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Chapter 17: Problem 23

A weak acid has a \(K_{\mathrm{a}}\) of \(6.5 \times 10^{-5} .\) What is the value of \(\mathrm{p} K_{\mathrm{a}}\) for the acid?

Short Answer

Expert verified
The value of \(pK_a\) is 4.1871.

Step by step solution

01

Understand the Relationship

Recall that the relationship between the acid dissociation constant \(K_a\) and its \( pK_a \) is given by the formula: \[ pK_a = -\log(K_a) \] This relationship shows how to convert the \(K_a\) value into the \(pK_a\) value by taking the negative logarithm.
02

Apply the Formula

Using the formula from Step 1, we can substitute the given \(K_a\) value:\[ pK_a = -\log(6.5 \times 10^{-5}) \] This step involves calculating the logarithm of the \(K_a\) value.
03

Calculate the Logarithm

Calculate \(-\log(6.5 \times 10^{-5})\). First, find \(\log(6.5)\) which is approximately \(0.8129\), and \(\log(10^{-5}) = -5\). So,\[ \log(6.5 \times 10^{-5}) = 0.8129 - 5 = -4.1871 \]
04

Determine the pK_a

To find \( pK_a \), take the negative of the result from Step 3:\[ pK_a = -(-4.1871) = 4.1871 \] Thus, the \( pK_a \) value is the positive version of \(-4.1871\).

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Key Concepts

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

Acid Dissociation Constant
The acid dissociation constant, denoted as \( K_a \), measures the strength of an acid in a solution. It represents the equilibrium constant for the dissociation of a weak acid into its ions. For a general weak acid, \( HA \), the dissociation can be represented as:\[ HA \rightleftharpoons H^+ + A^- \]The equilibrium expression for this reaction is:\[ K_a = \frac{[H^+][A^-]}{[HA]} \]Where:
  • \([H^+]\) is the concentration of hydrogen ions.
  • \([A^-]\) is the concentration of the anion.
  • \([HA]\) is the concentration of the undissociated acid.
The higher the \( K_a \), the stronger the acid, meaning it dissociates more in solution. Conversely, a lower \( K_a \) indicates a weaker acid, which is less dissociated.
Logarithmic Conversion
Logarithmic conversion is essential in chemistry to simplify large ranges of numbers into a manageable form. When dealing with \( K_a \), which can vary greatly in magnitude, we use the \( pK_a \) scale. This conversion involves calculating the negative logarithm of the \( K_a \):\[ pK_a = -\log(K_a) \]This negative logarithm scale is useful because it turns very small numbers (like \( 10^{-5} \) for weak acids) into more straightforward numbers (like 5) for easy comparison.
  • \( \log(6.5) \) gives you a value, for example, approximately 0.8129.
  • \( \log(10^{-5}) = -5 \), turning a very small exponent into a simple negative integer.
After finding \( \log(K_a) \), we take its negative to obtain the \( pK_a \). This simplifies our understanding of an acid鈥檚 strength using these concise, comparable values.
Weak Acid
A weak acid is an acid that does not completely dissociate in a solution. It is characterized by a low \( K_a \) value, indicating partial ionization in water. For instance, imagine a weak acid like acetic acid:\[ CH_3COOH \rightleftharpoons H^+ + CH_3COO^- \]The equilibrium nature means that both the acid and its ions are present in significant amounts in the solution. Here are some key aspects of weak acids:
  • They have high \( pK_a \) values due to their small \( K_a \) values. For instance, the example in the exercise shows a \( pK_a \) value of 4.1871, typical for weak acids.
  • Because they don鈥檛 release many \( H^+ \) ions, their solutions have higher pH values compared to strong acids.
  • In practical terms, this means weak acids are less aggressive and are often used as food preservatives or in buffer solutions due to their stability and mildness.
Understanding weak acids is crucial for studying equilibrium and reaction rates in chemistry.

