Molar conductivity

Theory

The aim of this lab course is to determine the specific conductivity of various electrolyte solutions. From this measurement, the limiting molar conductivity as function of the degree of dissociation is determined.

Resistance measurements in electrolyte solutions

Solutions can conduct electricity if they contain mobile charge carriers — e.g. ions. In this case, the solution is called electrolyte solution. Measuring the conductivity of an electrolyte solution allows to draw conclusions regarding the behavior of ions.

Two electrodes with an area $A$ are placed at distance $\ell$ into an electrolyte solution (figure ). If an alternating electrical voltage is applied, an electrical current flows. The ratio between applied voltage and measured current is the electrical resistance $R$ .

R = \frac{\tilde U}{I}

Schematic setup of an electrochemical cell for measurements of conductivity.

It is important to note that the equations described here only apply for alternating voltages. For direct (constant) voltages, an electrode-specific decomposition voltage $U_\text Z$ must first be reached until the electrolysis starts and current flows.

In the shows experimental setup the resistance $R$ is proportional to the electrode distance $\ell$ and anti proportional to the electrode area $A$ . The proportionality constant $\rho$ is called the specific resistance. The specific resistance depends on the electrolyte solution, but does not depend on the geometry of the experimental setup.

R = \rho\,\frac{\ell}{A}

The electric conductivity $G$ is the inverse of the electric resistance.

G = \frac{1}{R}

Similarly, the specific conductivity $\kappa$ is the inverse of the specific resistance.

\kappa = \frac{1}{\rho}

Since the specific conductivity depends on the concentration $c$ of the electrolyte solution, the molar conductivity $\Lambda$ is often considered :

\Lambda = \frac{\kappa}{c}

Assuming that the specific conductivity $\kappa$ scales linear with the concentration of ions we would expect that the molar conductivity $\Lambda$ is a constant that does not depend on the ion concentration. Experimentally, this is typically not observed.

Conductivity of weak electrolytes

In weak electrolytes — for example an aqueous acetic acid solution — only a portion of acetic acid molecules is present in their dissociated (and, therefore, electrically conducting) form. The following equilibrium reaction takes place:

\ce{HAc} \leftrightharpoons \ce{Ac^{-}} + \ce{H^{+}}

In this case, the equilibrium constant is called dissociation constant $K_\text D$ .

K_{\text D} = \frac{c_\ce{H^{+}}\,c_\ce{Ac^{-}}}{c_\ce{HAc}}

The degree of dissociation $\alpha$ is the concentration of dissociated molecules in relation to the total concentration $c$ of acetic acid.

\alpha = \frac{c_\ce{Ac^{-}}}{c} = \frac{c_\ce{H^{+}}}{c}

For the concentrations it follows that

\begin{aligned} c_\ce{H^{+}} &= \alpha\,c\\ c_\ce{Ac^{-}} &= \alpha\,c\\ c_\ce{HAc} &= (1 - \alpha)\,c \end{aligned}

Replacing these expressions in the mass action law yields the following expression for the degree of dissociation:

K_\text D = \frac{\alpha^2\,c}{1 - \alpha}.

Connection between the molar conductivity and the degree of dissociation

Assuming that there is a linear relationship between the molar conductivity and the degree of association we can write down the following equation:

\Lambda = \alpha\,\Lambda_\infty

The quantity $\Lambda_\infty$ is called limiting molar conductivity. It is the molar conductivity at infinite dilution, as here all molecules are dissociated ( $\alpha = 1$ ) and intermolecular interactions can be neglected.

Determination of the molar conductivity and the degree of dissociation

By inserting equation into equation the following equation is obtained:

K_\text D = \frac{\left(\frac{\Lambda}{\Lambda_\infty}\right)^2\cdot c}{1 - \frac{\Lambda}{\Lambda_\infty}}

Rearranging yields

\frac{c}{K_\text D} = \frac{\Lambda_\infty^2}{\Lambda^2}-\frac{\Lambda_\infty}{\Lambda}

By multiplying $\frac{\Lambda}{\Lambda_\infty^2}$ on both sides and former rearranging we obtain

\frac{1}{\Lambda} = \frac{c\,\Lambda}{K_\text D\Lambda_\infty^2} + \frac{1}{\Lambda_\infty}

From a plot of $\frac{1}{\Lambda}$ versus the product $c\,\Lambda$ yields the dissociation constant $K_\text D$ as well as the limiting molar conductivity $\Lambda_\infty$ from the slope and intercept of a linear fit according to equation .

