Question

At 25$^\circ$C, the concentration of H$^+$ ions in $ 1.00 \times 10^{-3} \, \text{M} $ aqueous solution of a weak monobasic acid having acid dissociation constant $ K_a = 4.00 \times 10^{-11} $ is $ X \times 10^{-7} \, \text{M} $. The value of $ X $ is ____. {Use: Ionic product of water $ K_w = 1.00 \times 10^{-14} $ at 25$^\circ$C

Hint:

For very weak acids, always include \( K_w \) when calculating \([H^+]\), especially if the \( K_a \) is comparable to \( K_w/C \). Use the formula: \([H^+] = \sqrt{CK_a + K_w}\).

Answer 1

Correct Answer: 2.23

We are given:

• Concentration of weak acid: \( C = 1.00 \times 10^{-3} \, \text{M} \)
• Acid dissociation constant: \( K_a = 4.00 \times 10^{-11} \)
• Ionic product of water: \( K_w = 1.00 \times 10^{-14} \)

Step 1: Use modified formula for \([H^+]\) in very weak acids
When \( K_a \) is very small (as in this case), we use: \[ [H^+] = \sqrt{C K_a + K_w} \] Step 2: Substitute the values \[ [H^+] = \sqrt{(1.00 \times 10^{-3})(4.00 \times 10^{-11}) + 1.00 \times 10^{-14}}\] \[= \sqrt{4.00 \times 10^{-14} + 1.00 \times 10^{-14}} = \sqrt{5.00 \times 10^{-14}} \] \[ [H^+] = 2.23 \times 10^{-7} \, \text{M} \Rightarrow X = \boxed{2.23} \]