How to calculate pKa from pKb? - pKb to pKa, Conversion, Formulas, Equations

pK_a determines the acidic strength of an aqueous solution, while pK_b marks the basicity present in an aqueous solution. In this way, pK_a and pK_b are two oppositely contrasting chemical attributes.

In this article, you will learn how to find pK_a from pK_b i.e (pK_b to pK_a conversion) through several solved examples.

But before that, let us introduce you to what is pK_a and pK_b and what their importance is in chemistry.

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What is pK_a?

The prefix p in pK_a stands for power. Just like pH determines the power of hydrogen ions present in an aqueous solution, pK_a measures the strength of an acidic solution as the power of the acid dissociation constant (K_a).

K_a measures the extent of ionization of an acid in an aqueous solution. A weak acid (HA) partially ionizes to release H⁺ and A^– ions in water, as shown by the reversible reaction below.

K_a for the above reaction is calculated at the equilibrium point using equation (i).

$Ka = \frac{[H^{+}][A^{-}]}{[HA]}$ ……….Equation (i)

Where;

[H⁺] = concentration of hydrogen ions released in the aqueous solution
[A^–] = concentration of conjugate base of the acid
[HA] = acid concentration at equilibrium

pK_a is calculated by taking the negative logarithm to the base 10 of the K_a value for acid, as shown in equation (ii).

pK_a = -log₁₀ K_a…………. Equation (ii)

pK_a is inversely related to the strength of an acid, just like pH.

The greater the strength of an acid, it undergoes dissociation to a large extent. Thus, a large number of H⁺ ions are released in the aqueous solution. This implies that the acid possesses a high K_a value; however, it has a low pK_a value as per equation (ii).

Strong mineral acids such as HCl that completely ionize in water have K_a above 1 and pK_a below 1.

Weak organic acids, such as acetic acid and benzoic acid, have K_a values below 1, while their pK_a values lie above 1.

What is pK_b?

pK_b denotes the power of the base dissociation constant (K_b). It measures the strength of a basic solution by taking the negative logarithm of K_b.

K_b measures the extent of ionization of a base in an aqueous solution. A base is defined as a chemical substance that liberates OH^– ions in water. It is also defined as a proton acceptor as per the Bronsted-Lowry theory.

A strong base completely ionizes in water, while a weak base (B) partially dissociates in an aqueous solution to yield OH^– ions and B⁺ ions. The B⁺ion accepts a proton from water to form a BH⁺ ion, as shown by the reversible reaction below.

K_b for the above reaction is calculated at the equilibrium point using equation (iii).

$Kb = \frac{[BH^{+}][OH^{-}]}{[B][H2O]}$ ……….Equation (III)

Where;

[BH⁺] = concentration of conjugate acid of the base
[OH^–] = hydroxide ion concentration in aqueous solution
[B]= concentration of base at equilibrium
[H₂O] = concentration of water

Considering the water concentration [H₂O] constant throughout the reaction, equation (iii) can be rearranged as equation (iv), shown below.

$Kb = \frac{[BH^{+}][OH^{-}]}{[B]}$ ……….Equation (iv)

pK_b is then calculated by taking the negative logarithm of K_b as shown in equation (v).

pK_b= -log₁₀ K_b…………. Equation (v)

K_bis directly related to the strength of a base. Greater the K_b value, the base undergoes dissociation to a greater extent thus, more OH^– ions are released in the aqueous solution.

However, a strongly basic solution has a low pK_b value as per equation (v), so pK_b is inversely related to the strength of a basic solution.

What is the relationship between pK_a and pK_b?

pK_a and pK_b are inversely related to each other.

The greater the acidic strength of an aqueous solution, the lower its pK_a value; however, it has a high pK_bvalue.

Conversely, the greater the basic strength of an aqueous solution, the lower its pK_b; however, it possesses a higher pK_a value.

How to find pK_afrom pK_b? – (pK_b to pK_a conversion)

Equation (vi) gives the formula that relates pK_a to pK_b.

pK_a+ pK_b= pK_w …………. Equation (vi)

In the above equation, pK_wrefers to the water dissociation constant.

A water (H₂O) molecule undergoes autoionization at room temperature (25°C) to release H⁺ and OH^– ions, as shown below.

The water dissociation constant (K_w) can be determined as per equation (vii).

