Question

Recall that \(\log (\mathrm{A} \times \mathrm{B})=\log \mathrm{A}+\log \mathrm{B}\). What is the relationship between \(\mathrm{pH}\), \(\mathrm{pOH}\), and \(\mathrm{p} K_{\mathrm{w} ?}\)

Short answer

The relationship between pH, pOH, and pKw is given by: \(pKw = pH + pOH\).

Step-by-step solution

Formulate known equations

We know the following: 1. \(pH = -\log[H^+]\) 2. \(pOH = -\log[OH^-]\) 3. \(pKw = -\log Kw\) where [H^+] is the concentration of hydrogen ions, [OH^-] is the concentration of hydroxide ions, and Kw is the product of [H^+] and [OH^-]. From properties of logarithms, we also know that \(\log(A \times B) = \log A + \log B\).

Calculate pKw from the ionic product of water

From the ionic product of water, we have Kw = [H+][OH-]. Taking the negative logarithm on both sides gives us \(pKw = -\log Kw = - \log([H+][OH-])\). Using the property \(\log(A \times B) = \log A + \log B\), we obtain \(pKw = -\log[H+] -\log[OH^-]\).

Link pH, pOH, and pKw

Knowing that pH = -\log[H+] and pOH = -\log[OH^-], we can substitute these into our equation from Step 2, yielding \(pKw = pH + pOH\). This expresses the relationship between pH, pOH, and pKw.

Key concepts

Logarithm Properties in Chemistry

In chemistry, logarithms are used to simplify the handling of multiplicative relationships, especially when dealing with concentrations of ions. One of the most important logarithm properties is

• \( \log(A \times B) = \log A + \log B \)

This property is instrumental in acid-base chemistry because pH and pOH are both logarithms.
Likewise, pOH, which indicates how basic a solution is, is given by \( pOH = -\log[OH^-] \).

The equation \( \log(A \times B) = \log A + \log B \) helps us see that when we have the product of two concentrations, like [H+] and [OH-], the logarithmic property can simplify the expression into a sum of their logarithms.
This is essential for understanding how pH and pOH relate to \( pK_w \), as they all interact through logarithmic transformations.

Ionic Product of Water

The ionic product of water, \( K_w \), is a fundamental concept in chemistry.
Water can dissociate into hydrogen ions, [H+], and hydroxide ions, [OH-], so for pure water at 25°C, \( K_w = [H^+][OH^-] = 1.0 \times 10^{-14} \).

Being a constant, \( K_w \) shows how water can self-ionize to a small extent.
Applying the concept of logarithms to the ionic product, we take the negative logarithm of \( K_w \), giving us:

• \( pK_w = -\log K_w \)

Since \( pH = -\log [H^+] \) and \( pOH = -\log [OH^-] \), combining these gives us \( pK_w = pH + pOH \).
This equation is vital as it allows us to understand that in any aqueous solution at 25°C, the sum of pH and pOH remains constant and equals 14.

Acid-Base Chemistry

Acid-base chemistry revolves around the concept of acids donating protons and bases accepting them.
The concentration of hydrogen ions, [H+], signifies the acidity of a solution, whereas [OH-] indicates its basicity.

When solving acid-base problems, the relationship between pH, pOH, and \( pK_w \) becomes crucial.
At 25°C, the neutral pH is 7, meaning [H+] equals [OH-] and pH + pOH = 14.
Given the formulas \( pH = -\log[H^+] \) and \( pOH = -\log[OH^-] \), the expression \( pK_w = pH + pOH \) helps determine unknowns when values are missing.

Understanding these interactions and transformations provides clarity when predicting the behavior of acids and bases in solution, allowing you to determine the acidic or basic nature of solutions confidently.