Which equation represents the change in internal energy?

The correct representation of the change in internal energy is captured in the relationship dU = TdS - pdV. This equation originates from the first law of thermodynamics, which states that the change in internal energy (dU) of a system can be expressed in terms of the heat exchanged (TdS) and the work done on or by the system (pdV).

In this context, TdS represents the reversible heat transfer to the system at temperature T during an infinitesimal reversible process. The term -pdV accounts for the work done by the system as it expands against a surrounding pressure p. The negative sign indicates that if the system expands (dV is positive), it does work on the surroundings and consequently reduces the internal energy of the system.

The other options do not capture the relationship correctly for internal energy changes in the context of thermodynamics. For instance, dU = dH + pdV incorrectly implies a connection between internal energy and enthalpy without appropriately factoring the nature of changes in volume. Similarly, the other formulations do not accurately reflect the necessary exchange of energy in the form of heat and work, which is essential for understanding internal energy changes in thermodynamic systems.