Problem 32 - Chemistry: The Central Science

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 \(\left[\mathrm{OH}^{-}\right]\).

Step-by-Step Solution

Answer

For the given solutions: (a) \([\mathrm{OH}^{-}] = 2.22 \times 10^{-12} \mathrm{M}\), acidic (b) \([\mathrm{OH}^{-}] = 6.67 \times 10^{-6} \mathrm{M}\), basic (c) \([\mathrm{OH}^{-}] = 1.00 \times 10^{-7.5} \mathrm{M}\), acidic

Step 1: Recall ion product constant of water

We'll use the ion product constant of water, which is given as: $$K_w = [\mathrm{H}^{+}][\mathrm{OH}^{-}] = 1.00 \times 10^{-14}$$ This equation will help us find \([\mathrm{OH}^{-}]\) for each solution.

Step 2: Calculate \([\mathrm{OH}^{-}]\) for solution (a)

We're given that \([\mathrm{H}^{+}] = 0.0045 \mathrm{M}\). Using the ion product constant of water, $$[\mathrm{OH}^{-}] = \frac{K_w}{[\mathrm{H}^{+}]} = \frac{1.00 \times 10^{-14}}{0.0045}$$ $$[\mathrm{OH}^{-}] = 2.22 \times 10^{-12} \mathrm{M}$$

Step 3: Determine if solution (a) is acidic, basic, or neutral

We compare the concentrations of hydrogen and hydroxide ions: - If \([\mathrm{H}^{+}] > [\mathrm{OH}^{-}]\), the solution is acidic. - If \([\mathrm{H}^{+}] < [\mathrm{OH}^{-}]\), the solution is basic. - If \([\mathrm{H}^{+}] = [\mathrm{OH}^{-}]\), the solution is neutral. In solution (a), \([\mathrm{H}^{+}] = 0.0045 \mathrm{M}\) and \([\mathrm{OH}^{-}] = 2.22 \times 10^{-12} \mathrm{M}\). Since \([\mathrm{H}^{+}] > [\mathrm{OH}^{-}]\), solution (a) is acidic.

Step 4: Calculate \([\mathrm{OH}^{-}]\) for solution (b)

We are given that \([\mathrm{H}^{+}] = 1.5 \times 10^{-9} \mathrm{M}\). Using the ion product constant of water, $$[\mathrm{OH}^{-}] = \frac{K_w}{[\mathrm{H}^{+}]} = \frac{1.00 \times 10^{-14}}{1.5 \times 10^{-9}}$$ $$[\mathrm{OH}^{-}] = 6.67 \times 10^{-6} \mathrm{M}$$

Step 5: Determine if solution (b) is acidic, basic, or neutral

In solution (b), \([\mathrm{H}^{+}] = 1.5 \times 10^{-9} \mathrm{M}\) and \([\mathrm{OH}^{-}] = 6.67 \times 10^{-6} \mathrm{M}\). Since \([\mathrm{H}^{+}] < [\mathrm{OH}^{-}]\), solution (b) is basic.

Step 6: Calculate \([\mathrm{OH}^{-}]\) for solution (c)

We are given that \([\mathrm{H}^{+}]\) is 10 times greater than \([\mathrm{OH}^{-}]\). Let's denote this condition as: $$[\mathrm{H}^{+}] = 10[\mathrm{OH}^{-}]$$ We'll use the ion product constant of water to find the concentration of hydroxide ions: $$K_w = [\mathrm{H}^{+}][\mathrm{OH}^{-}] = (10[\mathrm{OH}^{-}])([\mathrm{OH}^{-}]) = 10[\mathrm{OH}^{-}]^2$$ $$[\mathrm{OH}^{-}]^2 = \frac{K_w}{10} = \frac{1.00 \times 10^{-14}}{10}$$ $$[\mathrm{OH}^{-}]^2 = 1.00 \times 10^{-15}$$ $$[\mathrm{OH}^{-}] = 1.00 \times 10^{-7.5} \mathrm{M}$$

Step 7: Determine if solution (c) is acidic, basic, or neutral

In solution (c), \([\mathrm{H}^{+}] = 10[\mathrm{OH}^{-}]\). Since \([\mathrm{H}^{+}] > [\mathrm{OH}^{-}]\), solution (c) is acidic. In summary: - Solution (a): \([\mathrm{OH}^{-}] = 2.22 \times 10^{-12} \mathrm{M}\), acidic. - Solution (b): \([\mathrm{OH}^{-}] = 6.67 \times 10^{-6} \mathrm{M}\), basic. - Solution (c): \([\mathrm{OH}^{-}] = 1.00 \times 10^{-7.5} \mathrm{M}\), acidic.

Key Concepts

Ion Product Constant of Water

The ion product constant of water, commonly symbolized as K_w, is a basic principle in chemistry that plays a vital role in understanding the nature of aqueous solutions. This constant represents the equilibrium between hydrogen ions (H+) and hydroxide ions (OH-) in pure water. At 25°C, K_w has a value of 1.00 \times 10^{-14}.

This equilibrium concept helps us deduce the concentration of hydroxide ions when given the concentration of hydrogen ions and vice versa. It's the product of the molar concentrations of these ions and remains constant for water at a given temperature. In mathematical terms, it's expressed as: \[K_w = [H^+][OH^-] = 1.00 \times 10^{-14}\]
For instance, if you have a hydrogen ion concentration and need to find the corresponding hydroxide ion concentration, you simply divide K_w by the [H+] value. This principle is also foundational for understanding the pH scale, which measures the acidity or basicity of a solution.

Acidic and Basic Solutions

Solutions can generally be classified as acidic, basic (alkaline), or neutral based on their pH value, which is directly related to the concentrations of hydrogen ions [H+] and hydroxide ions [OH-].

An acidic solution has a higher concentration of hydrogen ions than hydroxide ions, resulting in a pH of less than 7. An example of this might be lemon juice or vinegar. Conversely, a basic solution has a lower concentration of hydrogen ions than hydroxide ions, leading to a pH above 7, such as with baking soda or bleach.

Identifying Acidity or Basicity

To determine whether a solution is acidic, basic, or neutral, compare the concentrations of [H+] and [OH-]. If [H^+] > [OH^-], the solution is acidic. If [H^+] < [OH^-], it's basic. Equal concentrations imply a neutral solution, typically found in pure water. This comparison forms the basis for calculating pH and helps in predicting the behavior of substances in different chemical reactions.

pH and pOH Calculations

pH and pOH are logarithmic scales that express the acidity and basicity of solutions, respectively. The pH scale ranges from 0 to 14 in aqueous solutions, with 7 as neutral. pH is defined as the negative base 10 logarithm of the hydrogen ion concentration: \[pH = -\log[H^+]\]

pOH serves a similar function for hydroxide ions, and is calculated as: \[pOH = -\log[OH^-]\]
The pOH can also be determined using the relationship between pH, pOH, and pK_w (the negative logarithm of K_w), where: \[pK_w = pH + pOH = 14\]
Under this concept, if you know the concentration of one type of ion, you can calculate its pH or pOH, and thus, deduce the opposite ion's concentration and its corresponding p-value. For example, if the [H+] of a solution is known, one can calculate the pH and then find the pOH value to calculate [OH-]. By mastering pH and pOH calculations, students can predict the properties of a solution and understand the balance between acidic and basic ions.