To determine which of the given relations is not correct, we need to analyze each of them in the context of thermodynamics:
\Delta H = \Delta U + P \Delta V - This is the correct expression for the change in enthalpy \Delta H, which is related to the change in internal energy \Delta U plus the work done at constant pressure P \Delta V.
\Delta U = q + W - This formula represents the first law of thermodynamics, which states that the change in internal energy \Delta U is equal to the heat added to the system q plus the work done on the system W. This is correct.
\Delta S_{\text{sys}} + \Delta S_{\text{surr}} \geq 0 - This expresses the second law of thermodynamics, which indicates that the total entropy change (system plus surroundings) is always greater than or equal to zero for a spontaneous process. This is also correct.
\Delta G = \Delta H - T \Delta S - This is the formula for the Gibbs free energy change, which relates the enthalpy change \Delta H, the temperature T, and the entropy change \Delta S. This formula is correct.
Upon analyzing all four relations, it is evident that the expression \Delta H = \Delta U - P \Delta V is incorrect. The correct relationship for enthalpy change should include a plus sign in the term P \Delta V, making it \Delta H = \Delta U + P \Delta V.
