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 \(6.67 \times 10^{-6}\).

Step-by-step solution

Understand the Relationship between Ka and Kb

The relation between the equilibrium constant for a base (K_b) and its conjugate acid (K_a) is given by the formula:\[K_a \times K_b = K_w\]where K_w is the ion product constant of water. Typically, at 25°C, K_w is 1.0 \times 10^{-14}.

Substitute Known Values

We know K_b = 1.5 \times 10^{-9} and K_w = 1.0 \times 10^{-14}. Substitute these values into the relation formula:\[K_a \times 1.5 \times 10^{-9} = 1.0 \times 10^{-14}\].

Solve for Ka

Rearrange the equation to solve for K_a:\[K_a = \frac{1.0 \times 10^{-14}}{1.5 \times 10^{-9}}\]This simplifies to:\[K_a = \frac{1.0}{1.5} \times 10^{-5}\]Calculate the values to find:\[K_a = 0.6667 \times 10^{-5} = 6.67 \times 10^{-6}\].

Key concepts

Understanding Weak Bases

Weak bases are substances that partially dissociate in water. They don't release many hydroxide ions \(OH^-\) compared to strong bases. This partial dissociation is why their equilibrium constant \(K_b\) is relatively small.
For instance, if you have a weak base with \(K_b = 1.5 \times 10^{-9}\), this value shows that the base has a low tendency to gain protons from acids compared to stronger bases.
Key characteristics of weak bases include:

• Partial ionization in water.
• Smaller \(K_b\) values, indicating less strength in base behavior.
• The presence of a corresponding conjugate acid when the base gains a proton.

Knowing how a weak base behaves helps in understanding various chemical reactions and their outcomes, especially in equilibrium calculations.

The Role of Conjugate Acids

When a weak base accepts a proton \(H^+\), it forms its conjugate acid. Conjugate acids are crucial because they help maintain the equilibrium in acid-base reactions. They provide insights into how the base will behave in different conditions.
This process is reversible. The conjugate acid can release the proton, turning back into the weak base, thus participating in equilibrium reactions.
Characteristics of conjugate acids:

• Derived from the base by gaining a proton.
• Possess a \(K_a\) value that indicates their acidic strength.
• Related to their corresponding base through the equation \(K_a \times K_b = K_w\).

Understanding conjugate acids helps predict how changes in conditions will affect equilibrium and the overall reaction between acids and bases.

Ion Product Constant of Water

The ion product constant of water, denoted as \(K_w\), is essential in acid-base chemistry. It's the product of the concentrations of hydrogen ions \(H^+\) and hydroxide ions \(OH^-\) in pure water. At 25°C, \(K_w\) is typically \(1.0 \times 10^{-14}\).
This constant remains crucial in calculating the \(K_a\) or \(K_b\) for weak acids and bases. By understanding that \(K_a \times K_b = K_w\), you can find the strength of an acid or base by knowing its conjugate partner.
Key points about \(K_w\):

• It reflects the extent of ionization in water.
• Changes with temperature, but \(1.0 \times 10^{-14}\) at 25°C is standard.
• Helps in solving equilibrium problems involving weak acids and bases.

Knowing \(K_w\) allows prediction and analysis of acid-base equilibriums, providing a fundamental understanding of solution chemistry.