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Most popular questions from this chapter

In each of the following acid-base reactions, identify the Bronsted acid and base on the left and their conjugate partners on the right. $$\begin{aligned}&\text { (a) } \mathrm{C}_{5} \mathrm{H}_{5} \mathrm{N}(\mathrm{aq})+\mathrm{CH}_{3} &\mathrm{C}_{5} \mathrm{H}_{5} \mathrm{NH}^{+}(\mathrm{aq})+\mathrm{CH}_{3} \mathrm{CO}_{2}^{-}(\mathrm{aq})\end{aligned}$$ (b) \(\mathrm{N}_{2} \mathrm{H}_{4}(\mathrm{aq})+\mathrm{HSO}_{4}^{-}(\mathrm{aq}) \rightleftarrows \mathrm{N}_{2} \mathrm{H}_{5}^{+}(\mathrm{aq})+\mathrm{SO}_{4}^{2-}(\mathrm{aq})\). (c) \(\left[\mathrm{Al}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}(\mathrm{aq})+\mathrm{OH}^{-}(\mathrm{aq}) \rightleftarrows\) \(\left[\mathrm{Al}\left(\mathrm{H}_{2} \mathrm{O}\right)_{5} \mathrm{OH}\right]^{2+}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\ell)\).

For each of the following cases, decide whether the pH is less than 7 , equal to 7 , or greater than 7 . (a) \(25 \mathrm{mL}\) of \(0.45 \mathrm{M} \mathrm{H}_{2} \mathrm{SO}_{4}\) is mixed with \(25 \mathrm{mL}\) of \(0.90 \mathrm{M}\) \(\mathrm{NaOH}\) (b) 15 mL of 0.050 M formic acid, \(\mathrm{HCO}_{2} \mathrm{H}\), is mixed with \(15 \mathrm{mL}\) of \(0.050 \mathrm{M} \mathrm{NaOH}\) (c) \(25 \mathrm{mL}\) of \(0.15 \mathrm{M} \mathrm{H}_{2} \mathrm{C}_{2} \mathrm{O}_{4}\) (oxalic acid) is mixed with \(25 \mathrm{mL}\) of \(0.30 \mathrm{M} \mathrm{NaOH}\) (Both \(\mathrm{H}^{+}\) ions of oxalic acid are removed with NaOH.)

Several acids are listed here with their respective equilibrium constants. $$\begin{array}{c} \mathrm{HF}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\ell) \rightleftarrows \mathrm{H}_{3} \mathrm{O}^{+}(\mathrm{aq})+\mathrm{F}^{-}(\mathrm{aq}) \\\K_{a}=7.2 \times 10^{-4} \\\\\mathrm{HPO}_{4}^{-}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\ell) \rightleftarrows \mathrm{H}_{3} \mathrm{O}^{+}(\mathrm{aq})+\mathrm{PO}_{4}^{3-}(\mathrm{aq}) \\\K_{a}=3.6 \times 10^{-13} \\\\\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\ell) \rightleftarrows \mathrm{H}_{3} \mathrm{O}^{+}(\mathrm{aq})+\mathrm{CH}_{3} \mathrm{CO}_{2}^{-}(\mathrm{aq}) \\\K_{\mathrm{a}}=1.8 \times 10^{-5}\end{array}$$. (a) Which is the strongest acid? Which is the weakest acid? (b) What is the conjugate base of the acid HF? (c) Which acid has the weakest conjugate base? (d) Which acid has the strongest conjugate base?

For each of the following reactions, predict whether the equilibrium lies predominantly to the left or to the right. Explain your prediction briefly.(a) \(\mathrm{NH}_{4}^{+}(\mathrm{aq})+\mathrm{Br}^{-}(\mathrm{aq}) \rightleftarrows \mathrm{NH}_{3}(\mathrm{aq})+\mathrm{HBr}(\mathrm{aq})\).(b) \(\mathrm{HPO}_{4}^{2-}(\mathrm{aq})+\mathrm{CH}_{3} \mathrm{CO}_{2}^{-}(\mathrm{aq}) \rightleftarrows\) \(\mathrm{PO}_{4}^{3-}(\mathrm{aq})+\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H}(\mathrm{aq})\).(c) \(\mathrm{SO}_{4}^{2-}(\mathrm{aq})+\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H}(\mathrm{aq}) \rightleftarrows\).\(\mathrm{HSO}_{4}^{-}(\mathrm{aq})+\mathrm{CH}_{3} \mathrm{CO}_{2}^{-}(\mathrm{aq})\)

Equal molar quantities of hydrochloric acid and sodium hypochlorite (NaCIO) are mixed. (a) Write the balanced, net ionic equation for the acid-base reaction that can, in principle, occur. (b) Does the equilibrium lie to the right or left?
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