Chapter 15: Problem 21

Calculate the \(\left[\mathrm{H}^{+}\right]\)for (a) black coffee, with a \(\mathrm{pH}\) of \(5.0\) (b) black tea with a pH of \(6.8\) (c) green tea with a pH of \(7.9\) (d) laundry detergent with a pH of \(9.7\)

Short Answer

Expert verified

Black Coffee: \([H^{+}] = 1 \times 10^{-5}\) M, Black Tea: \([H^{+}] \approx 1.58 \times 10^{-7}\) M, Green Tea: \([H^{+}] \approx 1.26 \times 10^{-8}\) M, Laundry Detergent: \([H^{+}] \approx 2 \times 10^{-10}\) M.

Step by step solution

- Understanding the Relationship between \(\text{pH}\) and \(\text{H}^{+}\)

The \(\text{pH}\) of a solution is related to the concentration of hydrogen ions \(\text{H}^{+}\) by the formula: \(\text{pH} = -\text{log}[\text{H}^{+}]\). To find \([\text{H}^{+}]\), we need to take the antilog (inverse log) of the negative \(\text{pH}\) value.

- Finding \([\text{H}^{+}]\) for Black Coffee (\text{pH} = 5.0)

Using the formula \([\text{H}^{+}] = 10^{-\text{pH}}\), substitute \(\text{pH} = 5.0\): \[ [\text{H}^{+}] = 10^{-5.0} = 1 \times 10^{-5} \text{ M} \]

- Finding \([\text{H}^{+}]\) for Black Tea (\text{pH} = 6.8)

Using the same formula, substitute \(\text{pH} = 6.8\): \[ [\text{H}^{+}] = 10^{-6.8} \] Using a calculator, find: \[ [\text{H}^{+}] \approx 1.58 \times 10^{-7} \text{ M} \]

- Finding \([\text{H}^{+}]\) for Green Tea (\text{pH} = 7.9)

Using the same formula, substitute \(\text{pH} = 7.9\): \[ [\text{H}^{+}] = 10^{-7.9} \] Using a calculator, find: \[ [\text{H}^{+}] \approx 1.26 \times 10^{-8} \text{ M} \]

- Finding \([\text{H}^{+}]\) for Laundry Detergent (\text{pH} = 9.7)

Using the same formula, substitute \(\text{pH} = 9.7\): \[ [\text{H}^{+}] = 10^{-9.7} \] Using a calculator, find: \[ [\text{H}^{+}] \approx 2 \times 10^{-10} \text{ M} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

acid-base chemistry

Acid-base chemistry is crucial in understanding how substances interact in solutions. Acids release hydrogen ions \(\text{H}^{+}\) when dissolved in water, while bases release hydroxide ions (\text{OH}^{-}). The pH scale, ranging from 0 to 14, measures the acidity or basicity of a solution. A pH less than 7 indicates an acidic solution, while a pH greater than 7 indicates a basic solution. Pure water, which is neutral, has a pH of 7. Understanding these fundamentals helps us predict how different substances will behave in aqueous environments.

hydrogen ion concentration

Hydrogen ion concentration \[ \text{H}^{+} \] determines the acidity of a solution. pH is the negative logarithm of the hydrogen ion concentration: \(\text{pH} = -\text{log}[\text{H}^{+}]\). If you know the pH, you can find \[ \text{H}^{+} \] by using the formula \[ [\text{H}^{+}] = 10^{-\text{pH}} \]. For example, black coffee with a pH of 5.0 has a hydrogen ion concentration of \[ 1 \times 10^{-5} \text{ M} \]. Similarly, for black tea with a pH of 6.8, \ [\text{H}^{+}] \ is approximately \[ 1.58 \times 10^{-7} \text{ M} \]. This relationship helps in determining the acidic or basic nature of different solutions.

logarithmic relationships in chemistry

Logarithmic relationships in chemistry simplify dealing with very large or very small numbers. The pH formula \(\text{pH} = -\text{log}[\text{H}^{+}]\) is a prime example. By using logarithms, we can compress a wide range of \[ \text{H}^{+} \] concentrations into a manageable scale between 0 and 14. Converting the pH back to \[ \text{H}^{+} \] might require a calculator for precision because the formula is \[ [\text{H}^{+}] = 10^{-\text{pH}} \]. For instance, a solution with a pH of 7.9 like green tea has a hydrogen ion concentration of approximately \[ 1.26 \times 10^{-8} \text{ M} \], demonstrating how we navigate between logarithmic and standard numerical forms.

chemical calculations

Chemical calculations often involve determining the pH or \[ \text{H}^{+} \] of a solution. It's critical to master the conversion between pH and hydrogen ion concentration. For example, laundry detergent with a pH of 9.7 has a \[ \text{H}^{+} \] concentration of \[ 2 \times 10^{-10} \text{ M} \]. Using the formula \([\text{H}^{+}] = 10^{-\text{pH}}\), you can calculate \[ \text{H}^{+} \] for any given pH accurately. These skills are not only applicable in academic settings but also in various industries like pharmaceuticals and environmental science, reinforcing the importance of precise chemical calculations.

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Calculate the \(\left[\mathrm{H}^{+}\right]\)for (a) black coffee, with a \(\mathrm{pH}\) of \(5.0\) (b) black tea with a pH of \(6.8\) (c) green tea with a pH of \(7.9\) (d) laundry detergent with a pH of \(9.7\)

Short Answer

Step by step solution

- Understanding the Relationship between \(\text{pH}\) and \(\text{H}^{+}\)

- Finding \([\text{H}^{+}]\) for Black Coffee (\text{pH} = 5.0)

- Finding \([\text{H}^{+}]\) for Black Tea (\text{pH} = 6.8)

- Finding \([\text{H}^{+}]\) for Green Tea (\text{pH} = 7.9)

- Finding \([\text{H}^{+}]\) for Laundry Detergent (\text{pH} = 9.7)

Key Concepts

acid-base chemistry

hydrogen ion concentration

logarithmic relationships in chemistry

chemical calculations

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