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, \(\mathscr{b}\), for the reaction. This is surprising because \(\mathscr{B}\) is an intensive property. How can the extensive property \(\Delta G\) be calculated from the intensive property \(\mathscr{E}\) ?

Short answer

An extensive property, like free energy change (\(\Delta G\)), depends on the amount of matter in a system, while an intensive property, like cell potential (\(\mathscr{E}\)), does not. Although they seem unrelated, the two can be connected using the equation \(\Delta G = -nFE\), where \(n\) is the number of electrons transferred in the reaction, and \(F\) is Faraday's constant. The product of \(n\) and \(F\) represents the total charge transferred, an extensive property. By substituting the values of \(n\), \(F\), and \(\mathscr{E}\) into the equation, we can calculate the extensive property \(\Delta G\) from the intensive property \(\mathscr{E}\), demonstrating that under certain conditions, extensive and intensive properties can be interconnected.

Step-by-step solution

Definition of Extensive Property

Extensive properties are the properties that depend on the amount of matter (mass or volume) in a system. These properties are additive, which means that the value of the property of the entire system is the sum of the values of the property for its individual components. Examples of extensive properties include mass, volume, and enthalpy.

Definition of Intensive Property

Intensive properties, on the other hand, do not depend on the amount of matter in the system. They are independent of the mass or volume and are characteristic of the substance itself. Intensive properties can help in identifying a substance as they remain the same regardless of the quantity present. Examples of intensive properties include temperature, pressure, and cell potential (\(\mathscr{E}\)).

Connecting Extensive and Intensive Properties

Though it might seem counterintuitive, extensive and intensive properties can be connected under certain conditions. In the case of calculating the free energy change (\(\Delta G\)) from the cell potential (\(\mathscr{E}\)), we use the following equation: \[\Delta G = -nFE\] In this equation, \(n\) represents the number of electrons transferred in the reaction, \(F\) is the Faraday's constant (which measures the charge of one mole of electrons), and \(\mathscr{E}\) is the cell potential. The product of \(n\) and \(F\) gives the total charge transferred during the reaction, which is an extensive property that depends on the amount of matter involved in the reaction. The cell potential, \(\mathscr{E}\), is an intensive property, as it doesn't depend on the amount of matter.

Calculating \(\Delta G\) from \(\mathscr{E}\)

To calculate the free energy change (\(\Delta G\)) from the cell potential (\(\mathscr{E}\)), we first need to determine the number of electrons (\(n\)) involved in the reaction and then multiply this value by the Faraday constant (\(F\)). After that, we can use the equation mentioned above: \[\Delta G = -nFE\] By plugging in the values of \(n\), \(F\), and \(\mathscr{E}\), we can calculate the extensive property \(\Delta G\) from the intensive property \(\mathscr{E}\). This shows that under certain conditions, extensive and intensive properties can be connected and used to determine each other.

Key concepts

Extensive Properties

Extensive properties are an essential concept in thermodynamics. They are properties that change when the size of the system changes. These properties depend on the amount of matter present in the system. Because they are additive, if you were to combine two identical systems, the value of an extensive property for the combined system would be twice that of a single system. Common examples include mass, volume, and enthalpy. This means if you double the amount of substance in a system, the mass and volume both double as well.

For instance:

• If you have a cup of water weighing 200 grams and add another 200 grams, the total is now 400 grams - showing that mass is an extensive property.

Intensive Properties

Intensive properties, unlike their extensive counterparts, are independent of the amount of material in a system. That means whether you have a droplet or an ocean of water, certain properties remain constant. These are properties intrinsic to the substance itself and do not change if you take a small sample from a larger amount.

Examples of intensive properties include:

• Temperature
• Pressure
• Cell potential ( \(\mathscr{E}\)

Intensive properties are quite helpful when it comes to identifying substances, as they are unique fingerprints of the material. For example, the boiling point of water is always 100°C at standard atmospheric pressure, whether you have a liter or a barrel of it.

Free Energy Change

A fascinating aspect of thermodynamics is the concept of free energy change, denoted as \(\Delta G\). This value tells us about the ability of a reaction to occur spontaneously. It is crucial as it lets us predict whether a reaction will proceed without external intervention. The free energy change is an extensive property because it depends on the scale of the reaction—meaning it's tied to the amount of material involved.

To bridge intensive and extensive properties, especially when trying to calculate \(\Delta G\) from an intensive property like cell potential, we employ a specific equation:

• \[ \Delta G = -nF\mathscr{E} \]

In this equation:

• \(n\) is the number of electrons transferred during the reaction.
• \(F\) is Faraday's constant, representing the electric charge per mole of electrons.
• \(\mathscr{E}\) stands for the cell potential, an intensive property.

The multiplication of \(n\) and \(F\) gives us the total charge, which transforms the intensive property (cell potential) into an extensive one (free energy change). This equation beautifully showcases how extensive and intensive properties can interconnect under specific conditions, guiding us to extract meaningful and actionable data from the properties of substances.