Problem 127 - Introductory Chemistry Atoms First

Question

Consider the following statement: "As the \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration in an aqueous solution increases, the \(\mathrm{OH}^{-}\) concentration must decrease." (a) Why is this true? (b) Is there any way to stop the \(\mathrm{OH}^{-}\) concentration from decreasing as you increase the \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration?

Step-by-Step Solution

Answer

(a) The statement is true because the ion product constant for water (\(K_w\)) is equal to the product of the concentrations of hydronium ions (\(\mathrm{H_3O^{+}}\)) and hydroxide ions (\(\mathrm{OH^{-}}\)) in any aqueous solution at a given temperature. Since \(K_w\) is constant, if the concentration of \(\mathrm{H_3O^+}\) increases, the concentration of \(\mathrm{OH^-}\) must decrease to maintain the constant value. (b) It is not possible to stop the \(\mathrm{OH^-}\) concentration from decreasing as you increase the \(\mathrm{H_3O^+}\) concentration, because this would violate the relationship established by \(K_w\), which is a fixed value at a given temperature and only changes with temperature variations.

Step 1: Understanding the Ion Product Constant for Water (\(K_w\))

In any aqueous solution, water molecules will undergo a process known as autoprotolysis or autoionization, which involves the transfer of a proton from one water molecule to another. This phenomenon can be represented by the following chemical equation: \[ \mathrm{H_2O} \leftrightarrows \mathrm{H_3O^{+}} + \mathrm{OH^{-}} \] The equilibrium constant for this reaction is called the ion product constant for water (\(K_w\)). It is equal to the product of the concentrations of hydronium ions (\(\mathrm{H_3O^{+}}\)) and hydroxide ions (\(\mathrm{OH^{-}}\)) in any aqueous solution at a given temperature: \[ K_w = [\mathrm{H_3O^{+}}] [\mathrm{OH^{-}}] \] The value of \(K_w\) varies with temperature, but at 25°C, \(K_w = 1.0 \times 10^{-14}\).

Step 2: Relationship between \(\mathrm{H_3O^+}\) and \(\mathrm{OH^-}\) concentrations

Since the value of \(K_w\) is constant at a given temperature, if the concentration of \(\mathrm{H_3O^+}\) increases, the concentration of \(\mathrm{OH^-}\) must decrease to maintain the constant value. Therefore, the given statement is true. #b) Evaluate if there is any way to stop the \(\mathrm{OH^-}\) from decreasing as the \(\mathrm{H_3O^+}\) increases#

Step 3: Possible scenarios to maintain both concentrations

Since the product of the two concentrations must equal to \(K_w\), if we were to increase the \(\mathrm{H_3O^+}\) concentration without changing the \(\mathrm{OH^-}\) concentration, we would violate the relationship established by \(K_w\). The only way to achieve simultaneous increase in both concentrations would be to somehow increase the value of \(K_w\). However, this is not feasible because \(K_w\) is a fixed value at a given temperature and only changes with temperature variations. Trying to stop the \(\mathrm{OH^-}\) concentration from decreasing when \(\mathrm{H_3O^+}\) concentration is increased goes against the fundamental principles of acid-base equilibrium in aqueous solutions. Therefore, it is not possible to stop the \(\mathrm{OH^-}\) concentration from decreasing as you increase the \(\mathrm{H_3O^+}\) concentration, in accordance with \(K_w\).

Key Concepts

Ion Product Constant for Water

The ion product constant for water, commonly denoted as \(K_w\), is a key concept in understanding the acid-base equilibrium in aqueous solutions. This constant arises from the autoprotolysis (or autoionization) of water where water molecules transfer protons between each other. This process can be simply described by the chemical equation: \[\mathrm{H_2O} \leftrightarrows \mathrm{H_3O^{+}} + \mathrm{OH^{-}}\]At any given temperature, the equilibrium constant for this reaction, \(K_w\), is defined as the product of the concentrations of hydronium ions (\(\mathrm{H_3O^{+}}\)) and hydroxide ions (\(\mathrm{OH^{-}}\)). For instance, at 25°C, \(K_w\) has a standard value of \(1.0 \times 10^{-14}\).

• Maintaining this equilibrium is crucial for the stability of aqueous solutions.
• Changes in temperature can cause variations in the value of \(K_w\).

Hence, a deep understanding of \(K_w\) helps comprehend why changes in one ion concentration affect the other.

Autoprotolysis

Autoprotolysis, also known as self-ionization of water, refers to the natural process in which water molecules donate and accept protons amongst themselves forming hydronium ions (\(\mathrm{H_3O^{+}}\)) and hydroxide ions (\(\mathrm{OH^{-}}\)). Although neutral water predominantly comprises water molecules, a tiny fraction undergoes autoprotolysis, establishing a balance between these ions.

• This equilibrium ensures that the product of the concentrations remains constant, governed by \(K_w\).
• The phenomenon is crucial for explaining the behavior of acids and bases in water.

Understanding autoprotolysis is essential as it forms the foundation for explaining the presence of \(\mathrm{H_3O^{+}}\) and \(\mathrm{OH^{-}}\), setting the ground for interpreting the characteristics of acidic and basic solutions.

Hydronium Ions

Hydronium ions \(\mathrm{H_3O^{+}}\) play a significant role in determining the acidity of aqueous solutions. They are formed when water molecules accept protons, fundamentally representing the presence of acids in water. The concentration of these ions in a solution directly affects the pH, which inversely reflects the acidity.

• An increase in \(\mathrm{H_3O^{+}}\) concentration typically corresponds to a decrease in pH, indicating a more acidic environment.
• Due to the ion product constant, an increase in \(\mathrm{H_3O^{+}}\) results in a decrease in \(\mathrm{OH^{-}}\).

This reciprocal relationship underscores the delicate balance maintained in an aqueous system, demonstrating why it is impossible to independently alter the concentration of \(\mathrm{H_3O^{+}}\) without affecting \(\mathrm{OH^{-}}\).

Hydroxide Ions

Hydroxide ions \(\mathrm{OH^{-}}\) are the counterparts to hydronium ions, primarily involved in defining the basicity of a solution. These ions manifest when water molecules lose protons, facilitating the establishment of a basic solution. The concentration of \(\mathrm{OH^{-}}\) outlines how basic or alkaline a solution is.

• Increasing the concentration of \(\mathrm{OH^{-}}\) will make the solution more basic, thus raising its pH.
• Due to \(K_w\), an increase in \(\mathrm{OH^{-}}\) necessitates a decrease in \(\mathrm{H_3O^{+}}\) to maintain equilibrium.

Appreciating this dynamic provides insight into the interconnected nature of acids and bases, reinforcing that any fluctuation in \(\mathrm{OH^{-}}\) or \(\mathrm{H_3O^{+}}\) affects the equilibrium constant, \(K_w\), which remains consistent at a specific temperature.