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Chapter 16: Problem 66

For solutions of a weak acid, a graph of \(\mathrm{pH}\) versus the logarithm of the initial acid concentration should be a straight line. What is the magnitude of the slope of that line?

Short Answer

Expert verified
The magnitude of the slope of the graph of pH versus the logarithm of the initial acid concentration for solutions of weak acids is \( \frac{1}{2} \).

Step by step solution

01

Write the ionization equation for a weak acid

Let's denote the weak acid as "HA". The ionization of a weak acid in water can be represented as: \[HA + H_2O \rightleftharpoons H_3O^+ + A^-\]
02

Write the expression for the ionization constant

The ionization constant, Ka, represents the equilibrium concentrations of the ions, which can be calculated as: \[ K_a = \frac{[H_3O^+][A^-]}{[HA]} \]
03

Simplify the expression for the concentrations

If we denote 'x' as the concentration of \(H_3O^+\), then \(A^-\) is also 'x' (since one HA produces one \(H_3O^+\) and one \(A^-\)). Thus, the concentration of HA remaining will be ([HA_initial] - x). Now we can rewrite the ionization constant expression as: \[K_a = \frac{x^2}{[HA_{initial}] - x}\]
04

Find the H3O+ concentration, x, in terms of Ka and the initial concentration of HA

Since the acid is weak, its ionization is small, and therefore we can assume that "x" is significantly smaller than the initial concentration of "HA": \[K_a = \frac{x^2}{[HA_{initial}]}\] Now we can solve for x: \[x = [H_3O^+] = \sqrt{K_a[HA_{initial}]}\]
05

Write the expression for pH and take the logarithm of both sides

The pH is related to the concentrations of the ions by the equation: \[ pH = -\log{[H_3O^+]}\] Substituting the expression we derived for x: \[pH = -\log{(\sqrt{K_a[HA_{initial}]})}\] Taking the logarithm of both sides, we get: \[\log{(pH)} = \log{(-(1/2))} \cdot (\log{(K_a)}) + (\log{([HA_{initial}] - \log{K_a})})\]
06

Find the magnitude of the slope

In the last expression, we can observe that the coefficient, '-(1/2)', is the slope of the graph. So, the magnitude of the slope is: \[ Slope = |- (\frac{1}{2})| = \frac{1}{2}\] Thus, the magnitude of the slope of the graph of pH versus the logarithm of the initial acid concentration for solutions of weak acids is 1/2.

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Most popular questions from this chapter

Explain the following observations: (a) HCl is a stronger acid than \(\mathrm{H}_{2} \mathrm{~S} ;\) (b) \(\mathrm{H}_{3} \mathrm{PO}_{4}\) is a stronger acid than \(\mathrm{H}_{3} \mathrm{AsO}_{4} ;\) (c) \(\mathrm{HBrO}_{3}\) is a stronger acid than \(\mathrm{HBrO}_{2}\); (d) \(\mathrm{H}_{2} \mathrm{C}_{2} \mathrm{O}_{4}\) is a stronger acid than \(\mathrm{HC}_{2} \mathrm{O}_{4}^{-} ;(\mathrm{e})\) benzoic acid \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COOH}\right)\) is a stronger acid than phenol \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{OH}\right)\).

At the freezing point of water \(\left(0^{\circ} \mathrm{C}\right), K_{w}=1.2 \times 10^{-15}\) Calculate \(\left[\mathrm{H}^{+}\right]\) and \(\left[\mathrm{OH}^{-}\right]\) for a neutral solution at this temperature.

The acid-dissociation constant for chlorous acid \(\left(\mathrm{HClO}_{2}\right)\) is \(1.1 \times 10^{-2} .\) Calculate the concentrations of \(\mathrm{H}_{3} \mathrm{O}^{+}, \mathrm{ClO}_{2}^{-}\) and \(\mathrm{HClO}_{2}\) at equilibrium if the initial concentration of \(\mathrm{HClO}_{2}\) is \(0.0125 \mathrm{M}\)

Calculate the percent ionization of hydrazoic acid \(\left(\mathrm{HN}_{3}\right)\) in solutions of each of the following concentrations \(\left(K_{a}\right.\) is given in Appendix D): (a) $0.400 \mathrm{M}\(, (b) \)0.100 \mathrm{M}\(, (c) \)0.0400 \mathrm{M}$.

Indicate whether each of the following statements is correct or incorrect. For those that are incorrect, explain why they are wrong. (a) Every Bronsted-Lowry acid is also a Lewis acid. (b) Every Lewis acid is also a Bronsted-Lowry acid. (c) Conjugate acids of weak bases produce more acidic solutions than conjugate acids of strong bases. (d) \(\mathrm{K}^{+}\) ion is acidic in water because it causes hydrating water molecules to become more acidic. (e) The percent ionization of a weak acid in water increases as the concentration of acid decreases.
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