Conductivity of strong electrolytes

The defining characteristic of strong electrolytes is that they dissociate completely, implying that $\alpha = 1$ . According to our model so far we would not expect a dependency of the molar conductivity on the concentration. However, since experimentally there often is an additional dependency on the concentration, we must extend the model by considering the Debye-Hückel theory. In this model, the electrostatic interaction of ions in solutions is considered in more detail.

Kohlrausch square root law

Kohlrausch empirically found that the molar conductivity $\Lambda$ decreases with increasing concentration $c$ according to

\Lambda = \Lambda_\infty - B\sqrt{c}

The limiting molar conductivity $\Lambda_\infty$ is the conductivity and infinite dilution.

The reduction of the molar conductivity with increasing concentration can be explained by hydrodynamic considerations and the Debye-Hückel-Onsager theory: As cations and anions move in opposite directions due to the external electrical field, friction between the solvation shells of the differently charged ions reduce the molar conductivity (electrophoretic effect ).

The Kohlrausch law of independent ion movement

Due to the full dissociation of strong electrolytes the square root law of Kohlrausch (equation ) is a useful model. The limiting molar conductivity $\Lambda_\infty$ can be determined from the intercept of a linear fit resulting from plotting the molar conductivity $\Lambda$ against the square root of the concentration $\sqrt{c}$ .

The molar conductivity of weak electrolytes shows a strong dependency of the concentration even at low electrolyte concentrations, making a determination of the limiting molar conductivity using equation rather imprecise.

A more precise determination of the limiting molar conductivity of weak electrolytes is possible with the law of independent ion movement (again by Kohlrausch) . The law states that the limiting molar conductivity of individual ions does not depend on its counter ion and that each individual ion contributes to the total conductivity. For single-charge ions we can write

\Lambda_\infty = \Lambda_\infty^+ + \Lambda_\infty^-

The limiting molar conductivity $\Lambda_\infty$ is the sum of the limiting molar conductivity of the cations $\Lambda_\infty^+$ and the anions $\Lambda_\infty^-$ .

Equation can be used to determine the limiting molar conductivity of weak electrolytes the form of A⁺D⁻. For this, the limiting molar conductivities of two strong electrolytes A⁺B⁻ and C⁺D⁻ which each contain the cation A⁺ and the anion D⁻ of the weak electrolyte as well as an additional electrolyte (with counter ions C⁺B⁻) are needed.

According to equation a solution of A⁺B⁻ and C⁺D⁻ contains the ions A⁺, B⁻, C⁺, D⁻, which contribute independently towards the total limiting molar conductivity. The values of the limiting molar conductivity of the weak electrolyte A⁺D⁻ can be obtained by subtracting the values obtained for C⁺B⁻ from the sum of A⁺B⁻ and C⁺D⁻.

Conductivity mechanism of hydroxy and oxonium ions

The limiting molar conductivity of a weak electrolyte that dissociates into $\ce{H_3O^+}$ ions is significantly larger than the one of a strong electrolyte. The reason for this originates from a different conductivity mechanism: Instead of moving $\ce{H_3O^+}$ ions through the solution the positive charge can be transferred to a surrounding water molecule. Thus, the transport is significantly faster.

Experimental setup

The setup consist of a conductometer. The sensor of the conductometer consists a thermometer and the two electrodes used for the conductivity measurement. The display shows the specific conductivity $\kappa$ in units of

\left[ \kappa \right] = \text S / \text{cm} = \Omega^{-1} \text{cm}^{-1}.

Since the specific conductivity depends on temperature, the samples are kept at a constant temperature using a tempering water bath.

Instructions

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Lab course instructions

Preparation

Turn on the thermostat and check whether the Celsius temperature of 25 °C is set by pressing T1. If not, this can be changed by pressing the arrow buttons. The changed temperature is accepted by pressing the ↲ button.
Turn on the conductometer and check that the devices displays the correct units for measuring the specific conductivity.

