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Chapter 16: Problem 33

By what factor does \(\left[\mathrm{H}^{+}\right]\)change for a pH change of (a) \(2.00\) units, (b) \(0.50\) units?

Short Answer

Expert verified
For a pH change of (a) 2.00 units, the [H+] concentration changes by a factor of 0.01. For a pH change of (b) 0.50 units, the [H+] concentration changes by a factor of 0.3162.

Step by step solution

01

Review the formula for pH

The formula for pH is given by: \( \textrm{pH} = -\log_{10} [\textrm{H}^+] \) Where pH is the measure of acidity or basicity, and [H+] represents the concentration of hydronium ions in the solution.
02

Write the formula for the change in pH

The change in pH is can be written as: \( \Delta \textrm{pH} = \textrm{pH}_{1} - \textrm{pH}_{2} \) Where \( \textrm{pH}_{1} \) is the initial pH, \( \textrm{pH}_{2} \) is the final pH, and \( \Delta \textrm{pH} \) is the change in pH.
03

Write the formula for the change in H+ concentration

We can rewrite the pH formula given in step 1 by taking the antilog to find the concentration of the hydronium ions: \( [\textrm{H}^+] = 10^{-\textrm{pH}} \) So, the change in the concentration of H+ ions can be written as: \( \frac{[\textrm{H}^+]_{2}}{[\textrm{H}^+]_{1}} = \frac{10^{-\textrm{pH}_{2}}}{10^{-\textrm{pH}_{1}}} \)
04

Calculate pH change and corresponding H+ concentration change

(a) For a 2.00 unit increase in pH: \( \Delta \textrm{pH} = 2.00 \) Substitute into the formula in step 3: \( \frac{[\textrm{H}^+]_{2}}{[\textrm{H}^+]_{1}} = \frac{10^{-\textrm{pH}_{2}}}{10^{-\textrm{pH}_{1}}} = 10^{\frac{-\textrm{pH}_{2}+\textrm{pH}_{1}}{}} = 10^{-2.00} = 0.01 \) In this case, the [H+] concentration changes by a factor of 0.01 (reduces). (b) For a 0.50 unit increase in pH: \( \Delta \textrm{pH} = 0.50 \) Substitute into the formula in step 3: \( \frac{[\textrm{H}^+]_{2}}{[\textrm{H}^+]_{1}} = \frac{10^{-\textrm{pH}_{2}}}{10^{-\textrm{pH}_{1}}} = 10^{\frac{-\textrm{pH}_{2}+\textrm{pH}_{1}}{}} = 10^{-0.50} = 0.3162 \) In this case, the [H+] concentration changes by a factor of 0.3162 (reduces).
05

Conclusion

For a pH change of (a) 2.00 units, the [H+] concentration changes by a factor of 0.01. For a pH change of (b) 0.50 units, the [H+] concentration changes by a factor of 0.3162.

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Key Concepts

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

Hydronium Ion Concentration
The hydronium ion, represented by \([H^+]\), is a key player in acid-base chemistry. Hydronium ions form when an acid dissolves in water and its hydrogen ions (H^+) associate with water molecules. The concentration of these ions is crucial in determining the acidity of a solution.

When it comes to measuring \([H^+]\) in a solution, we use the formula: \([H^+]= 10^{-\text{pH}}\), which connects directly to the pH value of the solution. A change in the pH directly impacts the concentration of the hydronium ions.
  • A decrease in pH represents an increase in \([H^+]\) concentration, indicating more acidic conditions.
  • An increase in pH points to a decrease in \([H^+]\) concentration, reflecting more basic or alkaline conditions.
Understanding this relationship is fundamental in predicting how a solution will behave chemically due to changes in hydronium ion levels.
Acid-Base Equilibrium
Acid-base equilibrium is a balancing act of sorts, where acids and bases maintain a particular concentration of ions in a solution. Acids provide hydrogen ions ( H^+ ), while bases supply hydroxide ions ( OH^鈭� ). The equilibrium is governed by how these ions interact.

The autoionization of water is a classic example, where water molecules dissociate into H^+ and OH^鈭� . The product of these concentrations is constant at a given temperature, symbolized by Kw, and is significant for maintaining equilibrium in pure water and solutions.
  • The formula Kw = [H+][OH鈭�] is fundamental, where Kw is approximately 1.0 脳 10^{-14} at 25掳C.
  • This equilibrium constant explains how, in a neutral solution, the concentrations of H^+ and OH^鈭� will be equal, each being 1.0 脳 10^{-7} M.
In practical terms, understanding these equilibriums helps chemists and students alike predict reaction outcomes and balance solution compositions.
pH Calculation
Calculating pH is a vital skill in chemistry, providing insight into the acidity or basicity of a solution. The formula for pH is \(\text{pH} = -\log_{10} [\text{H}^+]\), which allows us to determine the pH from the hydronium ion concentration.

To calculate pH effectively, follow these steps:
  • Determine the hydronium ion concentration, \([H^+]\), of the solution.
  • Apply the pH formula to find the pH value.
  • If needed, use the inverse operation, the antilog, to solve for \([H^+]\), when given the pH.

Let's say you have a solution with a hydronium ion concentration of 1.0 脳 10^{-3} M. Using the formula, you would find \(\text{pH} = -\log_{10} (1.0 脳 10^{-3})= 3\). Hence, the solution is acidic. Understanding this process aids in the analysis and classification of different substances, enhancing your comprehension of chemical properties and reactions.

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Most popular questions from this chapter

Carbon dioxide in the atmosphere dissolves in raindrops to produce carbonic acid \(\left(\mathrm{H}_{2} \mathrm{CO}_{3}\right)\), causing the \(\mathrm{pH}\) of clean, unpolluted rain to range from about \(5.2\) to \(5.6\). What are the ranges of \(\left[\mathrm{H}^{+}\right]\)and \(\left[\mathrm{OH}^{-}\right]\)in the raindrops?

If a substance is an Arrhenius base, is it necessarily a BronstedLowry base? Is it necessarily a Lewis base?

Citric acid, which is present in citrus fruits, is a triprotic acid (Table 16.3). (a) Calculate the pH of a \(0.040 \mathrm{M}\) solution of citric acid. (b) Did you have to make any approximations or assumptions in completing your calculations? (c) Is the concentration of citrate ion \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}^{3-}\right)\) equal to, less than, or greater than the \(\mathrm{H}^{+}\)ion concentration?

Indicate whether each of the following statements is true or false. For each statement that is false, correct the statement to make it true. (a) In general, the acidity of binary acids increases from left to right in a given row of the periodic table. (b) In a series of acids that have the same central atom, acid strength increases with the number of hydrogen atoms bonded to the central atom. (c) Hydrotelluric acid ( \(\left.\mathrm{H}_{2} \mathrm{Te}\right)\) is a stronger acid than \(\mathrm{H}_{2} \mathrm{~S}\) because Te is more electronegative than \(\mathrm{S}\).

Although the acid-dissociation constant for phenol \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{OH}\right)\) is listed in Appendix \(\mathrm{D}\), the base-dissociation constant for the phenolate ion \(\left(\mathrm{C}_{6} \mathrm{H}_{3} \mathrm{O}^{-}\right)\)is not. (a) Explain why it is not necessary to list both \(K_{a}\) for phenol and \(K_{b}\) for the phenolate ion. (b) Calculate \(K_{b}\) for the phenolate ion. (c) Is the phenolate ion a weaker or stronger base than ammonia?
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