Why is molar specific heat at constant volume (Cv) an extensive property?

The molar specific heat at constant volume, denoted as Cv, is considered an extensive property because it depends on the amount of substance present in a system. To understand this concept better, let’s break it down into simpler components and explore how Cv behaves in relation to the quantity of material involved.

Defining Extensive and Intensive Properties

Before diving into Cv, it’s essential to clarify the difference between extensive and intensive properties. Extensive properties are those that change when the size of the system changes. Examples include mass, volume, and total energy. In contrast, intensive properties remain constant regardless of the amount of substance present, such as temperature and pressure.

Understanding Molar Specific Heat at Constant Volume

Molar specific heat at constant volume (Cv) is defined as the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius while keeping the volume constant. This definition inherently ties Cv to the number of moles of the substance, making it an extensive property.

Why Cv is Extensive

• Dependence on Quantity: If you double the amount of a substance, you will require twice the amount of heat to achieve the same temperature change at constant volume. For instance, if it takes 25 J to raise the temperature of 1 mole of a gas by 1°C, it will take 50 J for 2 moles of the same gas.
• Calculation of Cv: The molar specific heat is calculated per mole, but when considering a larger sample, the total heat capacity (C) can be expressed as C = n * Cv, where n is the number of moles. This relationship illustrates how Cv scales with the amount of substance.
• Heat Transfer: When heat is added to a system, the total heat transfer is proportional to the number of moles present. Thus, Cv reflects the extensive nature of heat capacity in relation to the substance's quantity.

Examples to Illustrate

Consider a scenario where you have a gas in a rigid container. If you have 1 mole of gas, and it requires 25 J to increase its temperature by 1°C, then for 3 moles, it would require 75 J. This clear scaling with the number of moles demonstrates that Cv is extensive.

Implications of Cv Being Extensive

Recognizing Cv as an extensive property has practical implications in thermodynamics and engineering. For example, when designing systems that involve heat transfer, such as engines or refrigerators, engineers must account for the amount of substance involved to accurately predict how much energy will be needed to achieve desired temperature changes.

Final Thoughts

In summary, the molar specific heat at constant volume (Cv) is an extensive property because it scales with the amount of substance in a system. Understanding this characteristic is crucial for applications in various scientific and engineering fields, where precise calculations of heat transfer are necessary.