Question

The degree of dissociation of water at \(25^{\circ} \mathrm{C}\) is \(1.8 \times 10^{-7} \%\) and density is \(1.0 \mathrm{~g} \mathrm{~cm}^{-3}\). The ionic constant for water is (a) \(1.0 \times 10^{-14}\) (b) \(2.0 \times 10^{-16}\) (c) \(1.0 \times 10^{-16}\) (d) \(1.0 \times 10^{-8}\)

Short answer

The ionic constant for water (\text{Kw}) is \(1.0 \times 10^{-14}\). The correct answer is (a).

Step-by-step solution

Understanding the Degree of Dissociation

The degree of dissociation (\text{α}) of water represents the fraction of water molecules that dissociate into ions. It is given that \text{α} is \(1.8 \times 10^{-7} \%\). To use this value in calculations, convert it into decimal form by dividing by 100. Thus, \text{α} becomes \(1.8 \times 10^{-9}\).

Calculate the Concentration of Ions

Because the density of water is \(1.0 \mathrm{g/cm^3}\), and its molar mass is approximately \(18 \mathrm{g/mol}\), 1 liter (or 1000 cm3) weighs 1000 g and contains \(\frac{1000}{18}\) moles of water. The number of moles of ions will be the degree of dissociation times the number of moles of water, which means \(\text{α} \times \frac{1000}{18}\) for each of hydrogen and hydroxide ions.

Calculate the Ionic Product of Water (\text{Kw})

The ionic product for water, \text{Kw}, can be calculated using the concentrations of hydrogen (\text{H+}) and hydroxide ions (\text{OH-}). Since each water molecule dissociates into one \text{H+} and one \text{OH-} ion, their concentrations are the same: \(\text{α} \times \frac{1000}{18}\). The ionic product is then given by \text{Kw} = [\text{H+}][\text{OH-}] = \(\left( \text{α} \times \frac{1000}{18} \right)^2\).

Compute and Compare the Answer Choices

Substituting the value of \text{α} into the expression from Step 3 yields \text{Kw} = \(\left(1.8 \times 10^{-9} \times \frac{1000}{18} \right)^2 = 1.0 \times 10^{-14}\), which matches answer choice (a).

Key concepts

Degree of Dissociation

The degree of dissociation, often symbolized as α, is a measure of the extent to which a compound separates into its constituent ions in a particular solvent. Specifically, it demonstrates the fraction of a substance that breaks down into ions when dissolved. For example, in the case of water (H₂O), a small fraction of the molecules dissociate into hydrogen ions (H⁺) and hydroxide ions (OH^-).

Calculating the degree of dissociation involves dividing the number of moles of a substance that has dissociated by the total number of moles of the substance present before dissociation. The resulting number is generally very small for weak electrolytes like water and is therefore often expressed in percentage or decimal form to simplify subsequent calculations. Recognizing the degree of dissociation is key to understanding the nature of a compound in solution and predicting the concentration of its ions.

Molar Mass

The molar mass is the weight of one mole of a substance and is usually expressed in grams per mole (g/mol). It is equivalent to the atomic or molecular weight of a substance but scaled up to account for one mole's worth of particles. Molar mass allows chemists to convert between the mass of a substance and the amount of substance (moles).

For instance, the molar mass of water, H₂O, is approximately 18 g/mol—derived from the added atomic masses of two hydrogens (1 g/mol each) and one oxygen (16 g/mol). Molar mass is critical for calculating concentrations and participating in reactions because it bridges the gap between the microscopic scale of atoms and molecules and the macroscopic world where measurements are taken.

Molar Concentration

The molar concentration, also known as molarity and denoted by the symbol M, is a measure of the amount of a dissolved substance contained per unit volume of solution. It is expressed in terms of moles per liter (mol/L).

Molar concentration is found by taking the number of moles of solute and dividing it by the volume of the solution in liters. This measure is commonly used in chemistry to describe the concentration of a solution since it directly relates the quantity of solute to the volume of solvent, providing a clear understanding of how dilute or concentrated a solution is. Typical calculations with molar concentration involve determining how much solute is needed to prepare a solution of desired concentration and volume.

Dissociation Constant

The dissociation constant, designated as K_d, is a specific type of equilibrium constant that measures the propensity of a larger object to separate (dissociate) reversibly into smaller components, such as when a compound splits into ions in a solution. For water and other weak acids and bases, this constant is especially significant.

At a specific temperature, the dissociation constant is a fixed value that represents the ratio of the concentration of the dissociated form to the concentration of the undissociated form. In the case of water at room temperature, for instance, the dissociation constant (K_w), also known as the ionic product of water, is roughly 1.0 x 10^-14 at 25°C. Understanding K_w helps in calculating pH and predicting the behavior of acids and bases in aqueous solutions.