Module Notes: Heat Capacity, Molar Specific Heat, and Mass Specific Heat

When discussing heat capacity, it is crucial to specify whether it is at constant volume ($C_V$) or constant pressure ($C_P$), as they are fundamentally different.

Heat Capacity at Constant Volume ($C_V$)

• Definition: The heat required to change the temperature of a system while keeping its volume constant. At constant volume, no work is done ($W=0$), so by the First Law of Thermodynamics, the change in internal energy ($dE$) equals the heat added ($dQ$): $dE = dQ$.

• Mathematical Definition: $C_V = \left( \frac{dE}{dT} \right)_V$.

• Molar Specific Heat at Constant Volume ($c_{vm}$): This is heat capacity per mole. It is generalized by the number of degrees of freedom ($DOF$) as: $c_{vm} = \frac{DOF}{2}R$.

  • Monatomic gas: $DOF = 3$, so $c_{vm} = \frac{3}{2}R$.

  • Diatomic gas (normal temperatures): $DOF = 5$, so $c_{vm} = \frac{5}{2}R$.

  • Solids: $DOF = 6$ (three for kinetic and three for potential energy), so $c_{vm} = 3R$. These values align well experimentally for simple solids.

Mass Specific Heat ($c_v$)

• Definition: The heat needed to raise the temperature of one kilogram of a substance by one degree Kelvin at constant volume.

• Heat Transfer Equation: $Q = m c_v \Delta T$. This assumes $c_v$ is constant over the temperature range.

• Relationship to Molar Specific Heat: $c_v = \frac{1}{M} c_{vm}$ (where $M$ is molar mass).

Heat Capacity at Constant Pressure ($C_P$)

• Definition: The heat required to change the temperature of a system while keeping its pressure constant. In this process, some of the added heat can be converted into work (e.g., expansion of gas).

• Mayer's Relation: For ideal gases, the relationship between $C_P$ and $C_V$ is given by $C_P = C_V + nR$. For molar specific heats: $c_{pm} = c_{vm} + R$. This implies $C_P$ is always greater than $C_V$ because work is done, requiring more heat for the same temperature change.

  • Monatomic gas: $c_{pm} = \frac{5}{2}R$.

  • Diatomic gas: $c_{pm} = \frac{7}{2}R$.

• For solids and liquids, the difference between $C_P$ and $C_V$ is negligible due to their minimal volume changes.

Specific Heat of Water

• Liquid Water has a high specific heat ($c_{\text{water (liquid)}} = 4187 \text{ J/(kg}\cdot\text{K)}$), which is crucial for climate and biological systems due to its ability to absorb and release substantial heat with minimal temperature change.

• Ice ($2050 \text{ J/(kg}\cdot\text{K)}$) and Steam ($1996 \text{ J/(kg}\cdot\text{K)}$) have significantly lower specific heats than liquid water.