Enough of a monoprotic acid is dissolved in water to produce a 0.0192 M solution. The pH of the resulting solution is 2.53. Calculate the Ka for the acid.
The correct answer and explanation is :
To calculate the acid dissociation constant (( K_a )) for the monoprotic acid, we will use the following relationship:
[
\text{pH} = -\log[H^+]
]
Step 1: Determine the concentration of hydrogen ions ([H^+])
From the given pH of the solution, we can calculate the concentration of hydrogen ions. The pH is related to the concentration of hydrogen ions by the equation:
[
\text{pH} = -\log[H^+]
]
Given that the pH is 2.53, we can solve for ([H^+]):
[
2.53 = -\log[H^+]
]
[
[H^+] = 10^{-2.53} = 2.96 \times 10^{-3} \, \text{M}
]
Step 2: Set up the equilibrium expression for the dissociation of the acid
Let’s assume the acid is ( HA ), which dissociates as follows:
[
HA \rightleftharpoons H^+ + A^-
]
Let the initial concentration of ( HA ) be 0.0192 M, and at equilibrium, the concentration of ( H^+ ) will be ( 2.96 \times 10^{-3} ) M (as we calculated above).
Let’s define the changes in concentration during dissociation. Initially:
- ([HA] = 0.0192 \, \text{M})
- ([H^+] = 0)
- ([A^-] = 0)
At equilibrium:
- ([HA] = 0.0192 – x)
- ([H^+] = x)
- ([A^-] = x)
Where ( x ) is the amount of acid dissociated. Since we know ([H^+] = 2.96 \times 10^{-3} \, \text{M}), this means ( x = 2.96 \times 10^{-3} \, \text{M} ).
Thus, the equilibrium concentration of ( HA ) is:
[
[HA] = 0.0192 – 2.96 \times 10^{-3} = 0.01624 \, \text{M}
]
Step 3: Write the expression for ( K_a )
The acid dissociation constant, ( K_a ), is given by:
[
K_a = \frac{[H^+][A^-]}{[HA]}
]
Substitute the values we have:
[
K_a = \frac{(2.96 \times 10^{-3})(2.96 \times 10^{-3})}{0.01624}
]
[
K_a = \frac{8.76 \times 10^{-6}}{0.01624} = 5.4 \times 10^{-4}
]
Final Answer:
The acid dissociation constant ( K_a ) for the acid is ( 5.4 \times 10^{-4} ).
Explanation:
The process involves calculating the concentration of ( H^+ ) from the pH, then setting up an equilibrium expression to account for the dissociation of the acid. The value of ( K_a ) is determined using the equilibrium concentrations of the dissociated and undissociated species. This approach is common when determining the strength of an acid in solution, as ( K_a ) reflects the extent to which the acid dissociates.