1. Definition of Ionic Product ($K_w$): The ionic product of water is defined as the product of the molar concentrations of hydrogen ions $[H^+]$ and hydroxyl ions $[OH^-]$ in water or an aqueous solution.
$$K_w = [H^+][OH^-]$$
2. Dependence on Temperature: The value of $K_w$ is constant at a fixed temperature, regardless of the acidity or alkalinity of the solution. At the standard temperature of $25^\circ\text{C}$ (298 K), the value of $K_w$ for pure water and all dilute aqueous solutions is:
$$K_w = 1.0 \times 10^{-14} \text{ mol}^2/\text{L}^2$$
3. Application to the Problem: The concentration of the $HCl$ solution ($0.01\text{M}$) affects the individual concentrations of $[H^+]$ and $[OH^-]$, but their product remains equal to the constant $K_w$ value for that temperature.
• $[H^+]$ from $HCl = 10^{-2} \text{ M}$
• $[OH^-]$ in solution $= \frac{K_w}{[H^+]} = \frac{10^{-14}}{10^{-2}} = 10^{-12} \text{ M}$
• Product $[H^+][OH^-] = 10^{-2} \times 10^{-12} = 10^{-14}$
Thus, at $25^\circ\text{C}$, the ionic product remains $1.0 \times 10^{-14} \text{ mol}^2/\text{L}^2$.