K_w = [H⁺][OH^–]…………Equation (vii)

K_whas a fixed value at r.t.p i.e., K_w = 1 x 10^-14. So, pK_w can be calculated by taking the negative logarithm of this value as shown in equation (viii).

pK_w= -log₁₀K_w…………. Equation (viii)

pK_w= -log₁₀ (1 x 10^-14) = 14

Thus, putting the above-determined value of pK_winto equation (vi) and making pK_a the subject of the formula gives us the final equation ix, which we can use to find pK_a if the value of pK_b is known.

pK_a+ pK_b= 14

pK_a = 14 – pK_b…………. Equation (ix)

Now let’s see through the solved examples given below; how to apply equation (ix) and find pK_a from pK_b.

Solved examples of determining pK_awhen pK_b given

Example #1: The pK_bfor an acetic acid (CH₃COOH) solution is 9.23. What is its pK_a?

The pK_bvalue for acetic acid is given in the question statement, so we can use equation (ix) to find its relative pK_a, as shown below.

pK_a = 14 – pK_b…………. Equation (ix)

pK_a = 14 – 9.23

pK_a = 4.77

Result: The pK_a of the given acetic acid solution is 4.77.

Example #2: What is the relative pK_a for hydrofluoric acid (HF) if the pK_b is 10.9?

The pK_bvalue for hydrofluoric acid (HF) is given in the question statement, so we can easily apply equation (ix) to find the relative pK_a, as shown below.

pK_a = 14 – pK_b…………. Equation (ix)

pK_a = 14 – 10.9

pK_a = 3.1

Result: The pK_a of the given hydrofluoric acid solution is 3.1.

Example # 3: Comment using pK_a values calculated in examples 1 and 2; which out of the two given acids is stronger?

Lower the pK_avalue for an acidic solution, the greater the strength of the acid. As per the calculations made in the above two examples, pK_a (HF) = 3.1 < pK_a(CH₃COOH) = 4.77. Thus, hydrofluoric acid is stronger than acetic acid.

You may also note that pK_b (HF) = 10.9 > pK_b(CH₃COOH) = 9.23. As pK_b is directly related to acidic strength, so a higher pK_b value denotes hydrofluoric acid is a stronger acid than acetic acid, conforming to what we predicted above using pK_a values.

Example # 4: The K_b value for ammonia (NH₃) is 1.74 x 10^-5. What is the pK_a for an aqueous solution of NH₃?

The base dissociation constant (K_b) value for ammonia is given so we can use equation (v) to find its pK_b value.

pK_b= -log₁₀ K_b…………. Equation (v)

pK_b= -log₁₀ (1.74 x 10^-5) = 4.76

Now that the pK_bvalue is known, we can easily find pK_a using equation (ix), as shown below.

pK_a = 14 – pK_b…………. Equation (ix)

pK_a = 14 – 4.76 = 9.24

Result: The pK_a of the ammonia solution is 9.24.

Example # 5: Aspartic acid is an acidic amino acid; its pK_a value is 4.1. Find its pK_b.

As the pK_a for aspartic acid is given in the question statement, so we can easily find pK_b from the given pK_a using equation (ix).

pK_a = 14 – pK_b…………. Equation (ix)

pK_a = 14 – 4.1 = 9.9

Result: The pK_a for aspartic acid is 9.9.

Example # 6: The K_b for a chemical substance X is 1.32 x 10^-7. Is the aqueous solution of substance X more acidic or basic in nature?

We can determine the acidity or basicity of an aqueous solution by calculating its pK_a and pK_bvalues, respectively.

The base dissociation constant (K_b) for substance X is given so we can use equation (v) to find its pK_b value, as shown below.

pK_b= -log₁₀ K_b…………. Equation (v)

pK_b= -log₁₀ (1.32 x 10^-7) = 6.88

Now that the pK_b value is determined, we can conveniently use equation (ix) to calculate the relative pK_a value for substance X.

pK_a = 14 – pK_b…………. Equation (ix)

pK_a = 14 – 6.88 = 7.12

Result: pK_a = 7.12 > pK_b= 6.88 for substance X. This implies that substance X is slightly basic in nature.

Interesting fact: You may note from this example that an aqueous solution possesses both pK_a and pK_bvalues; however, it depends on the magnitude of these values that determines whether the solution is acidic or basic in nature.

Also, check:

FAQ

What is pK_a?

pK_a measures the acidity present in an aqueous solution. It is calculated by taking the negative logarithm to the base 10 of the acid dissociation constant (K_a).

∴ pK_a = -log₁₀ K_a

What is pK_b?

pK_b measures the basicity present in an aqueous solution. It is calculated by taking the negative logarithm to the base 10 of the base dissociation constant (K_b).

∴ pK_b = -log₁₀ K_b

How is pK_a related to the acidity of an aqueous solution?

pK_a is inversely related to the acidity of an aqueous solution.

