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.
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} = -log_{10} 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
- [H_{2}O] = concentration of water
Considering the water concentration [H_{2}O] 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 }= -log_{10} 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_{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_{b }value.
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_{a }from 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_{w }refers to the water dissociation constant.
A water (H_{2}O) 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 }= -log_{10}K_{w}…………. Equation (viii)
pK_{w }= -log_{10} (1 x 10^{-14}) = 14
Thus, putting the above-determined value of pK_{w }into 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_{a }when pK_{b} given
Example #1: The pK_{b }for an acetic acid (CH_{3}COOH) solution is 9.23. What is its pK_{a}? |
The pK_{b }value 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_{b }value 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_{a }value 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 }(CH_{3}COOH) = 4.77. Thus, hydrofluoric acid is stronger than acetic acid. You may also note that pK_{b} (HF) = 10.9 > pK_{b }(CH_{3}COOH) = 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 (NH_{3}) is 1.74 x 10^{-5}. What is the pK_{a} for an aqueous solution of NH_{3}? |
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 }= -log_{10} K_{b}…………. Equation (v) pK_{b }= -log_{10} (1.74 x 10^{-5}) = 4.76 Now that the pK_{b }value 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_{b }values, 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 }= -log_{10} K_{b}…………. Equation (v) pK_{b }= -log_{10} (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_{b }values; however, it depends on the magnitude of these values that determines whether the solution is acidic or basic in nature. |
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} = -log_{10} 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} = -log_{10} 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_{b }is 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_{a }is 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_{b }is 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_{b }are 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 [H_{2}O] stays constant overall, thus the above equation can be rewritten as: ∴ K_{w} = [H^{+}] [OH^{–}] pK_{w }is 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 }= -log_{10} K_{w} ∴ pK_{w }= -log_{10} (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} = -log_{10} 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} = -log_{10} 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.
About the author
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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