Problem 50
Question
Determine how many more times acidic the first substance is compared to the second substance. battery acid, \(\mathrm{pH}=1 ;\) lye, \(\mathrm{pH}=13\)
Step-by-Step Solution
Verified Answer
Battery acid is 10^{12} times more acidic than lye.
1Step 1: Understand pH scale
The pH scale measures how acidic or basic a substance is on a scale from 0 to 14. A lower pH value indicates higher acidity, while a higher pH value denotes greater basicity. A pH of 7 is neutral.
2Step 2: Recall the relationship between pH and hydrogen ion concentration
The pH value is calculated using the formula: \[\text{pH} = -\log_{10} [H^+]\]This means that a change of one unit on the pH scale corresponds to a tenfold change in the concentration of hydrogen ions.
3Step 3: Calculate hydrogen ion concentrations
We calculate the hydrogen ion concentration for both substances:\[ [H^+]_{\text{battery acid}} = 10^{-1} = 0.1 \, \text{mol/L} \] \[ [H^+]_{\text{lye}} = 10^{-13} \, \text{mol/L} \]
4Step 4: Determine how many times more acidic battery acid is
Find the ratio of the hydrogen ion concentrations:\[ \frac{[H^+]_{\text{battery acid}}}{[H^+]_{\text{lye}}} = \frac{0.1}{10^{-13}} = 10^{12} \]Thus, battery acid is \(10^{12}\) times more acidic than lye.
Key Concepts
Hydrogen Ion ConcentrationAcidity and BasicityLogarithmic Scale
Hydrogen Ion Concentration
The concentration of hydrogen ions \([H^+]\) in a solution plays a crucial role in determining its acidity. \([H^+]\) represents how many hydrogen ions are present in one liter of solution.
The more hydrogen ions in the solution, the more acidic it is. \(10^{-1} \text{mol/L}\) means 0.1 moles of hydrogen ions per liter, indicating high acidity. \(10^{-13} \text{mol/L}\) signifies very few hydrogen ions, showing low acidity or basicity.
To compare substances, understanding their respective \([H^+]\) is essential. This concentration helps in calculating and understanding how many times more acidic or basic one substance is compared to another.
The more hydrogen ions in the solution, the more acidic it is. \(10^{-1} \text{mol/L}\) means 0.1 moles of hydrogen ions per liter, indicating high acidity. \(10^{-13} \text{mol/L}\) signifies very few hydrogen ions, showing low acidity or basicity.
To compare substances, understanding their respective \([H^+]\) is essential. This concentration helps in calculating and understanding how many times more acidic or basic one substance is compared to another.
Acidity and Basicity
Acidity and basicity are two sides of the pH scale that defines where substances lie in terms of being acidic, neutral, or basic. A substance with a pH less than 7 is acidic, more than 7 is basic, and exactly 7 is neutral.
Each step down the pH scale from 7 indicates a tenfold increase in acidity. For example, pH 1 battery acid is highly acidic. Each step up indicates a tenfold increase in basicity. Here, lye with a pH of 13 is strongly basic.
Comparing substances involves finding their pH difference, which tells us how much more acidic or basic one is. For instance, a difference between pH 1 and pH 13 shows a significant contrast in acidity and basicity.
Each step down the pH scale from 7 indicates a tenfold increase in acidity. For example, pH 1 battery acid is highly acidic. Each step up indicates a tenfold increase in basicity. Here, lye with a pH of 13 is strongly basic.
Comparing substances involves finding their pH difference, which tells us how much more acidic or basic one is. For instance, a difference between pH 1 and pH 13 shows a significant contrast in acidity and basicity.
Logarithmic Scale
The pH scale is a logarithmic scale, meaning each whole number change reflects a tenfold change in hydrogen ion concentration. This nature of the scale helps handle the vast range of hydrogen ion concentrations in everyday substances.
For instance, moving from pH 1 to pH 2 indicates a decrease in acidity by a factor of 10. Hence, the log function condenses large or small numerical values into a more manageable form.
Understanding the logarithmic nature of the pH scale is essential because it simplifies the comparison of chemical substances with vastly different ion concentrations, enabling us to easily determine relative acidity or basicity.
For instance, moving from pH 1 to pH 2 indicates a decrease in acidity by a factor of 10. Hence, the log function condenses large or small numerical values into a more manageable form.
Understanding the logarithmic nature of the pH scale is essential because it simplifies the comparison of chemical substances with vastly different ion concentrations, enabling us to easily determine relative acidity or basicity.
Other exercises in this chapter
Problem 50
Either use factoring or the quadratic formula to solve the given equation. $$ 10^{2 x}-103\left(10^{x}\right)+300=0 $$
View solution Problem 50
Sketch the graph of the given piecewise-defined function \(f\). $$ f(x)=\left\\{\begin{array}{l} e^{-x}, x \leq 0 \\ -e^{x}, x>0 \end{array}\right. $$
View solution Problem 50
In Problems \(47-50\), the given function \(f\) is one-to-one. Find \(f^{-1}\) and give its domain and range. $$ f(x)=5+\log _{2} \frac{1}{x} $$
View solution Problem 51
In Problems \(51-56,\) find the \(x\) -intercepts of the graph of the given function. $$ f(x)=e^{x+4}-e $$
View solution