A strong Bronsted acid dissociates to a large extent in an aqueous solution by easily liberating its loosely held proton. It thus possesses a high K_a value.

Consequently, the negative logarithm of K_a i.e., pK_a, is lowered.

This implies that the greater the acidic strength, the lower its pK_a value and vice versa.

How is pK_b related to the acidity of an aqueous solution?

pK_bis directly related to the acidity of an aqueous solution. Greater the acidity, the higher the pK_b, and vice versa.

How is pK_a related to the basicity of an aqueous solution?

pK_ais directly related to the basic strength of an aqueous solution. Strongly basic solutions have high pK_a values and vice versa.

How is pK_b related to the basicity of an aqueous solution?

pK_bis inversely related to the basicity of an aqueous solution. Strongly basic solutions possess high K_b values, thus low pK_b values, and vice versa.

What is the relationship between pK_a and pK_b of an aqueous solution?

pK_a and pK_bare inversely related to each other as per the formula shown below.

∴ pK_a+ pK_b= pK_w

Where pK_w = water dissociation constant = 14 (at 25°C, 1 atm).

What is pK_w?

pK_w stands for the power of water dissociation constant (K_w). The autoionization of water at room temperature (25°C) liberates H⁺ and OH^– ions, as shown below.

The equilibrium constant (K_w) for the above reaction is calculated as follows:

As the concentration of water [H₂O] stays constant overall, thus the above equation can be rewritten as:

∴ K_w = [H⁺] [OH^–]

pK_wis calculated as a negative logarithm of K_w_.As the value of K_w= 1.00 x 10^-14 is fixed at r.t.p so,

∴ pK_w= -log₁₀ K_w

∴ pK_w= -log₁₀ (1 x 10^-14) = 14.

Can pK_a obtain both positive and negative values?

Yes. A negative pK_a value implies extremely strong acidic strength. For instance, pK_a for hydrochloric acid (HCl) = -6.30.

How to find pK_a from pK_b?

pK_a can be calculated from pK_b by substituting the given value into the equation shown below:

∴ pK_a= 14 – pK_b.

Summary

pK_a is the negative logarithm of the acid dissociation constant (K_a).
pK_a = -log₁₀ K_a.
pK_a measures the strength of an acidic solution. The lower the pK_a value, the higher the acidic strength.
pK_b is the negative logarithm of the base dissociation constant (K_b).
pK_b = -log₁₀ K_b.
pK_b measures the strength of a basic solution. Lower the pK_b value, the higher the basic strength of an aqueous solution.
pK_a is inversely related to pK_b by the formula: pK_a + pK_b = 14. The number 14 represents pK_w, where K_w is the water dissociation constant at room temperature (25°C).
If the value of pK_b for an aqueous solution is known, we can find its pK_a by substituting the given value into the formula: pK_a= 14 – pK_b.

References

“Converting Between pKa and pKb: Explanation.” Study.com, https://study.com/skill/learn/converting-between-pka-and-pkb-explanation.html.
Helmenstine, Anne Marie. “pH, pKa, Ka, pKb, and Kb Explained.” ThoughtCo, https://www.thoughtco.com/ph-pka-ka-pkb-and-kb-explained-4027791.
“pH, pKa, Ka, pKb, Kb.” Socratic.org, https://socratic.org/organic-chemistry-1/acids-and-bases-1/ph-pka-ka-pkb-kb.
“Equilibrium Constants (Ka and Kb, pKa and pKb).” Jack Westin MCAT Prep, https://jackwestin.com/resources/mcat-content/acid-base-equilibrium/equilibrium-constants-ka-and-kb-pka-pkb.

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Ammara Waheed

Ammara Waheed is a highly qualified and experienced chemist, whose passion for Chemistry is evident in her writing. With a Bachelor of Science (Hons.) and Master of Philosophy (M. Phil) in Physical and Analytical Chemistry from Government College University (GCU) Lahore, Pakistan, with a hands-on laboratory experience in the Pakistan Council of Scientific and Industrial Research (PCSIR), Ammara has a solid educational foundation in her field. She comes from a distinguished research background and she documents her research endeavors for reputable journals such as Wiley and Elsevier. Her deep knowledge and expertise in the field of Chemistry make her a trusted and reliable authority in her profession. Let's connect - https://www.researchgate.net/profile/Ammara-Waheed

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How to calculate pKa from pKb? - pKb to pKa, Conversion, Formulas, Equations

What is pK_a?

What is pK_b?

What is the relationship between pK_a and pK_b?

How to find pK_afrom pK_b? – (pK_b to pK_a conversion)

Solved examples of determining pK_awhen pK_b given