How to calculate Kb from Ka? - (Kb from Ka)

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Weak acids and bases dissociate to a small extent in an aqueous solution. A weak acid (HA) breaks down in the water to give H+ ions, while a weak base (B) dissociates to produce OH ions.

The extent of ionization of a weak acid or base in an aqueous solution can be determined by using ionization constants, i.e., Ka and Kb.

What is Ka?

Ka stands for acid dissociation constant.

The ionization equilibrium for the dissociation of a weak acid (HA) in an aqueous solution is represented as follows:

dissociation of weak acid in aqueous solution

The acid dissociation constant (Ka) for the above reaction can be represented as equation (i).

Ka = \frac{[H_{3}O^{+}][A^{-}]}{[HA][H_{2}O]}………. Equation (i)

Where;

  • [H3O+] = concentration of hydronium ions formed in the aqueous solution
  • [A] = concentration of conjugate base of the acid
  • [HA] = acid concentration at equilibrium
  • [H2O] = concentration of water

As water concentration stays constant throughout the reaction, while [H3O+] = [H+], i.e., the concentration of H+ ions released in the aqueous solution. So, equation (i) can be rearranged as equation (ii).

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

The greater the strength of an acid, the higher the Ka value for its aqueous solution and vice versa.

What is Kb?

Kb represents the base dissociation constant.

The ionization equilibrium for the dissociation of a weak base (B) in an aqueous solution is represented as:

dissociation of weak base in aqueous solution

The base dissociation constant (Kb) for the above reaction can be represented as equation (iii).

Kb = \frac{[BH^{+}][OH^{-}]}{[B][H_{2}O]}………. Equation (iii)

Where;

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

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

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

The greater the strength of a base, the higher the Kb value for its aqueous solution.

What is the relationship between Ka and Kb?

Ka and Kb can be interconverted using another chemical entity called the water dissociation constant (Kw), as shown in equation (v).

Ka. Kb = Kw………. Equation (v)

⇒ Kw= [H+] [OH]

The value of Kw is fixed at 25°C, i.e., room temperature. Kw = 1.0 x 10-14.

relationship between kb and ka

As per equation (v), if the value of Ka is known, we can easily determine the value of Kb by making Kb the subject of the formula.

formula to find Kb from Ka

Now let’s see how we can find Kb from Ka through the different examples given below.

Solved examples for finding Kb from Ka

Example #1: The acid dissociation constant (Ka) for acetic acid (CH3COOH) at room temperature is 1.8 x 10-5. What is the base dissociation constant (Kb) for sodium acetate (CH3COONa+)?

As per the above statement, Ka = 1.8 x 10-5. We already know the value of Kw at r.t.p, i.e., Kw = 1.0 x 10-14.

We can calculate the value of Kb as follows:

⇒ Kb = Kw/ka

∴ Kb =  \frac{1.0\times10^{-14}}{1.8\times10^{-5}} = 5.6 x 10-10.

Result: The base dissociation constant (Kb) for CH3COONa+ is 5.6 x 10-10.

Acetic acid and acetate ions are known as conjugate acid-base pairs.

Example #2:  Ammonium (NH4+) ion is a mildly strong conjugate acid of a weak base, i.e., ammonia (NH3). The Ka of NH4+ is 5.6 x 10-10. What is Kb for NH3?

As the Ka value is given so we can easily determine the Kb value for NH3 using the expression Kb = Kw/Ka.

⇒ Kb = Kw/Ka

∴ Kb = \frac{1.0\times10^{-14}}{5.6\times10^{-10}}  = 1.8 x 10-5.

Result: The base dissociation constant (Kb) for NH3, as per the above information, is 1.8 x 10-5.

Example # 3:  As per Peter’s experimental data, the pH of a 0.500 M formic acid (HCOOH) solution is measured to be 2.04. How can we determine the Kb value for formate ion (HCOO) using this data?

Formic acid and formate ion are conjugate acid-base pairs.

Formic acid dissociates in an aqueous solution, as shown below.

Formic acid dissociates in an aqueous solution

So if we first calculate the acid dissociation constant (Ka) value for formic acid, we can easily determine the Kb value for formate ion.

The Ka value of formic acid can be calculated using equation (ii), which we learned at the beginning of the article.

Ka = \frac{[H^{+}][A^{-}]}{[HA]}

For formic acid, the above equation can be transformed into:

Ka = \frac{[H^{+}][HCOO^{-}]}{[HCOOH]}

[H+] can be determined from the pH given in the question statement.

pH = -log [H+]

Making [H+] the subject of the formula.

