(a) Derive a general relation for \((\partial E / \partial p)_{T n}\) for electrochemical cells employing reactants in any state of matter. (b) E. Cohen and \(\mathrm{K}\). Piepenbroek (Z. Physik. Chem. 167A, 365 (1933)) calculated the change in volume for the reaction \(\mathrm{TlCl}(\mathrm{s})+\mathrm{CNS}^{-}(\mathrm{aq}) \rightarrow \mathrm{TlCNS}(\mathrm{s})+\mathrm{Cl}^{-}(\mathrm{aq})\) at \(30^{\circ} \mathrm{C}\) from density data and obtained \(\Delta_{1} \mathrm{~V}=-2.666 \pm 0.080 \mathrm{~cm}^{3} \mathrm{~mol}^{-1}\). They also measured the emf of the cell \(\mathrm{Tl}(\mathrm{Hg})|\mathrm{TICNS}(\mathrm{s})| \mathrm{KCNS}:|\mathrm{KCl}| \mathrm{TlCl} \mid \mathrm{Tl}(\mathrm{Hg})\) at pressures up to \(1500 \mathrm{~atm}\). Their results are given in the following table: $$ \begin{array}{llllllll} \text { p/atm } & 1.00 & 250 & 500 & 750 & 1000 & 1250 & 1500 \\ E / \mathrm{mV} & 8.56 & 9.27 & 9.98 & 10.69 & 11.39 & 12.11 & 12.82 \end{array} $$ From this information, obtain \((\partial E / \partial p)_{T n}\) at \(30^{\circ} \mathrm{C}\) and compare to the value obtained from \(\Delta_{1} \mathrm{~V} .\) (c) Fit the data to a polynomial for \(\mathrm{E}\) against \(\mathrm{p}\). How constant is \((\partial E / \partial p)_{T, n} ?\) (d) From the polynomial, estimate an effective isothermal compressibility for the cell as a whole.

Step by step solution

01

Understanding the Electrochemical Cell Relation

To derive a general relation for \((\partial E / \partial p)_{Tn}\) for electrochemical cells, we can use the fact that the Gibbs free energy \(G\) of the system is related to the electrode potential \(E\) by \(G = -nFE\), where \(n\) is the number of moles of electrons transferred and \(F\) is the Faraday constant. To find \((\partial E / \partial p)_{Tn}\), we can use \((\partial G / \partial p)_{Tn} = V\), where \(V\) is the volume of the system, and apply the chain rule to relate changes in \(G\) with changes in \(E\).

02

Calculating \((\partial E / \partial p)_{Tn}\) from Experimental Data

From the given data, the change in volume \(\Delta_{1} V\) for the reaction is \(-2.666 \pm 0.080 \text{~cm}^{3} \text{~mol}^{-1}\). This change in volume can be related to \((\partial E / \partial p)_{Tn}\) at constant temperature and moles by using the Maxwell relation \((\partial E / \partial p)_{Tn} = -(\partial V / \partial n)_{Tp}\). Since the volume change per mole is given, we can directly find the value of \((\partial E / \partial p)_{Tn}\) with the provided \(\Delta_{1} V\).

03

Fitting E vs. p to a Polynomial

To fit the emf \(E\) versus pressure \(p\) data to a polynomial, we can use regression analysis or a least-squares fit of the data to a suitable polynomial function. This will provide us with a mathematical expression for \(E(p)\), which we can differentiate to find \((\partial E / \partial p)_{Tn}\) as a function of pressure.

04

Calculating the Isothermal Compressibility

The isothermal compressibility \(\kappa_T\) is defined as \(\kappa_T = -\frac{1}{V}(\partial V / \partial p)_T\). Using the polynomial obtained in the previous step, we can estimate \((\partial V / \partial p)_T\) and since we have an estimate of the volume change from the reaction stoichiometry and density data, we can use these together to estimate the effective isothermal compressibility for the cell.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partial Molar Volume

The concept of partial molar volume plays a crucial role in understanding the changes occurring within an electrochemical cell. Imagine you have a blend of different substances; each component's partial molar volume is the volume that one mole of that substance contributes to the overall mixture. It becomes particularly important when considering reactions involving solids and liquids where significant volume changes may occur.

