Understanding the Relationship Between Molar Heat Capacity and Heat Capacity

When discussing thermal properties of materials, two crucial concepts often come into play: heat capacity and molar heat capacity. These terms are essential for comprehending how materials respond to changes in temperature, especially in the fields of thermodynamics and calorimetry.

Introduction to Heat Capacity and Molar Heat Capacity

Heat Capacity (C) is an extensive property that quantifies the amount of heat energy required to raise the temperature of a given substance by one degree Celsius (or one Kelvin). It is defined as the heat required to cause a unit change in the temperature of a substance and is typically measured in units of joules per degree Celsius (J/°C) or joules per Kelvin (J/K).

Molar Heat Capacity (Cm), on the other hand, is an intensive property that measures the heat energy required to raise the temperature of one mole of a substance by one degree Celsius or one Kelvin. Molar heat capacity is expressed in joules per mole per degree Celsius (J/mol·°C) or joules per mole per Kelvin (J/mol·K). The key distinction is that molar heat capacity is independent of the quantity of the substance.

The Relationship Between Molar Heat Capacity and Heat Capacity

The relationship between heat capacity (C) and molar heat capacity (Cm) is elegantly simple and can be mathematically described as:

Mathematical Expression:

C n × Cm

Where:

C - Heat capacity of the substance n - Number of moles of the substance Cm - Molar heat capacity of the substance

This equation highlights that the heat capacity of a substance is directly proportional to the number of moles of that substance, with the proportionality constant being the molar heat capacity.

Key Points and Differences

Extensive vs. Intensive Properties

Heat capacity (C) is considered an extensive property as it depends on the amount of substance present. Doubling the amount of a substance will double its heat capacity. In contrast, molar heat capacity (Cm) is an intensive property, meaning it remains constant regardless of the amount of substance.

Dependence on Substance

Molar heat capacity (Cm) varies from one substance to another and can also change with temperature and the state of the substance (solid, liquid, or gas).

Units and Application

When using the relationship, it is essential to ensure that the units used for heat capacity (C) and molar heat capacity (Cm) are consistent. For instance, if molar heat capacity is expressed per kilomole (kmol), the relationships for molar heat capacity at constant pressure and volume are given by:

Molar Heat Capacity at Constant Pressure (Cpm)

Cpm R × γ / (γ - 1)

Molar Heat Capacity at Constant Volume (Cvm)

Cvm R / (γ - 1)

Where:

R - Gas constant (R 8.314 J/mol·K) γ - Adiabatic index (for diatomic gases like air, γ 1.4; for monoatomic gases like helium, γ 1.667)

In the case of heat capacity calculations per kilogram (kg) of gas, the relations are adjusted by the molecular weight (M) of the gas as follows:

Heat Capacity at Constant Pressure (Cp)

Cp R × γ / (γ - 1) × M

Heat Capacity at Constant Volume (Cv)

Cv R / (γ - 1) × M

These relationships help in understanding how much heat is required to change the temperature of a sample based on its amount, making them invaluable tools in thermodynamics and calorimetry.