Aspirin Solution Equilibrium Calculation

Objective: Determine the equilibrium ratio of the conjugate base ($A^-$) to the undissociated acid ($HA$) for aspirin in an aqueous solution at a specific pH.

Given Information:

• Aspirin solution pH: $pH = 7.4$
• Acid dissociation constant ($K_a$) for aspirin: $K_a = 3.98 \times 10^{-4}$

Step 1: Calculate the Hydrogen Ion Concentration $[H^+]$

The pH is defined as $pH = -\log_{10}[H^+]$. We can find the hydrogen ion concentration $[H^+]$ using the given pH:

• $[H^+] = 10^{-pH}$
• $[H^+] = 10^{-7.4}$
• Calculating this value gives: $[H^+] \approx 3.981 \times 10^{-8}$ M

Step 2: Utilize the $K_a$ Expression for Ratio Calculation

The equilibrium expression for the dissociation of a weak acid ($HA$) is:

$K_a = \frac{[H^+][A^-]}{[HA]}$

To find the ratio $\frac{[A^-]}{[HA]}$, we rearrange the $K_a$ expression:

$\frac{[A^-]}{[HA]} = \frac{K_a}{[H^+]}$

Step 3: Substitute Values and Calculate the Ratio

Now, substitute the given $K_a$ value and the calculated $[H^+]$ concentration:

• $\frac{[A^-]}{[HA]} = \frac{3.98 \times 10^{-4}}{3.981 \times 10^{-8}}$
• Performing the division: $\frac{[A^-]}{[HA]} \approx 0.99975 \times 10^4$
• $\frac{[A^-]}{[HA]} \approx 9997.5$

Conclusion:

Rounding the result to the nearest integer, the ratio $\frac{[A^-]}{[HA]}$ is 9998. This value is consistent with the provided range of 9700 to 10000.