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

pK_b and pK_a are important chemical entities that help determine the basicity or acidity present in an aqueous solution, respectively. These two parameters are closely related and thus inconvertible, which means if we know pK_a, we can definitely find pK_b.

If you are interested to know how (and that is mainly why you are here) so, continue reading!

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_b is 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_b and pK_a?

pK_b and pK_a are inversely related to each other.

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

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

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

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

pK_a + pK_b = pK_w …………. Equation (vi)

In the above equation, pK_w refers to the negative logarithm of the water dissociation constant (K_w).

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_w has 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 = -log₁₀ K_w…………. Equation (viii)

pK_w = -log₁₀ (1 x 10^-14) = 14

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

pK_a + pK_b = 14

pK_b = 14 – pK_a…………. Equation (ix)

We have given you the following solved examples so that you can understand this concept better i.e., how to apply the above formula and find pK_b from pK_a.

Solved examples for finding pK_b from pK_a

Example # 1: The pKa for formic acid (HCOOH) is 3.75. What is its pKb?

As the pKa value for formic acid is given in the question statement, so we can easily use this value and equation (ix) to find the pKb value, as shown below. pKb = 14 – pKa…………. Equation (ix) pKb = 14 – 3.75 pKb = 10.25 Result: The pKb value for formic acid is 10.25.

Example # 2: The pKa value for carbonic acid (H2CO3) is 6.35. Calculate the pKb value.

As the pKa value for H2CO3 is given in the question statement, so we can easily use this value and equation (ix) to find the pKb value, as shown below. pKb = 14 – pKa…………. Equation (ix) pKb = 14 – 6.35 pKb = 7.65 Result: The pKb value for carbonic acid is 7.65.

Example # 3: The pKa value for oxalic acid (H2C2O4) is 1.2. Which of the following options gives the correct pKb value? A) 12.8     B) 11.2      C) 10.6    D) 13.5    E) 14.3

Answer: Option A (pKb = 12.8) gives the correct answer. Explanation: As the pKa value for H2C2O4 is given in the question statement, so we can easily use this value and equation (ix) to find the required pKb value, as shown below. pKb = 14 – pKa…………. Equation (ix) pKb = 14 – 1.2 pKb = 12.8

Example # 4: The acetate buffer contains a large concentration of acetic acid (CH3COOH) and acetate (CH3COO–) ions. The Ka value for CH3COOH is 1.58 x 10-5. Explain using pKa and pKb values why acetic acid is a weak acid that only partially ionizes in water.

CH3COOH partially dissociates in an aqueous solution, as shown below. Acetate (CH3COO–) is the conjugate base of acetic acid (CH3COOH). As the Ka value for CH3COOH is given in the question statement so we can use equation (ii) to find its pKa. pKa = -log10 Ka…………. Equation (ii) pKa = -log10 (1.58 x 10-5) = 4.80 Now that the pKa value for acetic acid is calculated, we can use equation (ix) to find the pKb for acetate ions. pKb = 14 – pKa…………. Equation (ix) pKb = 14 – 4.80 pKb = 9.20 As per the above calculations, CH3COOH has a comparatively higher pKa value compared to other acids, while CH3COO– has a low pKb value. This implies that CH3COO– is a strong base. Bases are proton acceptors, so CH3COO– readily accepts a proton and the above equilibrium reverts back. Consequently, a large amount of CH3COOH stays undissociated in water; it is thus a weak acid.

Example # 5: Which out of the following two is a stronger base? A) Cyanide (CN–)  B) Nitrite (NO2–) pKa (HCN) = 2.1 ;  pKa (NO2–) = 3.2

In the question statement, the pKa values for the conjugate acids of CN– and NO2– are given. We can use the given pKa values to calculate the pKb of the bases and compare their basicity, as shown below. pKb = 14 – pKa…………. Equation (ix) A)  pKb (CN–) = 14 – 2.1 = 11.9 B)  pKb (NO2–) = 14 – 3.2 = 10.8 Result: A lower pKb value denotes higher basicity and vice versa. So as per the above data, pKb (NO2–) = 10.8 < pKb (CN–) = 11.9. Hence, the nitrite (NO2–) ion is a stronger base than cyanide (CN–). Interesting fact: A strong acid forms a weak conjugate base, and vice versa i.e., a weak acid forms a strong conjugate base, as witnessed in this example.

Also, check:

• How to find K_a from K_b?
• How to find K_a from pK_a?
• How to find pK_a from K_a?
• How to find pK_a from pK_b?
• How to find K_b from K_a?
• How to find molar solubility from K_sp?
• How to find K_sp from molar solubility?
• How to find pOH from pH?
• How to find pOH from molarity?
• How to find OH^– from pH?
• How to find H⁺ from pH?
• How to find molarity from pH?
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• How to find K_a from pH?
• How to find pH from K_a?
• How to find pH from pK_a?
• How to find pK_a from pH?
• How to find pH from K_a and molarity?
• How to find concentration from absorbance?
• How to find K_a from the titration curve?
• How to find pK_a from the titration curve?
• How to find molarity from titration?

FAQ

What is pKb?

pKb 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 (Kb). pKb = -log10 Kb

What is pKa?

pKa 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 (Ka). pKa = -log10 Ka

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

pKb is inversely related to the basic strength of an aqueous solution. A base is defined as a proton acceptor or an electron pair donor. It is also defined as a chemical substance that breaks down to release OH– ions in water. The greater the base dissociation constant (Kb) value, the more OH– ions released in its aqueous solution, denoting a stronger base. However, the negative logarithm of Kb i.e., pKb, is lowered. This implies that the greater the basic strength, the lower its pKb value and vice versa.

What is the relationship between pKa and pKb of an aqueous solution?

pKa and pKb are inversely related to each other as per the formula given below. pKa + pKb = pKw Where pKw = negative logarithm of water dissociation constant = 14 (at 25°C, 1 atm).

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

pKb is directly related to the acidity of an aqueous solution. Greater the acidity, the higher the pKb and vice versa. A strong acid with low pKa gives a weak conjugate base with high pKb and vice versa.

How to find pKb from pKa?

pKb can be calculated from pKa by substituting the given value into the equation shown below: pKb = 14 – pKa.

 Summary

• 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 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 inversely related to pK_a 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_a for an aqueous solution is known, we can find its pK_b by substituting the given value into the formula: pK_b = 14 – pK_a.

References

1. “Converting Between pKa and pKb: Explanation.” Study.com, https://study.com/skill/learn/converting-between-pka-and-pkb-explanation.html.
2. Helmenstine, Anne Marie. “pH, pKa, Ka, pKb, and Kb Explained.” ThoughtCo, https://www.thoughtco.com/ph-pka-ka-pkb-and-kb-explained-4027791.
3. “pH, pKa, Ka, pKb, Kb.” Socratic.org, https://socratic.org/organic-chemistry-1/acids-and-bases-1/ph-pka-ka-pkb-kb.
4. “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.