Problem 42
Question
The pH value of a solution measures the concentration of hydrogen ions, denoted by \(\left[\mathrm{H}^{+}\right]\), and is defined as $$ \mathrm{pH}=-\log \left[\mathrm{H}^{+}\right] $$ Use calculus to decide whether the pH value of a solution increases or decreases as the concentration of \(\mathrm{H}^{+}\) increases.
Step-by-Step Solution
Verified Answer
The pH decreases as the concentration of \( \mathrm{H}^{+} \) increases.
1Step 1: Define the Relationship
The pH value is defined by the equation \( \text{pH} = -\log \left[ \mathrm{H}^{+} \right] \).This equation shows the relationship between pH and the hydrogen ion concentration \( \left[ \mathrm{H}^{+} \right] \).
2Step 2: Differentiate the pH Equation
To analyze how pH changes as \( \left[ \mathrm{H}^{+} \right] \) changes, we need the derivative.The derivative of \( -\log \left( x \right) \) with respect to \( x \) is \( \frac{-1}{x \ln(10)} \).Therefore, \( \frac{d}{d\left[ \mathrm{H}^{+} \right]}(-\log \left[ \mathrm{H}^{+} \right]) = \frac{-1}{\left[ \mathrm{H}^{+} \right]\ln(10)} \).
3Step 3: Analyze the Sign of the Derivative
The derivative \( \frac{-1}{\left[ \mathrm{H}^{+} \right]\ln(10)} \) is negative because \( \left[ \mathrm{H}^{+} \right] > 0 \) and \( \ln(10) > 0 \).This negative value indicates the function is decreasing.
4Step 4: Conclude Based on Derivative
Since the derivative of the pH function is negative, this implies that the pH value decreases as the concentration of hydrogen ions \( \left[ \mathrm{H}^{+} \right] \) increases.
Key Concepts
Understanding pH ValueHydrogen Ion ConcentrationDerivative Analysis in pH
Understanding pH Value
The pH value is a critical measurement in chemistry that helps determine the acidity or basicity of a solution. It is a scale used to quantify the concentration of hydrogen ions, \([ \mathrm{H}^{+} ]\), in a solution. The pH scale typically ranges from 0 to 14, where values less than 7 indicate an acidic solution, values greater than 7 depict a basic or alkaline solution, and a pH of 7 denotes neutrality. When handling chemical solutions, knowing the pH can be incredibly important, especially for biochemical reactions that are pH-sensitive.
pH is mathematically expressed as:
\( ext{pH} = -\log \left[ \mathrm{H}^{+} \right]\).
This logarithmic scale means that each whole number change represents a tenfold change in \([ \mathrm{H}^{+} ]\). Thus, a decrease in pH indicates an increase in hydrogen ion concentration, making the solution more acidic.
pH is mathematically expressed as:
\( ext{pH} = -\log \left[ \mathrm{H}^{+} \right]\).
This logarithmic scale means that each whole number change represents a tenfold change in \([ \mathrm{H}^{+} ]\). Thus, a decrease in pH indicates an increase in hydrogen ion concentration, making the solution more acidic.
Hydrogen Ion Concentration
Hydrogen ion concentration, denoted as \([ \mathrm{H}^{+} ]\), signifies the number of hydrogen ions present in a solution. These ions are responsible for the acidity of a solution. In a more intuitive sense, it's like measuring how many hydrogen protons are in a container of water or other solvents.
The greater the hydrogen ion concentration, the more acidic the solution becomes, which decreases the pH. Conversely, a lower concentration of \([ \mathrm{H}^{+} ]\) ions correlates to a higher pH and a more basic solution. This relationship is central in fields like environmental science, biochemistry, and medicine.
Understanding and managing \([ \mathrm{H}^{+} ]\) levels can be crucial for activities such as wastewater treatment or maintaining homeostasis in living organisms, where specific pH levels are required for enzyme function and biological processes.
The greater the hydrogen ion concentration, the more acidic the solution becomes, which decreases the pH. Conversely, a lower concentration of \([ \mathrm{H}^{+} ]\) ions correlates to a higher pH and a more basic solution. This relationship is central in fields like environmental science, biochemistry, and medicine.
Understanding and managing \([ \mathrm{H}^{+} ]\) levels can be crucial for activities such as wastewater treatment or maintaining homeostasis in living organisms, where specific pH levels are required for enzyme function and biological processes.
Derivative Analysis in pH
Calculus, especially derivative analysis, plays a pivotal role in predicting how changes in one variable affect another. In this context, using calculus helps us understand how the pH value of a solution varies with changing hydrogen ion concentration, \([ \mathrm{H}^{+} ]\).
To examine this relationship, we differentiate the pH function:
\( \frac{d}{d \left[ \mathrm{H}^{+} \right]}(-\log \left[ \mathrm{H}^{+} \right]) = \frac{-1}{\left[ \mathrm{H}^{+} \right]\ln(10)} \).
This derivative is negative across all positive values of \([ \mathrm{H}^{+} ]\) because \( \ln(10) \) is a positive constant. A negative derivative tells us that as the concentration of hydrogen ions increases, the pH decreases. This inverse relationship implies that a solution becomes more acidic as \([ \mathrm{H}^{+} ]\) levels rise.
This analytical method offers a mathematical way to predict how solutions will react in various chemical and biological processes, reinforcing the practical implications of calculus in scientific research and everyday applications.
To examine this relationship, we differentiate the pH function:
\( \frac{d}{d \left[ \mathrm{H}^{+} \right]}(-\log \left[ \mathrm{H}^{+} \right]) = \frac{-1}{\left[ \mathrm{H}^{+} \right]\ln(10)} \).
This derivative is negative across all positive values of \([ \mathrm{H}^{+} ]\) because \( \ln(10) \) is a positive constant. A negative derivative tells us that as the concentration of hydrogen ions increases, the pH decreases. This inverse relationship implies that a solution becomes more acidic as \([ \mathrm{H}^{+} ]\) levels rise.
This analytical method offers a mathematical way to predict how solutions will react in various chemical and biological processes, reinforcing the practical implications of calculus in scientific research and everyday applications.
Other exercises in this chapter
Problem 42
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Let $$ f(x)=\frac{2}{1+e^{-x}}, x \in \mathbf{R} $$ (a) Determine where \(f(x)\) is increasing and where it is decreasing. (b) Where is the function concave up
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In Problems \(41-46\), assume that \(a\) is a positive constant. Find the general antiderivative of the given function. $$ f(x)=\sin ^{2}\left(a^{2} x+1\right)
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Suppose that \(f(x)=x^{3} .\) Explain why there exists a point \(c\) in the interval \((-1,1)\) such that \(f^{\prime}(c)=1\).
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