Measurement

The aim is to measure four acetic acid, four sodium acetate, four sodium chloride and four hydrochloric acid solutions of different concentration. Create from a stock solution with concentration 0.1 mol/L three solutions diluted each by a factor of 10 for each substance.

In the dilution series, the pipette is wetted on the first withdrawal of 2 mL of solution and the contents discarded. A further 2 mL is taken and added to the next tube. After the solutions are ready the tube holder is immersed in the tempering bath of the thermostat. Also add approximately 20 mL of the calibrating solution and approximately 20 mL of water in two different tubes. After 10 min the temperature of the solutions should be constant. To calibrate the conductometer, press the button CAL until CAL CELL appears. Press OK and immerse the electrode into the calibrating solution. Confirm with OK and wait until AR does not blink anymore.

Measure the specific conductivity of pure water and of the samples with increasing concentration. Clean the electrode prior to each measurement with pure water and remove droplets carefully with a tissue. Move the electrode in the solution to ensure a complete immersion in the solution. After cleaning the electrode the next solution can be measured in a similar fashion. The specific conductivity of pure water only has to be measured once for every stock container of water used for the various solutions.

End of experiment

Switch off the conductometer, the thermostat as well as the cooling circulation. Discard all solutions in the sink. Clean all glassware with the pure water provided in the canister. Dry off the tubes and their holder.

Analysis

Note: Assume that the error/uncertainty of the concentration of the stock solution is negligible small.

Specific conductivity of water

Note that the water exhibits a non-zero specific conductivity $\kappa_{\mathrm{water}}$ . The movement of electrical current through the solution via water and the electrolyte can be considered as parallel circuit. Thus, the specific conductivities of the individual components can be added:

\kappa_{\mathrm{measured}} = \kappa_{\mathrm{electrolyte}} + \kappa_{\mathrm{water}}

Calculate from the measured conductivity of the solutions $\kappa_{\mathrm{measured}}$ and of pure water the conductivity of each electrolyte $\kappa_{\mathrm{electrolyte}}$ and use this quantity in the further analysis.

Plot in a single, large diagram the molar conductivity $\Lambda$ of the measured solutions against the concentration $c$ . Discuss the data and the concentration-dependence of the strong and weak electrolytes.

Strong electrolyte

Determine the limiting molar conductivity $\Lambda_\infty$ and the factor $B$ for sodium acetate, sodium chloride and hydrochloric acid in aqueous solution at 25 ℃.

Plot for all strong electrolytes the molar conductivity $\Lambda$ against the square root of the concentration $\sqrt{c}$ in a single diagram and perform a linear fit.
Use the slope and the intercept with equation to determine the limiting molar conductivity $\Lambda_\infty$ and the factor $B$ .

Weak electrolyte

Determine the limiting molar conductivity $\Lambda_\infty$ and the dissociation constant $K_\text D$ of acetic acid in aqueous solution at 25 ℃.

Plot $\frac{1}{\Lambda}$ against $c\,\Lambda$ and perform a linear fit.
From the slope and the intercept equation can be used to calculate the dissociation constant $K_\text D$ and the limiting molar conductivity $\Lambda_\infty$ .

Law of independent ion movement

Determine from the calculated limiting molar conductivity $\Lambda_\infty$ of sodium acetate, sodium chloride and hydrochloric acid the limiting molar conductivity $\Lambda_\infty$ of acetic acid.
Compare the calculated limiting molar conductivity $\Lambda_\infty$ of acetic acid with the experimentally determined value and the literature value.

The literature values are :

\begin{aligned} \Lambda_\infty (\text{NaAc}) &= 91{.}0\, \text S\, \text{cm}^2 / \text{mol}\\ \Lambda_\infty (\text{NaCl}) &= 126{.}5\, \text S\, \text{cm}^2 / \text{mol}\\ \Lambda_\infty (\text{HCl}) &= 426{.}2\, \text S\, \text{cm}^2 / \text{mol}\\ \Lambda_\infty (\text{HAc}) &= 390{.}6\, \text S\, \text{cm}^2 / \text{mol}\\ K_\text D (\text{HAc}) &= 1{.}79\cdot 10^{-5}\, \text{mol}/\text L \end{aligned}