Question

Calculate [OH \(^{-}\) ] for each of the following solutions, and indicate whether the solution is acidic, basic, or neutral: (a) \(\left[\mathrm{H}^{+}\right]=0.0045 \mathrm{M}\); (b) \(\left[\mathrm{H}^{+}\right]=1.5 \times 10^{-9} \mathrm{M} ;\) (c) a solution in which \(\left[\mathrm{H}^{+}\right]\) is 10 times greater than \([\mathrm{OH}]\).

Short answer

In summary, the \(\mathrm{OH}^-\) ion concentrations are: (a) \(2.22 \times 10^{-12}\) M (acidic), (b) \(6.67 \times 10^{-6}\) M (basic), and (c) \(1 \times 10^{-7}\) M (acidic).

Step-by-step solution

Find the concentration of OH\(^{-}\) ions for solution (a)

We are given the concentration of \(\mathrm{H}^{+}\) ions as 0.0045 M. We will use the ion product of water (\(K_\mathrm{w}\)) to find the concentration of \(\mathrm{OH}^{-}\) ions. \[[\mathrm{OH}^{-}] = \frac{K_\mathrm{w}}{[\mathrm{H}^{+}]} = \frac{1 \times 10^{-14}}{0.0045}\] \[ [\mathrm{OH}^{-}] = 2.22 \times 10^{-12}\]

Determine if solution (a) is acidic, basic, or neutral

Since \([\mathrm{H}^{+}]\) > \([\mathrm{OH}^{-}]\), solution (a) is acidic.

Find the concentration of OH\(^{-}\) ions for solution (b)

We are given the concentration of \(\mathrm{H}^{+}\) ions as \(1.5 \times 10^{-9}\) M. We will use the ion product of water (\(K_\mathrm{w}\)) to find the concentration of \(\mathrm{OH}^{-}\) ions. \[ [\mathrm{OH}^{-}] = \frac{K_\mathrm{w}}{[\mathrm{H}^{+}]} = \frac{1 \times 10^{-14}}{1.5 \times 10^{-9}} \] \[ [\mathrm{OH}^{-}] = 6.67 \times 10^{-6} \]

Determine if solution (b) is acidic, basic, or neutral

Since \([\mathrm{H}^{+}]\) < \([\mathrm{OH}^{-}]\), solution (b) is basic.

Find the concentration of OH\(^{-}\) ions for solution (c)

We are given that \([\mathrm{H}^{+}]\) is 10 times greater than \([\mathrm{OH}^{-}]\). Let \(x\) represent the concentration of \(\mathrm{OH}^{-}\) ions. We can write the equation: \[ [\mathrm{H}^{+}] = 10x \] Since the ion product of water (\(K_\mathrm{w}\)) is equal to \([\mathrm{H}^{+}][\mathrm{OH}^{-}]\), we can substitute the \([\mathrm{H}^{+}]\) in terms of \(x\) and solve for \(x\): \[ K_\mathrm{w} = (10x)(x) \] \[ 1 \times 10^{-14} = 10x^2 \] \[ x = \sqrt{\frac{1 \times 10^{-14}}{10}} \] \[ x = 1 \times 10^{-7} \]

Determine if solution (c) is acidic, basic, or neutral

Since \([\mathrm{H}^{+}]\) > \([\mathrm{OH}^{-}]\), solution (c) is acidic. In conclusion, the concentrations of \(\mathrm{OH}^{-}\) ions for the given solutions are: (a) \(2.22 \times 10^{-12}\) M, acidic solution (b) \(6.67 \times 10^{-6}\) M, basic solution (c) \(1 \times 10^{-7}\) M, acidic solution

Key concepts

Ion Product of Water

The ion product of water is a foundational concept in chemistry that plays a key role in understanding how solutions behave. Water, though seemingly neutral, has a characteristic known as the ion product, denoted by \(K_\mathrm{w}\). This value is crucial in determining how many hydrogen ions \([\mathrm{H}^{+}]\) and hydroxide ions \([\mathrm{OH}^{-}]\) are present in any given aqueous solution. Simply put, in water, the product of the concentrations of these ions is a constant at a given temperature.
In pure water at 25°C, this constant, \(K_\mathrm{w}\), is equal to \(1 \times 10^{-14}\). This concerted activity of ions allows us to calculate unknown concentrations when one of the ionic concentrations is known. The equation is straightforward:

• \([\mathrm{H}^{+}] \times [\mathrm{OH}^{-}] = K_\mathrm{w}\)

Many problems, like those from a textbook, use this relationship as a way to find either \([\mathrm{H}^{+}]\) or \([\mathrm{OH}^{-}]\) when one is already known. This intrinsic relationship showcases the dynamic equilibrium in which water remains even when acids or bases are present.

Acidic and Basic Solutions

The nature of a solution as acidic, basic, or neutral hinges on the balance between hydrogen ions and hydroxide ions. Solutions are deemed:

• Acidic: When the concentration of hydrogen ions \([\mathrm{H}^{+}]\) exceeds that of hydroxide ions \([\mathrm{OH}^{-}]\).
• Basic: When \([\mathrm{OH}^{-}]\) surpasses \([\mathrm{H}^{+}]\).
• Neutral: When both \([\mathrm{H}^{+}]\) and \([\mathrm{OH}^{-}]\) are equal, typically in pure water.

The use of the pH scale provides a convenient way to express the acidity or basicity of a solution:
- The pH scale ranges from 0 to 14, with 7 being neutral.- Values below 7 indicate an acidic solution, while values above 7 indicate a basic solution.
For example, if you compute the pH of a solution and find it to be below 7, you conclude the solution is acidic. Conversely, a pH above 7 denotes a basic solution. This can be a simple yet powerful tool in chemistry to understand the behavior of different solutions in various chemical reactions.

Concentration Calculations

Calculating concentrations of ions in a solution plays a crucial role in understanding chemical reactions and solution behavior. To compute the concentration of hydroxide ions \([\mathrm{OH}^{-}]\) given the concentration of hydrogen ions \([\mathrm{H}^{+}]\), you use the water ion product formula:
\[ [\mathrm{OH}^{-}] = \frac{K_\mathrm{w}}{[\mathrm{H}^{+}]} \]
This simple rearrangement allows you to determine the missing concentration in solutions where the hydrogen ion concentration is provided. For instance, if \([\mathrm{H}^{+}] = 0.0045 \mathrm{M}\), the calculation becomes:

• \([\mathrm{OH}^{-}] = \frac{1 \times 10^{-14}}{0.0045} = 2.22 \times 10^{-12} \mathrm{M}\)

These calculations are essential for understanding reactions where acidity or basicity greatly influences the chemical behavior and outcomes. Concentrations thus tell a more complete story about the balance of ions and offer insights into solution dynamics and reaction potential.