What is the difference between ∆H and ∆U? What is the sign of ∆H for exothermic and endothermic reactions? Under what circumstances ∆H =∆U?

∆H and ∆U are both terms used in thermodynamics to represent changes in different types of energy during a process. Here's the difference between them:

∆H (Delta H): ∆H represents the change in enthalpy of a system during a process. Enthalpy (H) is a thermodynamic quantity that combines the internal energy (U) of a system with the product of its pressure and volume. Mathematically, enthalpy is defined as H = U + PV, where P is the pressure and V is the volume. ∆H is expressed in units of energy (such as joules or calories).

∆U (Delta U): ∆U represents the change in internal energy of a system during a process. Internal energy (U) is the total energy of a system due to the motion and interactions of its particles. It does not include the energy associated with pressure-volume work. ∆U is also expressed in units of energy (joules or calories).

Now, let's discuss the sign of ∆H for exothermic and endothermic reactions:

Exothermic reactions: In an exothermic reaction, the system releases heat to the surroundings. This results in a decrease in the internal energy of the system. Since ∆H accounts for both internal energy (U) and pressure-volume work (PV), and in exothermic reactions, the decrease in internal energy is more significant than any changes in pressure-volume work, ∆H is generally negative (ΔH < 0) for exothermic reactions.

Endothermic reactions: In an endothermic reaction, the system absorbs heat from the surroundings. This leads to an increase in the internal energy of the system. Because the increase in internal energy outweighs any changes in pressure-volume work, ∆H is generally positive (ΔH > 0) for endothermic reactions.

Under what circumstances ∆H = ∆U:

∆H is equal to ∆U under conditions where the process does not involve any expansion or compression work. In other words, if the process occurs at constant volume (no pressure-volume work) and the only work done is "non-expansion" work (such as electrical work or shaft work), then ∆H and ∆U will be equal. This is because under constant volume conditions, PV work is zero, and the enthalpy change (∆H) only accounts for changes in internal energy (∆U).

To summarize:

∆H represents the change in enthalpy (internal energy plus PV work).
∆U represents the change in internal energy (excluding PV work).
∆H is negative for exothermic reactions and positive for endothermic reactions.
∆H = ∆U under constant volume conditions and no expansion work.