Calculate the \(\left[\mathrm{OH}^{-}\right]\) of each of the following solutions at \(25^{\circ} \mathrm{C}\). Identify each solution as neutral, acidic, or basic. a. \(\left[\mathrm{H}^{+}\right]=1.0 \times 10^{-7} M\) c. \(\left[\mathrm{H}^{+}\right]=12 M\) b. \(\left[\mathrm{H}^{+}\right]=8.3 \times 10^{-16} M\) d. \(\left[\mathrm{H}^{+}\right]=5.4 \times 10^{-5} M\)

Understanding the ion-product constant for water, commonly symbolized as \( K_w \), is crucial when studying the properties of water and aqueous solutions. At 25°C, the value for \( K_w \) is \( 1.0 \times 10^{-14} \). But what does this mean?

Essentially, water naturally dissociates into hydrogen ions (\( \text{H}^+ \)) and hydroxide ions (\( \text{OH}^- \)), albeit to a very small extent. The product of these ion concentrations is always constant (hence, 'ion-product constant') at a given temperature, and for pure water, they are equal to each other. Thus, the relationship can be summarized by the equation:

\[ K_w = [\text{H}^+] \times [\text{OH}^-] \]
Understanding \( K_w \) allows chemists to calculate the concentration of hydroxide ions if the hydrogen ion concentration is known, and vice versa. It serves as a foundational concept for further exploration into the properties of acids, bases, and the pH of solutions.

When dealing with solutions, pH and pOH are invaluable tools for representing the acidity or basicity of the solution. pH is the negative logarithm of the hydrogen ion concentration, while pOH is the negative logarithm of the hydroxide ion concentration. The formulas for these calculations are:

\[ \text{pH} = -\log([\text{H}^+]) \]\[ \text{pOH} = -\log([\text{OH}^-]) \]
In pure water or neutral solutions at 25°C, pH and pOH both equal 7 since the concentrations of \( \text{H}^+ \) and \( \text{OH}^- \) ions are both \( 1.0 \times 10^{-7} \) M. Intriguingly, pH and pOH are inversely related and always add up to 14, a direct consequence of the ion-product constant for water. This is expressed as:
\[ \text{pH} + \text{pOH} = 14 \]
With these calculations, scientists and students alike can quickly assess whether a solution is acidic, basic, or neutral without needing to delve into more complex chemical tests.

Identifying whether a solution is acidic or basic hinges on the concentration of hydrogen ions \( ([\text{H}^+]) \) and hydroxide ions \( ([\text{OH}^-]) \). A neutral solution has equal concentrations of \( [\text{H}^+] \) and \( [\text{OH}^-] \), like pure water. If the \( [\text{H}^+] \) is higher than \( [\text{OH}^-] \), the solution is acidic. Conversely, if the \( [\text{OH}^-] \) concentration exceeds that of the \( [\text{H}^+] \), then the solution is basic.

For practical identification, utilizing the pH scale is the standard approach. A pH less than 7 denotes an acidic solution, a pH of exactly 7 indicates neutrality, and a pH greater than 7 signifies a basic solution. By using the ion-product constant \( K_w \) and the relationships between pH and pOH, one can determine the nature of the solution which is an essential step in a wide range of chemical analyses and industrial processes.