Molar Heat Capacities — Core Principles

Core Principles

Molar heat capacity ($C$) quantifies the heat required to raise the temperature of one mole of a substance by one Kelvin or Celsius. For gases, it's crucial to distinguish between molar heat capacity at constant volume ($C_v$) and at constant pressure ($C_p$).

$C_v$ represents the heat used solely to increase internal energy, while $C_p$ includes additional heat for work done during expansion. Mayer's relation, $C_p - C_v = R$, links these two for ideal gases, with $R$ being the universal gas constant.

The values of $C_v$ and $C_p$ depend on the number of active degrees of freedom ($f$) of the gas molecules (translational, rotational, vibrational), as per the Law of Equipartition of Energy. For monoatomic gases, $f=3$, leading to $C_v = \frac{3}{2}R$.

For diatomic gases at room temperature, $f=5$, giving $C_v = \frac{5}{2}R$. The ratio $\gamma = C_p/C_v$ is also a key parameter, related to $f$ by $\gamma = 1 + \frac{2}{f}$.

Important Differences

vs Specific Heat Capacity

Aspect	This Topic	Specific Heat Capacity
Definition	Molar Heat Capacity ($C$): Heat required to raise the temperature of one mole of a substance by $1^{\circ}\text{C}$ (or $1\text{ K}$).	Specific Heat Capacity ($c$): Heat required to raise the temperature of one unit mass (e.g., $1\text{ kg}$ or $1\text{ g}$) of a substance by $1^{\circ}\text{C}$ (or $1\text{ K}$).
Units	$\text{J mol}^{-1}\text{ K}^{-1}$ (or $\text{cal mol}^{-1}\text{ K}^{-1}$)	$\text{J kg}^{-1}\text{ K}^{-1}$ (or $\text{cal g}^{-1}\text{ K}^{-1}$)
Dependence on Amount	Intensive property (independent of the amount of substance, as it's per mole).	Intensive property (independent of the amount of substance, as it's per unit mass).
Relation to each other	$C = M_m \times c$, where $M_m$ is the molar mass of the substance.	$c = C / M_m$, where $M_m$ is the molar mass of the substance.
Context of Use	More common in thermodynamics, physical chemistry, and when dealing with gases, as it relates directly to the number of molecules and their degrees of freedom.	More common in general heat transfer calculations, calorimetry, and when dealing with solids and liquids where mass is often the primary measure.

While both molar heat capacity and specific heat capacity quantify a substance's ability to store thermal energy, they differ in their normalization. Molar heat capacity is defined per mole, making it ideal for comparing substances based on the number of constituent particles and their molecular structure, especially for gases where molecular motion (degrees of freedom) is critical. Specific heat capacity, defined per unit mass, is more practical for everyday applications and calorimetry involving bulk materials. The two are interconvertible using the molar mass of the substance.