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Chapter 15: Problem 67

Obtain the \(\mathrm{pH}\) corresponding to the following hydronium-ion concentrations. a. \(1.0 \times 10^{-8} M\) b. \(5.0 \times 10^{-12} M\) c. \(7.5 \times 10^{-3} M\) d. \(6.35 \times 10^{-9} M\)

Short Answer

Expert verified
a. 8; b. 11.3; c. 2.12; d. 8.20

Step by step solution

01

Understand the relationship between pH and hydronium-ion concentration

The pH of a solution is a measure of how acidic or basic it is. The pH is calculated using the negative logarithm (base 10) of the hydronium-ion ([H_3O^+]) concentration. The formula is:\[\text{pH} = -\log_{10}([H_3O^+])\]
02

Calculate pH for each concentration

Use the formula from Step 1 to calculate the pH for each given hydronium-ion concentration.
03

Step 3a: Calculate pH for 1.0 脳 10^{-8} M

For concentration \([H_3O^+] = 1.0 \times 10^{-8} \text{ M}\), calculate:\[\text{pH} = -\log_{10}(1.0 \times 10^{-8}) = 8\]
04

Step 3b: Calculate pH for 5.0 脳 10^{-12} M

For concentration \([H_3O^+] = 5.0 \times 10^{-12} \text{ M}\), calculate:\[\text{pH} = -\log_{10}(5.0 \times 10^{-12}) = 11.3\]
05

Step 3c: Calculate pH for 7.5 脳 10^{-3} M

For concentration \([H_3O^+] = 7.5 \times 10^{-3} \text{ M}\), calculate:\[\text{pH} = -\log_{10}(7.5 \times 10^{-3}) \approx 2.12\]
06

Step 3d: Calculate pH for 6.35 脳 10^{-9} M

For concentration \([H_3O^+] = 6.35 \times 10^{-9} \text{ M}\), calculate:\[\text{pH} = -\log_{10}(6.35 \times 10^{-9}) \approx 8.20\]

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Key Concepts

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

Hydronium-ion concentration
Hydronium ions, represented as \([H_3O^+]\), are crucial in determining the acidity of a solution. When an acid dissolves in water, it donates a proton \((H^+)\) to a water molecule, creating the hydronium ion. The concentration of these ions indicates the acidic strength of the solution:
  • Higher hydronium-ion concentration means stronger acidity.
  • Lower concentration implies weaker acidity.
The hydronium-ion concentration is typically expressed in moles per liter \((M)\), making it easier to handle mathematically. In exercises like finding \( ext{pH}\), we often convert this concentration into a more convenient scale, using logarithmic calculations.
Understanding hydronium-ion concentration helps bridge the gap between acid strength and pH, allowing us to see whether a solution leans towards being acidic or basic.
Acidic and basic solutions
Acidity and basicity are key properties of solutions defined by their \( ext{pH}\) values. The term "pH" means "potential of hydrogen" and represents how acidic or basic a solution is:
  • An acidic solution has a \( ext{pH}\) less than 7. This indicates a higher concentration of hydronium ions.
  • A basic (or alkaline) solution has a \( ext{pH}\) greater than 7. This signifies higher hydroxide ion concentrations than hydronium ions.
  • A neutral solution, such as pure water, exhibits a \( ext{pH}\) of exactly 7, where hydronium and hydroxide ion concentrations are equal.
The scale ranges from 0 to 14, making it an easy reference for determining the nature of solutions. Understanding \( ext{pH}\) facilitates many real-world applications, from formulating cleaning agents to ensuring the suitable conditions for aquatic life.
Logarithms in chemistry
Logarithms, powerful mathematical tools, are crucial in chemistry for simplifying calculations that involve concentrations, like \( ext{pH}\). The logarithmic scale allows us to express very large or very small numbers in a compact form.
The \( ext{pH}\) calculation uses the base-10 logarithm due to its capacity to easily express the changes in hydronium-ion concentrations:
  • The formula \( ext{pH} = -\log_{10}([H_3O^+])\) takes the logarithm of the hydronium concentration, making it simpler to grasp.
  • The negative sign flips the scale to ensure higher \([H_3O^+]\) leads to lower \( ext{pH}\).
Thus, each unit change in \( ext{pH}\) reflects a tenfold change in the hydronium-ion concentration. This highlights the sensitivity of the \( ext{pH}\) scale and aligns it with the way reactions behave in chemistry.
Mastering logarithms enriches one's ability to communicate effectively about chemical properties and reactions, especially in exercises involving pH calculations.

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Most popular questions from this chapter

Why do medicines frequently taste bitter?

Suppose you have \(557 \mathrm{~mL}\) of \(0.0300 \mathrm{M} \mathrm{HCl}\), and you want to make up a solution of \(\mathrm{HCl}\) that has a \(\mathrm{pH}\) of \(1.831 .\) What is the maximum volume (in liters) that you can make of this solution?

Some lemon juice has a hydronium-ion concentration of \(5.0 \times 10^{-3} M\). What is the \(\mathrm{pH}\) of the lemon juice?

For the following reactions, label each species as an acid or a base. Indicate the species that are conjugates of one another. a. \(\mathrm{HSO}_{4}^{-}+\mathrm{NH}_{3} \rightleftharpoons \mathrm{SO}_{4}^{2-}+\mathrm{NH}_{4}^{+}\) b. \(\mathrm{HPO}_{4}^{2-}+\mathrm{NH}_{4}^{+} \rightleftharpoons \mathrm{H}_{2} \mathrm{PO}_{4}^{-}+\mathrm{NH}_{3}\) c. \(\mathrm{Al}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}^{3+}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{Al}\left(\mathrm{H}_{2} \mathrm{O}\right)_{5} \mathrm{OH}^{2+}+\mathrm{H}_{3} \mathrm{O}^{+}\) d. \(\mathrm{SO}_{3}^{2-}+\mathrm{NH}_{4}^{+} \rightleftharpoons \mathrm{HSO}_{3}^{-}+\mathrm{NH}_{3}\)

You make solutions of ammonia and sodium hydroxide by adding the same moles of each solute to equal volumes of water. Which solution would you expect to have the higher pH?
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