For instance, the given reaction involving TlCl and CNS^- shows a negative partial molar volume change, \(\Delta_1 V\). This indicates that the volume of the system decreases as the reaction progresses at constant temperature and pressure, likely due to the more compact arrangement of ions in the solid products compared to the reactants. Through understanding partial molar volumes, students can better comprehend volume-related properties of mixtures and their response to external changes such as pressure and temperature.

Electrode Potential

Electrode potential is a measure of the electric potential difference created by a chemical reaction occurring at an electrode within an electrochemical cell. It is one of the primary factors influencing the performance and efficiency of cells. This potential is determined by the tendency of a chemical species to oxidize or reduce.

For example, by measuring the change in electrode potential under various pressures, we can insight into the effect of pressure on the reaction occurring within the cell. In the context of the exercise, the electrode potential changes with pressure, which is essential information for calculating thermodynamic properties using fundamental equations and experimental data.

Gibbs Free Energy

Gibbs free energy, often symbolized as \(G\), is a thermodynamic quantity that serves as a predictor for the direction of spontaneous chemical processes. It incorporates both the enthalpy and entropy of a system and is directly related to the work that can be extracted, especially within electrochemical cells.

The relationship \(G = -nFE\) illustrates how this free energy is connected to the electrode potential \(E\), where \(n\) is the number of moles of electrons transferred and \(F\) is the Faraday constant. By understanding Gibbs free energy, students can predict the feasibility of chemical reactions and analyze the potential work output of electrochemical systems.

Maxwell Relations

Maxwell relations are a set of equations in thermodynamics derived from the symmetry of second derivatives and the fundamental thermodynamic potentials. These relations enable us to convert a partial derivative of a thermodynamic quantity that may be difficult to measure into another derivative that is easier to determine experimentally.

In the context of the exercise, one of the Maxwell relations \(\left(\partial E / \partial p\right)_{Tn} = -\left(\partial V / \partial n\right)_{Tp}\) was used. This pivotal transformation allows students to take experimentally acquired volume change information and use it to find how the electrode potential varies with pressure—this understanding aids in connecting theoretical principles with practical observations.

Isothermal Compressibility

Isothermal compressibility \(\kappa_T\) is defined as the relative change in volume a substance undergoes due to pressure changes, provided the temperature is kept constant. It's a measure of a material's 'squeeze-ability' under compression.

In the case of the electrochemical cell in the exercise, calculating the isothermal compressibility requires an understanding of the system's volume response under varying pressures. It gives insight into how the entire cell may change dimensions with pressure, thereby influencing the efficiency and safety of electrochemical technologies. Students can apply this knowledge to predict how certain reactions can withstand pressure changes without compromising the cell's integrity.

Polynomial Regression Analysis

Polynomial regression analysis is a form of regression analysis in which the relationship between the independent variable and the dependent variable is modeled as an nth degree polynomial. In the electrochemical context, it can be used to fit a curve to experimental emf versus pressure data, which helps to predict how emf will behave under different conditions.

A polynomial fit allows students to extract more detailed patterns from the data, such as how the electrode potential's rate of change varies with pressure. This is precisely what was done in the exercise with the emf data to estimate how constant \(\left(\partial E / \partial p\right)_{T, n}\) is across different pressures.

Electromotive Force (EMF)

Electromotive force (emf) refers to the voltage generated by an electrochemical cell or a single electrode when no current is flowing. It is a measure not of force, but rather the potential to do work or drive electric current. Emf is fundamental in characterizing and designing batteries, fuel cells, and other electrochemical systems.

In our electrochemical cell scenario under varying pressures, noticing how the emf changes provides valuable insight into the reaction's harnessable energy and its dependence on environmental factors such as pressure. Understanding emf is essential for students and professionals working with electrochemical systems, as it is a key factor in analyzing the performance and efficiency of these cells.