Using the Ka’s for HC2H3O2 and HCO3- calculate the Kb’s for the C2H3O2- and CO3-2 ions.
The Correct Answer and Explanation is :
To calculate the base dissociation constant (( K_b )) for the conjugate bases of acetic acid (( \text{HC}_2\text{H}_3\text{O}_2 )) and bicarbonate (( \text{HCO}_3^- )), we use the relationship:
[
K_w = K_a \times K_b
]
where:
- ( K_w ) is the ionization constant of water (( 1.0 \times 10^{-14} ) at 25°C),
- ( K_a ) is the acid dissociation constant,
- ( K_b ) is the base dissociation constant.
Given ( K_a ) values:
- ( K_a ) for acetic acid (( \text{HC}_2\text{H}_3\text{O}_2 )) = ( 1.8 \times 10^{-5} )
- ( K_a ) for bicarbonate (( \text{HCO}_3^- )) = ( 4.8 \times 10^{-11} )
Calculating ( K_b ):
- For the acetate ion (( \text{C}_2\text{H}_3\text{O}_2^- )):
[
K_b = \frac{K_w}{K_a} = \frac{1.0 \times 10^{-14}}{1.8 \times 10^{-5}}
]
[
K_b = 5.56 \times 10^{-10}
] - For the carbonate ion (( \text{CO}_3^{2-} )):
- ( K_a ) for bicarbonate (( \text{HCO}_3^- )) corresponds to the second dissociation step of carbonic acid.
- The conjugate base of bicarbonate is carbonate (( \text{CO}_3^{2-} )). [
K_b = \frac{K_w}{K_a} = \frac{1.0 \times 10^{-14}}{4.8 \times 10^{-11}}
] [
K_b = 2.08 \times 10^{-4}
]
Explanation:
The ( K_b ) values represent the strength of the conjugate bases. Since acetate (( \text{C}_2\text{H}_3\text{O}_2^- )) is derived from a weak acid (acetic acid), it has a low ( K_b ), making it a weak base. On the other hand, carbonate (( \text{CO}_3^{2-} )) is derived from bicarbonate, which is a much weaker acid, leading to a higher ( K_b ), making carbonate a stronger base.
This calculation is crucial in acid-base chemistry to understand the behavior of conjugate acid-base pairs in aqueous solutions.