

The Henderson-Hasselbalch equation makes use of the approximations [A-] = [base]initial and [HA] = [acid]initial. We can make the approximation when the concentrations of the acid and the base are very high.

The Henderson-Hasselbalch equation can also be used when you know the Ka and are working with a weak acid. If you are given Kb and are working with a weak base, you will need to use the appropriate equation for the Kb.