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Chapter 43: Problem 9

Based on the relationships given in the Information sections, what is the relationship between \(\mathrm{pH}, \mathrm{pOH}\), and \(\mathrm{p} K_{\mathrm{w} ?}\)

Short Answer

Expert verified
The relationship between pH, pOH, and \(pK_w\) is given by the equation \(pK_w\) = pH + pOH. This relationship states that at a constant temperature, the sum of the pH and the pOH of a solution is equal to the \(pK_w\) of that solution.

Step by step solution

01

Definition of pH and pOH

Define pH as the negative logarithm (base 10) of the molar concentration of dissolved \(H^+\) and pOH as the negative logarithm of the molar concentration of \(OH^-\). Mathematically, these definitions are expressed as pH = -log[H\(^+\)] and pOH = -log[OH\(^-\)].
02

Definition of \(pK_w\)

Define \(pK_w\) as the negative logarithm (base 10) of the ion product of water. From this definition, we get \(pK_w\) = -log(\([H^+][OH^-]\)).
03

Substituting definitions and simplifying

Substitute the definitions of pH and pOH into the equation for \(pK_w\). This leads to \(pK_w\) = pH + pOH. This is the derived relationship between pH, pOH, and \(pK_w\).

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Most popular questions from this chapter

Show that the pH reading in the model is correct (given that the hydronium ion concentration is \(5.0 \times 10^{-4} \mathrm{M}\) ) and explain why the value is not \(3.3\).

Consider a neutral aqueous solution. a) What is \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) in a neutral aqueous solution? b) What is the \(\mathrm{pH}\) of a neutral aqueous solution? c) What is the pOH of a neutral aqueous solution?

For a weak acid, HA: $$ \mathrm{HA}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O} \rightleftarrows \mathrm{H}_{3} \mathrm{O}^{+}(\mathrm{aq})+\mathrm{A}^{-}(\mathrm{aq}) $$ a) Write the equilibrium expression for \(K_{\mathrm{a}}\). b) Show that \(\mathrm{pH}=\mathrm{p} K_{\mathrm{a}}-\log \frac{[\mathrm{HA}]}{\left[\mathrm{A}^{-}\right]}\)

What values of \(\mathrm{pH}\) characterize: a) an acidic solution? b) a basic solution?

a) Show that \(\log \left(10^{5} \times 10^{-5}\right)=0\) without calculating a value for \(10^{5} \times 10^{-5}\). b) Identify which relationship you used from the Information to perform part a.
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