Heat Capacities — Core Principles

Core Principles

Heat capacity (C) quantifies the heat required to change a substance's temperature by one unit, measured in J/K. It's an extensive property, meaning it depends on the amount of substance. To make it an intrinsic material property, we use specific heat capacity (c), which is per unit mass (J/(kg·K)), or molar heat capacity ($C_m$), which is per unit mole (J/(mol·K)).

For gases, heat capacity varies with the process: $C_V$ (constant volume) and $C_P$ (constant pressure). $C_P$ is always greater than $C_V$ because at constant pressure, some heat is used for work done by expansion, in addition to increasing internal energy.

Mayer's relation states $C_P - C_V = R$ for an ideal gas. The values of $C_V$, $C_P$, and their ratio $\gamma = C_P/C_V$ depend on the gas's degrees of freedom (translational, rotational, vibrational) as per the equipartition theorem.

Monoatomic gases have 3 degrees of freedom, diatomic 5 (at moderate T), and polyatomic 6 (non-linear). These concepts are fundamental to the First Law of Thermodynamics and crucial for understanding energy transfer.

Important Differences

vs Specific Heat Capacity vs. Molar Heat Capacity

Aspect	This Topic	Specific Heat Capacity vs. Molar Heat Capacity
Definition	Specific Heat Capacity (c): Heat required to raise the temperature of one unit mass of a substance by one degree.	Molar Heat Capacity ($C_m$): Heat required to raise the temperature of one mole of a substance by one degree.
Formula	$c = \frac{Q}{m\Delta T}$	$C_m = \frac{Q}{n\Delta T}$
Units	J/(kg·K) or J/(g·°C)	J/(mol·K) or J/(mol·°C)
Applicability	More commonly used for solids and liquids, where mass is a convenient measure.	More commonly used for gases, where moles are a fundamental unit in gas laws and chemical reactions.
Relationship	Related to molar heat capacity by multiplying by molar mass: $C_m = c \times M$ (where M is molar mass).	Related to specific heat capacity by dividing by molar mass: $c = \frac{C_m}{M}$.

While both specific heat capacity and molar heat capacity quantify the heat required for a unit temperature change, they differ in the 'unit' of substance considered. Specific heat capacity refers to a unit mass, making it suitable for materials where mass is the primary measure. Molar heat capacity, on the other hand, refers to a unit mole, which is particularly useful for gases and chemical contexts where the number of particles (moles) is more relevant. Their interconversion is straightforward via the molar mass of the substance.