Chapter 7: Problem 4
The degree of dissociation of pure water at \(25^{\circ} \mathrm{C}\) is found to be \(1.8 \times 10^{-9}\). The dissociation constant, \(K_{\mathrm{d}}\) of water, at \(25^{\circ} \mathrm{C}\) is (a) \(10^{-14}\) (b) \(1.8 \times 10^{-16}\) (c) \(5.56 \times 10^{-13}\) (d) \(1.8 \times 10^{-14}\)
Short Answer
Expert verified
The dissociation constant, \(K_{\mathrm{d}}\) of water at \(25^\circ \mathrm{C}\) is \(10^{-14}\).
Step by step solution
01
Understand the Degree of Dissociation
The degree of dissociation, usually denoted by \(\alpha\), is the fraction of the original solute molecules that have dissociated. For water, \(\alpha = 1.8 \times 10^{-9}\) at \(25^\circ \mathrm{C}\).
02
Write the Dissociation Reaction for Water
Water dissociates according to the following equilibrium reaction: \(2\mathrm{H_2O} \rightleftharpoons \mathrm{OH^-} + \mathrm{H_3O^+}\). For each mole of water that dissociates, one mole of hydroxide \(\mathrm{OH^-}\) ions and one mole of hydronium \(\mathrm{H_3O^+}\) ions are produced.
03
Derive the Expression for Dissociation Constant
The dissociation constant \(K_{\mathrm{d}}\) can be given by the expression \[K_{\mathrm{d}} = \frac{[\mathrm{OH^-}][\mathrm{H_3O^+}]}{[\mathrm{H_2O}]^2}\] Because the concentration of water is so large compared to the concentration of the ions, it is treated as a constant and incorporated into the dissociation constant, simplifying the expression to \[K_w = [\mathrm{OH^-}][\mathrm{H_3O^+}]\].
04
Calculate the Concentrations of Ions
Since \(\alpha\) is the degree of dissociation, the concentration of hydroxide and hydronium ions can be given by \(\alpha \times [\mathrm{H_2O}]\). The concentration of water, \(\mathrm{H_2O}\), in pure water is approximately \(55.5 \text{ mol/L}\) since the density of water is \(1 \text{ g/mL}\) and the molar mass is \(18 \text{ g/mol}\). Therefore, \[ [\mathrm{OH^-}] = [\mathrm{H_3O^+}] = \alpha \times 55.5 \text{ mol/L} \]
05
Apply Values to Calculate the Dissociation Constant
Plugging the values into the dissociation constant expression: \[K_w = [\mathrm{OH^-}][\mathrm{H_3O^+}] = (\alpha \times 55.5)^2 \] Substituting \(\alpha = 1.8 \times 10^{-9}\) we get \[K_w = (1.8 \times 10^{-9} \times 55.5)^2 = (1.8 \times 10^{-9} \times 55.5) \times (1.8 \times 10^{-9} \times 55.5)\] \[K_w = (99.9 \times 10^{-9}) \times (99.9 \times 10^{-9})\] \[K_w = (100 \times 10^{-9}) \times (100 \times 10^{-9})\] \[K_w = 10^{-14} \] Hence the dissociation constant of water at \(25^\circ \mathrm{C}\) is \(10^{-14}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Degree of Dissociation
The degree of dissociation, denoted as \(\alpha\), is crucial for understanding how substances like water split into ions. It's a comparison of the number of molecules that dissociate to the total number initially present. In simpler terms, it tells us how much of the water has separated into its ionic components, hydroxide \(\mathrm{OH^-}\) and hydronium \(\mathrm{H_3O^+}\) ions. For pure water at room temperature \(25^\circ \mathrm{C}\), \(\alpha\) is incredibly small: \(1.8 \times 10^{-9}\), indicating that only a tiny fraction of the water molecules are dissociated at any given time.
Equilibrium Constant
The equilibrium constant, which for the dissociation of water is symbolized as \(K_w\), is a number that expresses the ratio of the concentrations of products to reactants when a reaction is at equilibrium. In a balanced equation, this constant helps predict the extent of the reaction. For water's dissociation, \(K_w\) considers the concentrations of the ions produced, \(\mathrm{OH^-}\) and \(\mathrm{H_3O^+}\), but ignores the concentration of water itself since it's in such large excess that it remains practically constant.
Ionic Product of Water
The ionic product of water, \(K_w\), is a special name for the equilibrium constant of water's self-ionization. It's a static value at a given temperature, representing the product of the hydroxide and hydronium ion concentrations. At \(25^\circ \mathrm{C}\), \(K_w\) is always \(10^{-14}\), irrespective of whether water is pure or in a solution. This constancy underpins many calculations in chemistry, particularly those involving pH and pOH, which measure the acidity and basicity of solutions.
Hydroxide Concentration
Hydroxide concentration refers to the amount of hydroxide ions (\mathrm{OH^-}) in a solution. In the case of water, each dissociated molecule produces one hydroxide ion. At room temperature, because the degree of dissociation is \(1.8 \times 10^{-9}\), the concentration of hydroxide ions is also \(1.8 \times 10^{-9} \times 55.5\) moles per liter, signifying a direct relationship with the degree of dissociation. Hydroxide concentration is a fundamental parameter in calculating the pH of a solution, which is a measure of its basicity.
Hydronium Concentration
Hydronium concentration, exemplified by the number of hydronium ions (\mathrm{H_3O^+}) present, is equally important for assessing the acidic nature of solutions. In pure water at \(25^\circ \mathrm{C}\), each dissociation yields one hydronium ion, similar to the hydroxide ion concentration. Hence, the hydronium concentration will be the same as the hydroxide concentration, confirming water's neutral nature. However, in acidic or basic solutions, the hydronium concentration will differ, determining the solution's pH level.