What is defined as C_V = (∂U/∂T)_V?

The equation ( C_V = \left(\frac{\partial U}{\partial T}\right)_V ) defines the heat capacity at constant volume, represented by ( C_V ). This relationship is fundamentally tied to the first law of thermodynamics and thermodynamic definitions.

Heat capacity represents the amount of heat that must be added to a system to increase its temperature by one degree while maintaining specific constraints, in this case, constant volume. The internal energy ( U ) of a system is a function of temperature ( T ) and can change as temperature changes. By holding the volume constant and observing how internal energy changes with temperature, we can precisely quantify how much heat is required to raise the system's temperature, which defines the heat capacity at constant volume.

In contrast, the other options refer to different thermodynamic concepts. The heat capacity at constant pressure, enthalpy change, and internal energy change do not directly pertain to the definition involving the derivative of internal energy with respect to temperature under constant volume conditions. Thus, the correct choice accurately identifies ( C_V ) in the context of thermodynamic principles.