Understanding Specific Heat at Constant Volume

Specific heat is a fundamental property of a substance that describes how much heat energy is required to raise the temperature of a unit mass of the substance by one degree. It is often denoted by the symbol \(c\). The value of specific heat can vary depending on the conditions under which the heat is added, particularly whether the process occurs at constant volume or constant pressure.

Definition of Specific Heat at Constant Volume

When heat is added to a substance while its volume is kept constant, the work done by or on the substance is zero. According to the first law of thermodynamics, the heat added in a constant volume process is equal to the change in the internal energy of the substance.

The specific heat at constant volume, denoted as \(c_v\), is defined as the rate of change of the specific internal energy with respect to temperature when the volume is held constant. Specific internal energy refers to the internal energy per unit mass of the substance.

Mathematically, the specific heat at constant volume is expressed as:

\(c_v = \left(\frac{\partial u}{\partial T}\right)_v\)

Where:

• \(u\) is the specific internal energy (internal energy per unit mass)
• \(T\) is the absolute temperature
• The subscript \(v\) indicates that the partial derivative is taken at constant specific volume.

Analyzing the Options

Let's look at the given options in light of the definition:

• Option 1: "the rate of change of specific pressure with respect to temperature when the volume is held constant." This describes a property related to the equation of state of the substance, not its specific heat capacity. Pressure change with temperature at constant volume is not the definition of specific heat.
• Option 2: "the rate of change of specific enthalpy with respect to temperature when the volume is held constant." Specific enthalpy ($h = u + Pv$) is often used in thermodynamics, and its rate of change with temperature at constant pressure is the specific heat at constant pressure ($c_p$). The definition involves specific enthalpy, but the condition is constant volume, not constant pressure, and the definition of $c_v$ is based on internal energy.
• Option 3: "the rate of change of specific internal energy with respect to temperature when the volume is held constant." This statement directly matches the definition of specific heat at constant volume ($c_v = (\frac{\partial u}{\partial T})_v$).
• Option 4: "the rate of change of specific volume with respect to temperature when the volume is held constant." This option is contradictory, as specific volume is already being held constant, so its rate of change with respect to temperature would be zero. This is not the definition of specific heat.

Based on the fundamental definitions in thermodynamics, the specific heat at constant volume is precisely the rate at which the specific internal energy changes with temperature when the volume does not change.