Question

A weak base has \(K_{\mathrm{b}}=1.5 \times 10^{-9} .\) What is the value of \(K_{\mathrm{a}}\) for the conjugate acid?

Short answer

The value of \(K_{a}\) is approximately \(6.67 \times 10^{-6}\).

Step-by-step solution

Understanding the Relationship between Ka and Kb

For a conjugate acid-base pair, the product of the ionization constant of the acid (Ka) and the base (Kb) is equal to the ion product of water (Kw), which is 1.0 imes 10^{-14} at 25°C. This can be expressed as:\[K_{a} \times K_{b} = K_{w}\]

Rearranging the Formula to Solve for Ka

From the relationship \(K_{a} \times K_{b} = K_{w}\), you can find \(K_{a}\) by rearranging the formula:\[K_{a} = \frac{K_{w}}{K_{b}}\]

Substituting Known Values

Now, substitute the known values into the formula:\[K_{a} = \frac{1.0 \times 10^{-14}}{1.5 \times 10^{-9}}\]

Calculating Ka

Perform the division to find \(K_{a}\):\[K_{a} = \frac{1.0 \times 10^{-14}}{1.5 \times 10^{-9}} \approx 6.67 \times 10^{-6}\]

Key concepts

Weak Base

A weak base is a base that does not fully ionize in solution. Unlike strong bases, which completely dissociate in water, weak bases only partially separate into ions. This means that the equilibrium lies to the left. When a weak base dissolves in water, it forms a conjugate acid. The degree of ionization of the weak base in water is described by its ionization constant, denoted as \( K_b \).Some key characteristics of weak bases are:

• Low tendency to donate electron pairs
• Partial ionization in aqueous solution
• Presence of an equilibrium state between the base and its conjugate acid

Understanding weak bases helps in predicting how they behave in chemical reactions and in calculating the pH of their solutions.

Conjugate Acid

When a weak base receives a hydrogen ion (proton), it becomes its conjugate acid. This relationship is crucial in acid-base equilibrium. The conjugate acid is formed when the weak base accepts a proton from water, which is behaving as a Bronsted-Lowry acid in this instance.The transformation of a base into its conjugate acid is denoted by the reversible equation:\[B(aq) + H_2O(l) \leftrightarrow BH^+(aq) + OH^-(aq)\]Where:

• \( B \) represents the weak base
• \( BH^+ \) represents the conjugate acid
• \( OH^- \) is the hydroxide ion

Conjugate acids have their own equilibrium constants, \( K_a \), which relate back to their corresponding \( K_b \) constants for their base forms.

Ionization Constant

The ionization constant, or equilibrium constant, is essential in the study of acids and bases. For weak bases, the ionization constant is expressed as \( K_b \). It quantifies the strength of the base in water, showing how much of the base breaks into ions at equilibrium.The formula for the base ionization constant \( K_b \) is:\[K_b = \frac{\left [ BH^+ \right ] \cdot \left [ OH^- \right ]}{\left [ B \right ]}\]A low \( K_b \) value, such as \( 1.5 \times 10^{-9} \), indicates a weak base. This means only a small fraction of the base molecules ionize in water.
For its conjugate acid, the ionization constant \( K_a \) describes its reverse process. Both constants \( K_a \) and \( K_b \) are related by the ion product of water.

Kw (Ion Product of Water)

Kw, the ion product of water, is a critical constant in the study of chemical reactions in water. At 25°C, \( K_w \) is always \( 1.0 \times 10^{-14} \). It's the product of the concentrations of hydrogen ions \( [H^+] \) and hydroxide ions \( [OH^-] \) in pure water.Its expression is as follows:\[K_w = [H^+] \cdot [OH^-]\]\( K_w \) plays a pivotal role in understanding the equilibrium between acids and bases in aqueous solutions. When dealing with conjugate acid-base pairs, \( K_a \times K_b = K_w \).
This relationship enables us to find the ionization constant of an acid \( K_a \) if the base constant \( K_b \) is known, and vice versa.For example, given \( K_b = 1.5 \times 10^{-9} \), the \( K_a \) of its conjugate acid can be calculated using \( K_a = \frac{K_w}{K_b} \), which equals approximately \( 6.67 \times 10^{-6} \). This highlights the delicate balance maintained by water's autoionization.