Heat Capacity — Core Principles

Core Principles

Heat capacity ($C$) is a measure of a substance's ability to absorb heat energy for a given temperature change. It's an extensive property, meaning it depends on the amount of substance. To make it an intensive property, we use **specific heat capacity ($c$)**, which is heat capacity per unit mass ($J,g^{-1},K^{-1}$), or **molar heat capacity ($C_m$)**, which is heat capacity per unit mole ($J,mol^{-1},K^{-1}$).

The amount of heat ($q$) absorbed or released can be calculated using $q = mcDelta T$ or $q = nC_mDelta T$.

Crucially, heat capacity depends on the conditions: **$C_V$ (at constant volume)** relates to the change in internal energy ($C_V = (\frac{partial U}{partial T})_V$), while **$C_P$ (at constant pressure)** relates to the change in enthalpy ($C_P = (\frac{partial H}{partial T})_P$).

For ideal gases, $C_P - C_V = nR$ (or $C_{P,m} - C_{V,m} = R$), where $R$ is the ideal gas constant. This difference arises because at constant pressure, some energy is used for expansion work. The values of $C_V$ and $C_P$ are influenced by the molecular degrees of freedom (translational, rotational, vibrational), which vary for monatomic, diatomic, and polyatomic gases, affecting the ratio $gamma = C_P/C_V$.

Important Differences

vs Heat Capacity at Constant Pressure ($C_P$) vs. Heat Capacity at Constant Volume ($C_V$)

Aspect	This Topic	Heat Capacity at Constant Pressure ($C_P$) vs. Heat Capacity at Constant Volume ($C_V$)
Definition	Rate of change of enthalpy with temperature at constant pressure: $C_P = (\frac{\partial H}{\partial T})_P$	Rate of change of internal energy with temperature at constant volume: $C_V = (\frac{\partial U}{\partial T})_V$
Work Done	System can do P-V work (expansion/contraction) against surroundings.	No P-V work is done by or on the system.
Heat Supplied	Heat supplied increases internal energy AND does expansion work.	All heat supplied directly increases the internal energy.
Magnitude (for gases)	Always greater than $C_V$ ($C_P = C_V + nR$ for ideal gases).	Always less than $C_P$.
Relevance	Relevant for most chemical reactions and processes occurring in open containers (constant atmospheric pressure).	Relevant for processes occurring in rigid, sealed containers (e.g., bomb calorimeter).
Measurement	Measured using calorimeters open to atmosphere.	Measured using bomb calorimeters.

The primary distinction between heat capacity at constant pressure ($C_P$) and constant volume ($C_V$) lies in the work done by or on the system. At constant pressure, a system (especially a gas) can expand, performing work against the surroundings. Consequently, the heat supplied not only raises the internal energy but also accounts for this expansion work, making $C_P$ larger than $C_V$. At constant volume, no such work is possible, so all the heat directly contributes to increasing the internal energy. This fundamental difference is quantified by the relationship $C_P - C_V = nR$ for ideal gases, where $R$ is the ideal gas constant.