PH and pOH Knowledge-Based Study

Study notes

1. The pH scale provides a convenient logarithmic method to express the hydronium ion concentration, $[\text{H}_3\text{O}^+]$, in an aqueous solution.

   Mathematical Definition

   pH is defined as the negative base-10 logarithm of the hydronium ion concentration:

   $$\text{pH} = -\log[\text{H}_3\text{O}^+]$$

   Because the scale is logarithmic, each whole pH unit represents a tenfold change in hydronium ion concentration:

   • A solution with $\text{pH} = 3$ has 10 times the hydronium ion concentration of a solution with $\text{pH} = 4$.
   • A solution with $\text{pH} = 2$ has 100 times the hydronium ion concentration of a solution with $\text{pH} = 4$.

   Similar to pH, pOH expresses the alkalinity or hydroxide ion concentration, $[\text{OH}^-]$, of an aqueous solution.

   Mathematical Definition

   pOH is defined as the negative base-10 logarithm of the hydroxide ion concentration:

   $$\text{pOH} = -\log[\text{OH}^-]$$

   Key characteristics of the pOH scale include:

   • A smaller pOH value indicates a higher concentration of hydroxide ions, meaning the solution is more basic.
   • A larger pOH value indicates a lower concentration of hydroxide ions.

   Pure water undergoes autoionization to a small extent:

   $$2\text{H}_2\text{O}(l) \rightleftharpoons \text{H}_3\text{O}^+(aq) + \text{OH}^-(aq)$$

   At 25 °C, the ion-product constant for water ($K_w$) is constant:

   $$K_w = [\text{H}_3\text{O}^+][\text{OH}^-] = 1.0 \times 10^{-14}$$

   Taking the negative logarithm of both sides yields the interconversion equation:

   $$\text{pH} + \text{pOH} = 14.00 \quad (\text{at } 25\ ^\circ\text{C})$$

   Classification of Solutions at 25 °C

   • Acidic: $[\text{H}_3\text{O}^+] > [\text{OH}^-]$, resulting in $\text{pH} < 7.00$ and $\text{pOH} > 7.00$.
   • Neutral: $[\text{H}_3\text{O}^+] = [\text{OH}^-] = 1.0 \times 10^{-7}\text{ M}$, resulting in $\text{pH} = 7.00$ and $\text{pOH} = 7.00$.
   • Basic: $[\text{OH}^-] > [\text{H}_3\text{O}^+]$, resulting in $\text{pH} > 7.00$ and $\text{pOH} < 7.00$.

   When the pH or pOH of a solution is known, the respective ion concentrations can be determined using the inverse logarithm (powers of 10):

   • $[\text{H}_3\text{O}^+] = 10^{-\text{pH}}$
   • $[\text{OH}^-] = 10^{-\text{pOH}}$

   Summary of Four-Way Interconversions

   Given any one of $[\text{H}_3\text{O}^+]$, $[\text{OH}^-]$, $\text{pH}$, or $\text{pOH}$, the other three values can be found using these key relationships:

   1. $\text{pH} = -\log[\text{H}_3\text{O}^+]$ and $[\text{H}_3\text{O}^+] = 10^{-\text{pH}}$
   2. $\text{pOH} = -\log[\text{OH}^-]$ and $[\text{OH}^-] = 10^{-\text{pOH}}$
   3. $\text{pH} + \text{pOH} = 14.00$ (at 25 °C)
   4. $[\text{H}_3\text{O}^+][\text{OH}^-] = 1.0 \times 10^{-14}$ (at 25 °C)