Question

The free energy change for a reaction, \(\Delta G,\) is an extensive property. What is an extensive property? Surprisingly, one can calculate \(\Delta G\) from the cell potential, \(\mathscr{E},\) for the reaction. This is surprising because \(\mathscr{E}\) is an intensive property. How can the extensive property \(\Delta G\) be calculated from the intensive property \(\mathscr{E} ?\)

Short answer

An extensive property depends on the amount of matter present in a system, while an intensive property is independent of the amount of matter present. The free energy change (\(\Delta G\)) is an extensive property, whereas the cell potential (\(\mathscr{E}\)) is an intensive property. They are related through the Nernst equation: \[\Delta G = -nFE\] The term \(nF\) in the equation represents the number of moles of electrons transferred (\(n\)) and the Faraday constant (\(F\)), which are extensive properties. By multiplying the cell potential (\(\mathscr{E}\)) by the extensive term \(nF\), we can calculate the extensive property \(\Delta G\) from the intensive property \(\mathscr{E}\).

Step-by-step solution

Define Extensive Property

An extensive property is a property that depends on the amount of matter present in the system. Examples of extensive properties include mass, volume, and energy. In contrast, intensive properties are independent of the amount of matter present and include properties like temperature, density, and cell potential.

Define \(\Delta G\) and \(\mathscr{E}\)

The free energy change for a reaction, \(\Delta G\), represents the difference in free energy between the products and reactants in a chemical reaction. It measures the maximum useful work that can be obtained from a process occurring at constant temperature and pressure. The cell potential, \(\mathscr{E}\), on the other hand, is the difference in potential energy between the two electrodes in an electrochemical cell. It is a measure of the driving force for a redox reaction to occur in the cell and is an intensive property, as it only depends on the identity and state of the reacting species, not their amounts.

Relate \(\Delta G\) and \(\mathscr{E}\)

The relationship between the change in free energy, \(\Delta G\), and the cell potential, \(\mathscr{E}\), can be found using the Nernst equation: \[\Delta G = -nFE \] Here, \(\Delta G\) is the change in free energy, \(n\) is the number of moles of electrons transferred in the redox reaction, \(F\) is the Faraday constant (approximately \(96,485 C/mol\)), and \(\mathscr{E}\) is the cell potential.

Explain How an Extensive Property Can Be Calculated from an Intensive Property

Since \(\Delta G\) is an extensive property and \(\mathscr{E}\) is an intensive property, it may seem counterintuitive that they are related. However, the key here is the term \(nF\) in the Nernst equation. The number of moles of electrons transferred (\(n\)) represents the amount of matter involved in an electrochemical reaction, which makes this term extensive. The Faraday constant (\(F\)) links the amount of charge/electrons to a measurable quantity (the change in free energy). By multiplying the cell potential (\(\mathscr{E}\)) by the extensive term \(nF\), we are effectively scaling the intensive property to account for the amount of matter involved in the reaction. This makes it possible to calculate the extensive property \(\Delta G\) from the intensive property \(\mathscr{E}\).

Key concepts

Extensive and Intensive Properties

When studying the properties of matter, we encounter terms such as 'extensive' and 'intensive' properties. Extensive properties, as the name suggests, are dependent on the quantity or extent of matter in a sample. For example, mass and volume are extensive because their values increase as you have more substance. Imagine filling a balloon with helium; the more helium you add, the larger the volume of the balloon gets.

In contrast, intensive properties do not change with the amount of matter. These properties define the nature of the material, irrespective of how much exists. Common intensive properties include temperature and density. If you cut a magnet in half, the strength of magnetism per unit mass (an intensive property) remains the same in both halves. Interestingly, cell potential, denoted as \(\text{\(\mathscr{E}\)}}\), is an intensive property as well, because it characterizes the chemical potential energy per unit charge in a reaction, regardless of the scale at which the reaction occurs.

Understanding the distinction between these properties is essential as they often interplay in scientific equations and concepts, providing a more profound comprehension of chemical thermodynamics and material behavior.

Cell Potential

The concept of cell potential, represented by \(\mathscr{E}\), is pivotal in electrochemistry. It refers to the electrical potential difference between two electrodes of an electrochemical cell. When we talk about the cell potential, we're discussing the driving force or 'push' that electrons feel to move from the anode (where oxidation occurs) to the cathode (where reduction occurs).

This potential arises from the different tendencies of electrodes to lose or gain electrons, essentially the strength of the redox reaction taking place. Since cell potential doesn't depend on the size of the electrodes or the amount of electrolyte used, but rather on the inherent characteristics of the materials involved, it is an intensive property. Analyzing cell potential enables us to predict whether a reaction will occur spontaneously and aids in calculating the electrical energy that can be harvested from chemical reactions.

Nernst Equation

The Nernst Equation establishes a relationship between the cell potential (\(\mathscr{E}\)), the standard cell potential (\(\mathscr{E}^0\)), and the reaction quotient (\(Q\)). It is a fundamental equation in electrochemistry, allowing us to calculate the potential of an electrochemical cell under non-standard conditions:

\[ \mathscr{E} = \mathscr{E}^0 - \frac{RT}{nF} \ln(Q) \]
where \(R\) is the gas constant, \(T\) is the temperature in Kelvin, \(n\) is the number of moles of electrons transferred in the redox reaction, and \(F\) is the Faraday constant. The Nernst equation illustrates how cell potential varies with concentration, temperature, and pressure, providing deeper insight into the dynamics of electrochemical cells.

Furthermore, the equation bridges the gap between the thermodynamic quantities of free energy change (\(\Delta G\)) and the cell potential, enabling calculations that tie together these extensive and intensive properties in a meaningful way.