The Kb value for pyridine, C5H5N is 1.7×10-9. What is Ka for C5H5NH+
The correct answer and explanation is :
To calculate the ( K_a ) (acid dissociation constant) for ( \text{C}_5\text{H}_5\text{NH}^+ ) (pyridinium ion), we can use the relationship between the ( K_a ) of a conjugate acid and the ( K_b ) of its conjugate base. This relationship is derived from the equation for the ionization of water and the equilibrium expressions for the acid and base dissociation constants.
The general relationship is:
[
K_a \times K_b = K_w
]
where:
- ( K_a ) is the acid dissociation constant for the conjugate acid (( \text{C}_5\text{H}_5\text{NH}^+ )),
- ( K_b ) is the base dissociation constant for the conjugate base (pyridine, ( \text{C}_5\text{H}_5\text{N} )),
- ( K_w ) is the ionization constant for water, which is ( 1.0 \times 10^{-14} ) at 25°C.
Step 1: Write the relationship
Given:
- ( K_b ) for pyridine is ( 1.7 \times 10^{-9} ),
- ( K_w ) is ( 1.0 \times 10^{-14} ).
We can rearrange the equation to solve for ( K_a ):
[
K_a = \frac{K_w}{K_b}
]
Step 2: Substitute the known values
[
K_a = \frac{1.0 \times 10^{-14}}{1.7 \times 10^{-9}}
]
Step 3: Calculate ( K_a )
[
K_a = 5.88 \times 10^{-6}
]
Thus, the acid dissociation constant ( K_a ) for pyridinium ion ( \text{C}_5\text{H}_5\text{NH}^+ ) is approximately ( 5.88 \times 10^{-6} ).
Explanation:
Pyridine, ( \text{C}_5\text{H}_5\text{N} ), is a weak base that accepts a proton (H+) to form the pyridinium ion, ( \text{C}_5\text{H}_5\text{NH}^+ ). When pyridine accepts a proton, it becomes a conjugate acid. The equilibrium between the base and its conjugate acid is given by the following:
[
\text{C}_5\text{H}_5\text{N} + \text{H}^+ \rightleftharpoons \text{C}_5\text{H}_5\text{NH}^+
]
The base dissociation constant ( K_b ) measures the extent to which pyridine (the base) reacts with water to produce hydroxide ions and pyridinium ions. Conversely, ( K_a ) measures the extent to which the pyridinium ion dissociates to release a proton and form pyridine. Since ( K_b ) and ( K_a ) are related by the water dissociation constant ( K_w ), knowing ( K_b ) allows us to calculate ( K_a ) using the formula above.