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Chapter 16: Problem 51

Lactic acid $\left(\mathrm{CH}_{3} \mathrm{CH}(\mathrm{OH}) \mathrm{COOH}\right)\( has one acidic hydrogen. A \)0.10 \mathrm{M}$ solution of lactic acid has a pH of 2.44. Calculate \(K_{a}\)

Short Answer

Expert verified
The \(K_{a}\) value for lactic acid in the given solution is approximately \(1.3 \times 10^{-4}\).

Step by step solution

01

Calculate the H+ ion concentration

We can use the formula pH = \(-\log_{10}([\mathrm{H}^+])\) to calculate the concentration of \(\mathrm{H}^+\) ions in the solution. Rearranging the formula for calculating \([\mathrm{H}^+]\): \([\mathrm{H}^+] = 10^{-\text{pH}}\) Now, let's plug in pH value given in the problem into the above formula: \([\mathrm{H}^+] = 10^{-2.44} \approx 3.6 \times 10^{-3} \, \mathrm{M}\)
02

Write the equilibrium expression for lactic acid

Next, let's write out the equilibrium of lactic acid dissociation: \(\mathrm{CH}_{3} \mathrm{CH}(\mathrm{OH})\mathrm{COOH} \rightleftharpoons \mathrm{CH}_{3} \mathrm{CH}(\mathrm{OH})\mathrm{COO}^{-} + \mathrm{H}^{+}\) As given, the initial concentration of lactic acid is \(0.10 \; \mathrm{M}\). Let x represent the change in concentrations at equilibrium: [CH3CH(OH)COOH] = 0.10 - x [CH3CH(OH)COO-] = x [H+] = x + 3.6 x 10^{-3} Since we know the concentration of H+ from Step 1, we can find the concentrations of the dissociated components at equilibrium.
03

Express Ka in terms of x and given concentrations

The equilibrium expression is given by the formula for \(K_{a}\): \(K_{a} = \dfrac{[\mathrm{CH}_{3} \mathrm{CH}(\mathrm{OH})\mathrm{COO}^{-}][\mathrm{H}^+]}{[\mathrm{CH}_{3} \mathrm{CH}(\mathrm{OH})\mathrm{COOH}]}\) Now we can substitute the values that we found in Step 2 into the equilibrium expression: \(K_{a} = \dfrac{(x)(x + 3.6 \times 10^{-3})}{0.10 - x}\)
04

Approximations and solving for Ka

Since the pH is 2.44, the dissociation of lactic acid is very small, and thus we can assume that x is much smaller than the initial concentration, i.e., \(x << 0.10\). We can further simplify the expression and solve for \(K_{a}\). \(K_{a} = \dfrac{(x)(x + 3.6 \times 10^{-3})}{0.10}\) Now, we know the H+ concentration from Step 1, which is approximately 3.6 x 10^{-3}: \(K_{a} = \dfrac{(3.6 \times 10^{-3})(3.6 \times 10^{-3})}{0.10} \) Finally, let's calculate the K\(_{a}\) value: \(K_{a} \approx 1.3 \times 10^{-4}\) Therefore, the \(K_{a}\) value for lactic acid in the given solution is approximately \(1.3 \times 10^{-4}\).

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Most popular questions from this chapter

Predict whether aqueous solutions of the following compounds are acidic, basic, or neutral: $(\mathbf{a}) \mathrm{NH}_{4} \mathrm{Br},(\mathbf{b}) \mathrm{FeCl}_{3},$ (c) \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) (e) \(\mathrm{NaHC}_{2} \mathrm{O}_{4}\). (d) \(\mathrm{KClO}_{4}\),

Ammonia, \(\mathrm{NH}_{3}\), acts as an Arrhenius base, a Brønsted-Lowry base, and a Lewis base, in aqueous solution. Write out the reaction \(\mathrm{NH}_{3}\) undergoes with water and explain what properties of ammonia correspond to each of the three definitions of "base."

Oxalic acid \(\left(\mathrm{H}_{2} \mathrm{C}_{2} \mathrm{O}_{4}\right)\) is a diprotic acid. By using data in Appendix \(\mathrm{D}\) as needed, determine whether each of the following statements is true: (a) \(\mathrm{H}_{2} \mathrm{C}_{2} \mathrm{O}_{4}\) can serve as both a Bronsted-Lowry acid and a Brønsted-Lowry base. (b) $\mathrm{C}_{2} \mathrm{O}_{4}^{2-}\( is the conjugate base of \)\mathrm{HC}_{2} \mathrm{O}_{4}^{-}$. (c) An aqueous solution of the strong electrolyte \(\mathrm{KHC}_{2} \mathrm{O}_{4}\) will have \(\mathrm{pH}<7\).

Predict the stronger acid in each pair: (a) \(\mathrm{HCl}\) or HF; (b) \(\mathrm{H}_{3} \mathrm{PO}_{4}\) or \(\mathrm{H}_{3} \mathrm{AsO}_{4} ;\) (c) \(\mathrm{HBrO}_{3}\) or \(\mathrm{HBrO}_{2}\) (d) \(\mathrm{H}_{2} \mathrm{C}_{2} \mathrm{O}_{4}\) or \(\mathrm{HC}_{2} \mathrm{O}_{4} \overline{;} ;(\mathbf{e})\) benzoic acid \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COOH}\right)\) or phenol \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{OH}\right) .\)

(a) Write a chemical equation that illustrates the autoionization of water. (b) Write the expression for the ionproduct constant for water, $K_{w} .(\mathbf{c})$ If a solution is described as basic, which of the following is true: (i) \(\left[\mathrm{H}^{+}\right]>\left[\mathrm{OH}^{-}\right]\), (ii) \(\left[\mathrm{H}^{+}\right]=\left[\mathrm{OH}^{-}\right],\) or (iii) \(\left[\mathrm{H}^{+}\right]<\left[\mathrm{OH}^{-}\right] ?\)
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