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Mobile App Chapter 15: Problem 33
What is the \(\mathrm{OH}^{-}\) concentration in a solution having a pH of \(5.55\) ? (Hint: Use the \(K_{w}\) expression.)
Short Answer
Expert verified
The concentration of $\mathrm{OH}^{-}$ ions in a solution with a pH of 5.55 is approximately \(2.82 \times 10^{-9} M\).
Step by step solution
01
Understanding the ion product constant of water (Kw)
The ion product constant of water (Kw) is calculated by multiplying the concentration of hydrogen ions (H+) and hydroxide ions (OH-) in a solution. Mathematically, it can be represented as:
\[K_w = [H^+][OH^-]\]
At 25°C, the value of Kw is approximately \(1.0 \times 10^{-14}\).
02
Finding the concentration of H+ ions using the given pH value
The pH value represents the acidity or basicity of a solution. The pH is defined as the negative logarithm of the concentration of H+ ions in the solution. Mathematically, it can be represented as:
\[pH = -\log[H^+]\]
Given the pH value of 5.55, we can find the concentration of H+ ions as follows:
\[H^+ = 10^{-pH}\]
\[H^+ = 10^{-5.55}\]
03
Calculating the concentration of OH- ions using the Kw expression
Since we now have the concentration of H+ ions, we can use the Kw expression to find the concentration of OH- ions in the solution. Using the expression:
\[K_w = [H^+][OH^-]\]
\[OH^- = \frac{K_w}{[H^+]}\]
Substitute the values of Kw and H+ into the equation:
\[OH^- = \frac{1.0 \times 10^{-14}}{10^{-5.55}}\]
04
Solving for the concentration of OH- ions
Calculate the concentration of OH- ions in the solution:
\[OH^- ≈ 2.82 \times 10^{-9}\]
Therefore, the concentration of OH- ions in the solution with a pH of 5.55 is approximately \(2.82 \times 10^{-9} M\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Ion Product Constant of Water (Kw)
Understanding the ion product constant of water, denoted as Kw, is essential when discussing the pH and the OH- concentration of a solution. The Kw is a special equilibrium constant for water at a particular temperature, specifically around 25°C, where it has a value of approximately (1.0 \times 10^{-14}).
This constant is derived from the autoionization of water, where water molecules dissociate into hydrogen ions ((H^+)) and hydroxide ions ((OH^-)) in a reversible reaction. The equation representing this constant is: \[K_w = [H^+][OH^-]\]
The constant nature of Kw signifies that the product of the concentrations of (H^+) and (OH^-) in pure water is always the same at a given temperature, even when the individual concentrations might change in acidic or basic solutions. Hence, understanding Kw is vital for calculating (OH^-) concentrations when the pH is known.
This constant is derived from the autoionization of water, where water molecules dissociate into hydrogen ions ((H^+)) and hydroxide ions ((OH^-)) in a reversible reaction. The equation representing this constant is: \[K_w = [H^+][OH^-]\]
The constant nature of Kw signifies that the product of the concentrations of (H^+) and (OH^-) in pure water is always the same at a given temperature, even when the individual concentrations might change in acidic or basic solutions. Hence, understanding Kw is vital for calculating (OH^-) concentrations when the pH is known.
Hydrogen Ions Concentration
The pH of a solution essentially measures the concentration of hydrogen ions, (H^+), present in that solution. The pH scale quantifies acidity or basicity based on these ion concentrations.
The formula to find the hydrogen ion concentration from the pH value is an application of the definition of pH: \[pH = -log([H^+])\]
which can be rearranged to find ([H^+]) by taking the antilog (or inverse log): \[[H^+] = 10^{-pH}\]
From the given pH value, one can easily calculate the (H^+) concentration. For instance, a pH of 5.55 translates to a (H^+) concentration of \[10^{-5.55}\], which is crucial for further calculations involving pH or (OH^-) concentrations.
The formula to find the hydrogen ion concentration from the pH value is an application of the definition of pH: \[pH = -log([H^+])\]
which can be rearranged to find ([H^+]) by taking the antilog (or inverse log): \[[H^+] = 10^{-pH}\]
From the given pH value, one can easily calculate the (H^+) concentration. For instance, a pH of 5.55 translates to a (H^+) concentration of \[10^{-5.55}\], which is crucial for further calculations involving pH or (OH^-) concentrations.
Hydroxide Ions Concentration
Similarly to the (H^+) concentration, the concentration of hydroxide ions, (OH^-), can be essential for characterizing a solution's properties. While the pH scale is a direct way to comprehend the (H^+) concentration, we often need to compute the (OH^-) concentration as well, especially in a basic solution.
The relationship between the (Kw), (H^+), and (OH^-) concentrations allows us to do just that. With the established expression \[OH^- = \frac{K_w}{[H^+]}\]
we can find the specific (OH^-) concentration once the (H^+) concentration is known, utilizing the known value of (Kw). For example, if we have a (H^+) concentration from our previous calculations, we can then determine that the (OH^-) concentration is determined by dividing the constant (Kw) by the calculated (H^+) concentration.
The relationship between the (Kw), (H^+), and (OH^-) concentrations allows us to do just that. With the established expression \[OH^- = \frac{K_w}{[H^+]}\]
we can find the specific (OH^-) concentration once the (H^+) concentration is known, utilizing the known value of (Kw). For example, if we have a (H^+) concentration from our previous calculations, we can then determine that the (OH^-) concentration is determined by dividing the constant (Kw) by the calculated (H^+) concentration.
pH Calculation
Calculating pH is a common and straightforward process if the (H^+) concentration is known. The pH is calculated using the formula: \[pH = -log([H^+])\]
This logarithmic scale means that for every change of one in pH, the (H^+) concentration changes by an order of magnitude, making it easier to compare the relative acidity or basicity of solutions. Conversely, if you need to find the (H^+) concentration from a known pH, you simply need to reverse the process, which has been demonstrated earlier.
Additionally, understanding pH calculation is vital in many chemical and biological processes, as optimal pH levels are often needed for reactions to occur correctly or for organisms to thrive.
This logarithmic scale means that for every change of one in pH, the (H^+) concentration changes by an order of magnitude, making it easier to compare the relative acidity or basicity of solutions. Conversely, if you need to find the (H^+) concentration from a known pH, you simply need to reverse the process, which has been demonstrated earlier.
Additionally, understanding pH calculation is vital in many chemical and biological processes, as optimal pH levels are often needed for reactions to occur correctly or for organisms to thrive.
