Question

The relation $\left[H^{+}O\right]=\frac{K_w}{\left[H_3{O}^{+}\right]}+{\left[HCl\right]}_0$ for an aqueous solution of $HCl$ can be reduced to:

• A. $\left[H_3{O}^{+}\right]={\left[HCl\right]}_{0}if\left[H_3{O}^{+}\right]\ge {10}^{-6}$
• B. $\left[H_3{O}^{+}\right]={\left[HCl\right]}_{0}if\frac{K_w}{\left[H_3{O}^{+}\right]}\ge {10}^{-8}$
• C. $\left[H_3{O}^{+}\right]=\frac{{\left[HCl\right]}_0\pm \sqrt{{\left[HCl\right]}_0^2+4K_w}}{2}if\left[H_3{O}^{+}\right]\le {10}^{-6}$
• D. $\left[H_3{O}^{+}\right]={\left[HCl\right]}_{0}if{\left[HCl\right]}_0\ge {10}^{-6}$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $\left[H_3{O}^{+}\right]={\left[HCl\right]}_{0}if\left[H_3{O}^{+}\right]\ge {10}^{-6}$
B $\left[H_3{O}^{+}\right]={\left[HCl\right]}_{0}if\frac{K_w}{\left[H_3{O}^{+}\right]}\ge {10}^{-8}$
C $\left[H_3{O}^{+}\right]=\frac{{\left[HCl\right]}_0\pm \sqrt{{\left[HCl\right]}_0^2+4K_w}}{2}if\left[H_3{O}^{+}\right]\le {10}^{-6}$
D $\left[H_3{O}^{+}\right]={\left[HCl\right]}_{0}if{\left[HCl\right]}_0\ge {10}^{-6}$

$HCl+H_{2}O⟶H_3^{+}O+{Cl}^{-}$
$\left[H_3{O}^{+}\right]=\frac{Kw}{\left[H_3{O}^{+}\right]}+{\left[HCl\right]}_o$

(A)When $\left[H_3{O}^{+}\right]\ge {10}^{-6}$

$\frac{Kw}{\left[H_3{O}^{+}\right]}\approx 0$

So $\left[H_3{O}^{+}\right]={\left[HCl\right]}_o$

(B) When $\frac{Kw}{\left[H_3{O}^{+}\right]}\ge {10}^{-8}\approx 0$

So $\left[H_3{O}^{+}\right]={\left[HCl\right]}_o$

(C)When $\left[H_3{O}^{+}\right]\le {10}^{-6}$

Then ${\left[H_3{O}^{+}\right]}^2=Kw+{\left[HCl\right]}_o\left[H_3{O}^{+}\right]$

${\left[H_3{O}^{+}\right]}^2-{\left[HCl\right]}_o\left[H_3{O}^{+}\right]-Kw=0$

Compare with $a{x}^2-bx+c=0$
$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$\left[H_3{O}^{+}\right]=\frac{{\left[HCl\right]}_o\pm \sqrt{{\left[HCl\right]}_o^2+4Kw}}{2}$

(D) When ${\left[HCl\right]}_o\ge {10}^{-6}$

Here acid concentration is very high so automatically dissociation of water is negligible

$\left[H_3{O}^{+}\right]={\left[HCl\right]}_o$