Acids and Bases
Ionic Product of Water ( $ K_w $ ) - AP Chemistry Rundown
The ionic product of water ( $ K_w $ ) is a fundamental concept in acid-base chemistry, especially when dealing with aqueous solutions. It describes the equilibrium constant for the autoionization of water.
Autoionization of Water
Water is amphoteric, meaning it can act as both an acid and a base. Therefore, water molecules can react with each other in a process called autoionization (or self-ionization). This reaction involves the transfer of a proton ( $ H^+ $ ) from one water molecule to another:
$$ H_2O(l) + H_2O(l) \rightleftharpoons H_3O^+(aq) + OH^-(aq) $$
- One water molecule acts as an acid, donating a proton to form hydroxide ion ( $ OH^- $ ).
- The other water molecule acts as a base, accepting a proton to form hydronium ion ( $ H_3O^+ $ ).
Simplified Autoionization It is often simplified to: $ H_2O(l) \rightleftharpoons H^+(aq) + OH^-(aq) $ While technically correct, it’s important to remember that free $ H^+ $ ions don’t exist in water; they are always associated with water molecules as $ H_3O^+ $ .
The Ionic Product of Water ( $ K_w $ )
The equilibrium constant for the autoionization of water is called the ionic product of water, denoted by $ K_w $ . It is defined as:
$$ K_w = [H_3O^+][OH^-] = [H^+][OH^-] $$
- The brackets denote molar concentrations at equilibrium.
- $ K_w $ is temperature-dependent.
Value of $ K_w $ at 25°C
At 25°C (298 K), the experimentally determined value of $ K_w $ is:
$$ K_w = 1.0 \times 10^{-14} $$
Neutral Water In pure water at 25°C, the concentrations of hydronium and hydroxide ions are equal:
$$ [H_3O^+] = [OH^-] = \sqrt{K_w} = \sqrt{1.0 \times 10^{-14}} = 1.0 \times 10^{-7} M $$
This is the definition of neutral water: $ [H_3O^+] = [OH^-] $ .
Temperature Dependence of $ K_w $
The autoionization of water is an endothermic process (it absorbs heat). Therefore, according to Le Chatelier’s principle, increasing the temperature will shift the equilibrium to the right, favoring the formation of more hydronium and hydroxide ions. This means that $ K_w $ increases with increasing temperature.
- At temperatures above 25°C, $ K_w > 1.0 \times 10^{-14} $ .
- At temperatures below 25°C, $ K_w < 1.0 \times 10^{-14} $ .
Important Note about Neutrality Even though $ K_w $ changes with temperature, neutral water always has $ [H_3O^+] = [OH^-] $ . However, the value of those concentrations will change with temperature. For example, at a higher temperature, neutral water will have $ [H_3O^+] = [OH^-] > 1.0 \times 10^{-7} M $ , but it is still neutral.
Implications of $ K_w $
$ K_w $ provides a crucial relationship between the concentrations of hydronium and hydroxide ions in any aqueous solution. This relationship is fundamental to understanding acid-base chemistry.
- Acidic Solution: $ [H_3O^+] > [OH^-] $ (and $ [H_3O^+] > 1.0 \times 10^{-7} M $ at 25°C)
- Basic Solution: $ [H_3O^+] < [OH^-] $ (and $ [OH^-] > 1.0 \times 10^{-7} M $ at 25°C)
- Neutral Solution: $ [H_3O^+] = [OH^-] $
Calculating Ion Concentrations Knowing either $ [H_3O^+] $ or $ [OH^-] $ allows you to calculate the other using the $ K_w $ expression:
$ [H_3O^+] = \frac{K_w}{[OH^-]} $
$ [OH^-] = \frac{K_w}{[H_3O^+]} $
Relationship to pH and pOH
$ K_w $ is directly related to pH and pOH:
- $ pH = -log[H_3O^+] $
- $ pOH = -log[OH^-] $
- $ pK_w = -log(K_w) $
At 25°C:
$ pK_w = -log(1.0 \times 10^{-14}) = 14 $
Therefore:
$ pH + pOH = pK_w = 14 $ (at 25°C)
Temperature and pH Scale It’s crucial to remember that the pH scale is temperature-dependent due to the temperature dependence of $ K_w $ . While pH = 7 is neutral at 25°C, it is not neutral at other temperatures. Neutrality is always defined by $ [H_3O^+] = [OH^-] $ .
Common Mistakes to Avoid
- Assuming pH = 7 is always neutral: Remember that neutrality is defined by equal concentrations of hydronium and hydroxide ions, not a specific pH value. pH = 7 is only neutral at 25°C.
- Forgetting the temperature dependence of $ K_w $ : Use the value of $ K_w = 1.0 \times 10^{-14} $ only at 25°C. Problems might give you a different temperature and a different $ K_w $ value.
- Incorrectly using the $ K_w $ equation: Make sure you are using molar concentrations in the $ K_w $ expression.
- Confusing pH and pOH: Understand the definitions of pH and pOH and how they relate to hydronium and hydroxide ion concentrations. Remember $ pH + pOH = 14 $ is only true at 25°C.