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 material or substance present, while an intensive property does not. The free energy change (ΔG) is an extensive property, whereas cell potential (ℰ) is an intensive property. It may seem counterintuitive, but ΔG can be calculated from ℰ due to the relationship between the two properties in electrochemical reactions. The equation for this relationship is ΔG = -nFℰ, where n represents the number of moles of electrons involved in the redox reaction, and F is the Faraday constant. Since n is an extensive property, the product of n, F, and ℰ gives us the extensive property ΔG.

Step-by-step solution

Understand Extensive and Intensive Properties

An extensive property is a property that depends on the amount of material or substance present. It scales with the size or quantity of the system. Examples of extensive properties include mass, volume, and energy. On the other hand, an intensive property is a property that does not depend on the amount of material or substance. It is independent of the size or quantity of the system. Examples of intensive properties include temperature, pressure, and density.

Examine the Free Energy Change, ΔG

The free energy change (ΔG) for a reaction is the change in the Gibbs free energy, which is a thermodynamic potential that measures the maximum reversible work that may be performed by a system at constant temperature and pressure. In essence, ΔG can be understood as the capacity of a system to perform useful work or to drive chemical reactions. Since ΔG depends on the energy content of the reactants and products in a reaction, it is an extensive property.

Define Cell Potential, ℰ

Cell potential (ℰ) is the measure of the potential difference between the two electrodes in an electrochemical cell when no current flows through the cell. It is related to the driving force of the chemical reaction and can be thought of as the "push" that drives electrons from the anode to the cathode in a redox reaction. Cell potential is determined by the composition of materials in the cell and the temperature at which the cell is operating. As it is independent of the amount of material, ℰ is an intensive property.

Relate ΔG and ℰ

It may seem counterintuitive that an extensive property like ΔG can be calculated from an intensive property like ℰ. However, this is possible due to the relationship between the two properties in electrochemical reactions. The relationship between ΔG and ℰ is given by the equation: ΔG = -nFℰ In this equation, n is the number of moles of electrons involved in the redox reaction, and F is the Faraday constant (approximately 96485 C/mol), which represents the electric charge per mole of electrons. Since n is an extensive property (related to the amount of material), the product of n, F, and ℰ gives us an extensive property - the free energy change, ΔG. This relationship enables us to calculate the extensive property ΔG by using the intensive property ℰ, considering the number of moles of electrons involved in the reaction and the Faraday constant.

Key concepts

Extensive Property

An extensive property is a characteristic that changes with the amount of substance in a system. Consider mass, volume, and energy. These properties fluctuate when the size of the system changes, which is why they are termed as "extensive".

• Mass: If you double the amount of a substance, its mass also doubles.
• Volume: Two different amounts of water will have different volumes.
• Energy: The energy of a system will increase if more material is added.

This property is applicable in scenarios where the size of the system plays a crucial role, such as calculating free energy change (\(\Delta G\)), where the energy content varies with the number of molecules involved in a reaction.
Understanding how extensive properties scale with system size helps in analyzing reactions and processes on a larger scale, ensuring accurate calculations for energy, mass, and volume based on the quantity present.

Intensive Property

Unlike extensive properties, intensive properties do not depend on the amount of material in a system. These properties remain constant regardless of the system's size, making them essential for identifying and characterizing substances.

• Temperature: A cup of tea is as hot as a pot full of tea.
• Pressure: Pressure within a tire remains uniform regardless of the volume of air.
• Density: A small gold bar and a large gold block have the same density.

One key intensive property in electrochemistry is the cell potential (\(\mathscr{E}\)), which remains the same whether it is measured for a single cell or multiple cells connected in a series.
Utilizing intensive properties allows scientists and engineers to determine material-specific characteristics that are invariant with scale, ensuring that measurements and predictions are consistent across different situations.

Cell Potential

Cell potential (\(\mathscr{E}\)) measures the electrical potential difference between two electrodes in an electrochemical cell. It indicates the driving force behind electron flow from the anode to the cathode, akin to the "push" in a battery that powers devices.
The value of \(\mathscr{E}\) does not vary with the size or amount of material in the cell, categorizing it as an intensive property. Its magnitude depends on factors like the nature of the electrodes and the temperature of the system. This potential difference is essential in understanding the efficiency and performance of batteries, fuel cells, and other electrochemical devices.
In practical applications, knowing the cell potential helps predict how much work can be done by the cell and determines the feasible voltage under different operating conditions.

Electrochemical Reactions

Electrochemical reactions involve the transfer of electrons between chemical species. These reactions occur in devices like batteries and electrochemical cells, where they enable the conversion of chemical energy into electrical energy and vice versa.
During an electrochemical reaction, oxidation takes place at the anode (loss of electrons) and reduction occurs at the cathode (gain of electrons). The cell potential (\(\mathscr{E}\)) serves as the force that initiates this transfer. The extensive property, free energy change (\(\Delta G\)), in such reactions is connected to \(\mathscr{E}\) through the equation: \[ \Delta G = -nF\mathscr{E} \]where:

• \(n\) is the number of moles of electrons exchanged.
• \(F\) is the Faraday constant (~96485 C/mol).

This relationship indicates that while \(\mathscr{E}\) alone is an intensive property, the product with the extensive property \(nF\) results in the extensive property \(\Delta G\). Understanding these principles is crucial for designing and optimizing devices that harness and store energy efficiently.