[H+] = 10-pH

[H+] = 10-2.04 = 9.12 x 10-3 M

As per the above reaction equation, [H+] = [HCOO] = 9.12 X 10-3 M

[HCOOH] = formic acid concentration at equilibrium = initial acid concentration – [H+]

The initial formic acid concentration is given in the question statement, i.e., 0.500 M.

[HCOOH]equilibrium = 0.500 – (9.12 X 10-3) = 0.491 M.

Finally, we need to plug in all the values determined above into the Ka formula.

 Ka = \frac{(9.12\times10^{-3})(9.12\times10^{-3})}{(0.491)} = 1.7 x 10-4

Now that we have the Ka of formic acid, we can easily determine the Kb value for formate ion using Kb = Kw/Ka.

Kb = \frac{(1.0\times10^{-14})}{(1.7\times10^{-4})} = 5.9 × 10-11.

Result: The base dissociation constant (Kb) value for formate ion as per the above data is 5.9 x 10-11.

Example # 4:  The pKa for hydrofluoric acid (HF) at 25°C is 3.36. How can we use this information to determine Kb for F (the conjugate base of HF)?

The pKa value for the acidic solution is related to its Ka by the formula given below.

pKa = -log Ka or Ka = 10-pKa

As given in the question statement, pKa for HF = 3.36 so, its Ka = 10-3.36 = 4.4 x 10-4.

Now that we know Ka for HF, we can easily find Kb for F as follows:

∴ Kb = Kw/ka

∴ Kb = \frac{(1.0\times10^{-14})}{(4.4\times10^{-4})}  = 2.2 x 10-11.

Result: The base dissociation constant (Kb) value for fluoride (F) ion is 2.2 x 10-11 at 25°C.

FAQ

What is Ka?

Ka is defined as the acid dissociation constant. It determines the extent of ionization of usually a weak acid in an aqueous solution.

A higher Ka value denotes that the respective acid breaks down to a large extent in water, i.e., it has a higher strength.

What is Kb?

Kb stands for the base dissociation constant that determines the extent of ionization of a base in an aqueous solution.

How do you differentiate between an acid and a base? 

Acids break down to release H+ ions in water, while a base breaks down to produce OH ions in water.

Acids are also defined as proton donors, while bases are proton acceptors as per the Bronsted Lowry acid-base theory.

How to differentiate between an acid and a base

Also read:

What is the relationship between Ka and Kb?

The product of Ka and Kb is equal to Kw, i.e., the water dissociation constant.

Kw = Ka. Kb

How to find Kb from Ka?

The value of Kw is fixed at room temperature (25°C), i.e., Kw = 1.0 x 10-14. If the value of Ka of acid is known, then the value for its conjugate base can be determined by rearranging the expression Kw = Ka. Kb, making Kb the subject of the formula, as shown below.

∴ Kb = Kw/Ka

What is a conjugate acid-base pair?

According to the Bronsted-Lowry theory of acids and bases, a weak acid, such as acetic acid (CH3COOH), dissociates to a small extent in an aqueous solution by releasing H+ ions.

The acid molecule is converted into an ion by the loss of a proton. This ion is known as the conjugate base of the acid.

For e.g. CH3COOH dissociates into acetate (CH3COO) by losing an H+ ion.

Consequently, CH3COOH and CH3COO is known as a conjugate acid-base pair.

How are Ka and Kb related to the strength of an acid or a base? 

The higher the Ka or Kb value, the greater the strength of the respective acid or base.

Is there a numerical relation between the pH of an acidic solution and its Ka value?

Yes. The Ka of an acidic solution can be used to find pKa as follows:

⇒ pKa = -log Ka

pKa is directly related to pH as follows:

⇒ pH = pKa + log [A]/[HA]

In short, the pH of an acidic solution is inversely related to its Ka value.

The greater the acidic strength, the higher the Ka value, but the lower its pH.

Summary

  • The extent of ionization of a weak acid or base in an aqueous solution can be determined using their Ka and Kb values, respectively.
  • Ka represents the acid dissociation constant.
  • Kb stands for base dissociation constant.
  • Greater the strength of an acid or base, the higher the respective dissociation constant values because the latter ionizes to a larger extent in an aqueous solution.
  • Ka is related to Kb by the equation Kw = Ka.Kb.
  • If the value of Ka for acid is known, the Kb value for its conjugate base can be determined by rearranging the expression: Kb = Kw/Ka.
  • Kw = 1.0 x 10-14 at 25°
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