General Chemistry: Principles and Modern Applications (10th Edition)

Published by Pearson Prentice Hal
ISBN 10: 0132064529
ISBN 13: 978-0-13206-452-1

Chapter 16 - Acids and Bases - Example 16-8 - Determining Percent Ionization as a Function of a Weak Acid Concentration - Page 717: Practice Example A

Answer

Percent ionization of 0.20 M HF: 5.5% Percent ionization of 0.020 M HF: 16.5%

Work Step by Step

1. Draw the ICE table for this equilibrium: $$\begin{vmatrix} Compound& [ HF ]& [ F^- ]& [ H_3O^+ ]\\ Initial& 0.20 & 0 & 0 \\ Change& -x& +x& +x\\ Equilibrium& 0.20 -x& 0 +x& 0 +x\\ \end{vmatrix}$$ 2. Write the expression for $K_a$, and substitute the concentrations: - The exponent of each concentration is equal to its balance coefficient. $$K_a = \frac{[Products]}{[Reactants]} = \frac{[ F^- ][ H_3O^+ ]}{[ HF ]}$$ $$K_a = \frac{(x)(x)}{[ HF ]_{initial} - x}$$ 3. Assuming $ 0.20 \gt\gt x:$ $$K_a = \frac{x^2}{[ HF ]_{initial}}$$ $$x = \sqrt{K_a \times [ HF ]_{initial}} = \sqrt{ 6.6 \times 10^{-4} \times 0.20 }$$ $x = 0.011 $ Percent ionization: $$\frac{ 0.011 }{ 0.20 } \times 100\% = 5.5 \%$$ The assumption is approximately valid, around 5%. ------- 1. Draw the ICE table for this equilibrium: $$\begin{vmatrix} Compound& [ HF ]& [ F^- ]& [ H_3O^+ ]\\ Initial& 0.020 & 0 & 0 \\ Change& -x& +x& +x\\ Equilibrium& 0.020 -x& 0 +x& 0 +x\\ \end{vmatrix}$$ 2. Write the expression for $K_a$, and substitute the concentrations: - The exponent of each concentration is equal to its balance coefficient. $$K_a = \frac{[Products]}{[Reactants]} = \frac{[ F^- ][ H_3O^+ ]}{[ HF ]}$$ $$K_a = \frac{(x)(x)}{[ HF ]_{initial} - x}$$ 3. Assuming $ 0.020 \gt\gt x:$ $$K_a = \frac{x^2}{[ HF ]_{initial}}$$ $$x = \sqrt{K_a \times [ HF ]_{initial}} = \sqrt{ 6.6 \times 10^{-4} \times 0.020 }$$ $x = 3.6 \times 10^{-3} $ 4. Test if the assumption was correct: $$\frac{ 3.6 \times 10^{-3} }{ 0.020 } \times 100\% = 18.0 \%$$ The percent is greater than 5%, therefore, the approximation is invalid. 5. Return for the original expression and solve for x: $$K_a = \frac{x^2}{[ HF ]_{initial} - x}$$ $$K_a [ HF ] - K_a x = x^2$$ $$x^2 + K_a x - K_a [ HF ] = 0$$ $$x_1 = \frac{- 6.6 \times 10^{-4} + \sqrt{( 6.6 \times 10^{-4} )^2 - 4 (1) (- 6.6 \times 10^{-4} ) ( 0.020 )} }{2 (1)}$$ $$x_1 = 3.3 \times 10^{-3} $$ $$x_2 = \frac{- 6.6 \times 10^{-4} - \sqrt{( 6.6 \times 10^{-4} )^2 - 4 (1) (- 6.6 \times 10^{-4} )( 0.020 )} }{2 (1)}$$ $$x_2 = -4.0 \times 10^{-3} $$ - The concentration cannot be negative, so $x_2$ is invalid. $$x = 3.3 \times 10^{-3} $$ Percent ionization: $$\frac{ 3.3 \times 10^{-3} }{ 0.020 } \times 100\% = 16.5 \%$$
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