Question

Conductivity measurements were one of the first methods used to determine the autoionization constant of water. The autoionization constant of water is given by the following equation: \\[K_{w}=a_{H^{+}} a_{O H^{-}}=\left(\frac{\left[H^{+}\right]}{1 M}\right)\left(\frac{\left[O H^{-}\right]}{1 M}\right)\\] where \(a\) is the activity of the species, which is equal to the actual concentration of the species divided by the standard state concentration at infinite dilution. This substitution of concentrations for activities is a reasonable approximation given the small concentrations of \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\) that result from autoionization. a. Using the expression provided, show that the conductivity of pure water can be written as \\[\Lambda_{m}\left(\mathrm{H}_{2} \mathrm{O}\right)=(1 M) K_{w}^{1 / 2}\left(\lambda\left(\mathrm{H}^{+}\right)+\lambda\left(\mathrm{OH}^{-}\right)\right)\\] b. Kohlrausch and Heydweiller measured the conductivity of water in 1894 and determined that \(\Lambda_{m}\left(\mathrm{H}_{2} \mathrm{O}\right)=5.5 \times\) \(10^{-6} \mathrm{S} \mathrm{m}^{-1}\) at \(298 \mathrm{K} .\) Using the information in Table 34.2 determine \(\mathrm{K}_{w}\).

Short answer

The autoionization constant of water, K_w, can be determined using the given conductivity equation and experimental data. By relating molar conductivity to specific conductivity and the autoionization constant, we find: \[\Lambda_{m}(H_2O) = (1\ M) \sqrt{K_w} \left(\lambda_{H^+}+\lambda_{OH^-}\right)\] Using the measured conductivity, \(\Lambda_{m}(H_2O) = 5.5 \times 10^{-6}\: S\cdot m^{-1}\) at \(298 K\), and the molar conductivities of hydronium and hydroxide ions from Table 34.2, we can calculate the autoionization constant to be: \(K_w \approx 1.39 \times 10^{-14}\) at \(298 K\).

Step-by-step solution

Write down the autoionization equation for water

The autoionization equation of water is: \[2H_2O \leftrightarrows H_3O^+ + OH^-\]

Consider that conductivity is determined by the sum of ions

Here, the molar conductivity of water, \(\Lambda_m(H_2O)\) is the sum of molar conductivities of hydronium ions (\(\Lambda_m(H^+)\)) and hydroxide ions (\(\Lambda_m(OH^-)\)).

Write down the relationship between conductivity and concentration

We know that molar conductivity(\(\Lambda_m\)) can be related to specific conductivity(\(\kappa\)), concentration(C), and the molar conductivities of the ions: \[\kappa = C\left(\lambda_{H^+}+\lambda_{OH^-}\right)\]

Relate conductivity and K_w

Rewrite the specific conductivity equation by replacing the concentration on the right-hand side with the concentration of \(H^+\) and \(OH^-\) ions, taking into account that \(C = [H^+] = [OH^-]\), as well as the equilibrium equation for \(K_w\) as given in the problem statement: \[C =\sqrt{K_w}\Rightarrow K_w = C^2\] Now, replace the autoionization constant, K_w, in the equation: \[\kappa = \Lambda_{m}(H_2O)\] \[\Rightarrow\Lambda_{m}(H_2O) = (1\ M) \sqrt{K_w} \left(\lambda_{H^+}+\lambda_{OH^-}\right)\] b.

Use experimental data to find K_w

We can determine \(K_w\) from the measured conductivity, which is given as \(\Lambda_{m}(H_2O)\) = 5.5x\(10^{-6} S m^{-1}\) at 298 K. We also have the λ(H+) =' 349.81 S cm² mol⁻¹ ') and λ(OH⁻) =' 198.34 S cm² mol⁻¹ ') from Table 34.2. Plug in the values, and convert them to the common unit (m²/mol): \[5.5 \times 10^{-6}\: S\cdot m^{-1} = 1\: M \times \sqrt{K_w} \left(\frac{349.81\times10^{-4}\: S\cdot m^{2} \cdot mol^{-1} + 198.34\times10^{-4}\: S\cdot m^{2} \cdot mol^{-1}}{N\cdot A}\right)\] Where N is Avogadro's constant and is equal to \(6.022\times 10^{23} mol^{-1}\). In our case, A cancel out, so we do not use it. Now we can solve for K_w: \[K_w = \left(\frac{5.5 \times 10^{-6}\: S\cdot m^{-1}}{(349.81\times10^{-4}\: S\cdot m^{2} \cdot mol^{-1} + 198.34\times10^{-4}\: S\cdot m^{2} \cdot mol^{-1})}\right)^2\] \[K_w \approx 1.39 \times 10^{-14}\] Therefore, the autoionization constant of water, K_w, is approximately 1.39 x 10^{-14} at 298K.

Key concepts

Conductivity Measurements

Conductivity measures a solution's ability to conduct electricity. This is achieved through the movement of ions in the solution. The higher the concentration of ions, the better the conductivity. In the case of water, conductivity measurements help reveal its autoionization properties. Autoionization is the spontaneous formation of ions, which are responsible for conducting electricity in water. In pure water, despite the low concentration of ions, these measurements are sensitive enough to detect even minor conductivity.

Pure water has a very low conductivity. This is because it only ionizes to a small extent. However, even with this low ionization, instruments are capable of measuring the electrical properties of water by summing up the conductivity of both the hydronium (\(H_3O^+\) or \(H^+\)) and hydroxide (\(OH^-\)) ions.

Autoionization Constant

The autoionization constant,\(K_w\), is critical because it quantifies the extent of water's autoionization.

This constant is defined by the product of the concentrations of hydronium and hydroxide ions.\[K_w = [H^+][OH^-]\]Due to the simplicity of this equilibrium in pure water,\(K_w\) can be determined using conductivity measurements.

Interestingly, the small values of ion concentrations make using activities (the effective concentration) a valid approximation. Capture this important property using the relation:\[K_w = ( ext{concentration of ions})^2\] Understanding this relation helps in explaining how seemingly pure water has the ability to conduct electricity due to its ion formation.

Molar Conductivity

Molar conductivity (\(\Lambda_m\)) describes how well an ion conducts electricity in a solution at a specific concentration. For water, it can be defined for individual ions, namely hydronium and hydroxide, and their influence is additive.

Mathematically, the molar conductivity is related to specific and ion conductivities by:\[\kappa = C \cdot (\lambda_{H^+} + \lambda_{OH^-})\]Where \(C\) represents the concentration of ions in the solution. Though the concentration of ions in pure water is low, defining molar conductivity helps us understand the effectiveness of each ion's ability to conduct electricity.

This knowledge becomes practical and measurable using conductivity meters, which historically provided the first insights into water's autoionization.

Specific Conductivity

Specific conductivity (\(\kappa\)) measures the actual electrical conductivity of a solution. It is dependent on the concentration of ions and the ionic conductivities in the solution. In simple terms, it's a way to quantify how many ions a solution can transport across a set distance.

In the context of water, specific conductivity provides insights into the extent of autoionization. While pure water naturally has a very low specific conductivity, It’s valued at \(5.5 \times 10^{-6} \, S \, m^{-1}\) at 298K, reflecting its ionic components.

Understanding specific conductivity is crucial as it serves as a bridge between theoretical ion concentrations and practical electrical conductivity. It truly describes the interaction of \(H^+\) and \(OH^-\) ions in a measurable way, further deepening the concept of water's self-ionization.