What is the difference between \(\Delta H\) and \(\Delta E\) ?

Enthalpy, represented by the symbol 'H', is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the energy required to make room for it by displacing its environment and establishing its volume and pressure.

The change in enthalpy (\textbf{\(\Delta H\)}) during a process is termed as the enthalpy change. It's a crucial concept in chemistry and physics, especially when dealing with heat exchange in chemical reactions or phase changes. In a reaction, \textbf{\(\Delta H\)} is positive in endothermic processes, where heat is absorbed from the surroundings, and negative in exothermic processes, where heat is released to the surroundings. Understanding enthalpy change is necessary for fields such as chemical engineering, material science, and thermodynamics, as it helps predict the energy exchange involved in various processes.

Internal energy is a bit less intuitive than enthalpy because it's not something you feel directly like you might feel heat coming off an object. It's the sum of all the kinetic and potential energy of the particles in a system. This includes everything from the electrons moving around nuclei to the vibrations of atoms in a piece of metal. The symbol 'E' denotes internal energy.

The change in internal energy (\textbf{\(\Delta E\)}) is the difference in a system's internal energy from the initial state to the final state. It accounts for the energy changes happening inside the system due to chemical reactions or physical transformations, not involving the external work done by the system or heat transfer. For instance, when a chemical bond breaks, the system absorbs energy, increasing the \textbf{\(\Delta E\)}, and when a bond forms, it releases energy, decreasing the \textbf{\(\Delta E\)}. Variations in \textbf{\(\Delta E\)} are central to fields such as statistical mechanics and quantum chemistry.

Thermodynamics is essentially the science of energy, heat, work, and how they interplay within the physical world. It deals with the concepts of energy transfer and transformations and provides a framework for understanding phenomena such as why ice melts when left out at room temperature or why your car engine gets hot when running.

At the core of thermodynamics lie several laws that dictate how these energy transformations can occur, and they tell us things like energy cannot be created or destroyed (the First Law) and that the disorder, or entropy, of the universe tends to increase over time (the Second Law). Thermodynamics is not just theoretical—it's immensely practical. It helps us design engines, refrigerators, and even predict the spontaneous direction of chemical reactions—among a myriad of other practical applications.

Pressure-volume work (\textbf{\(pV\)} work) is a concept in thermodynamics that occurs when a force is applied to change the volume of a system against an external pressure. Imagine pushing down on a syringe: here, you're doing work against the air pressure to decrease the volume. In a chemical system, pressure-volume work happens when gases expand or contract within a reaction.

Mathematically, it can be expressed as \textbf{\(w = -P\Delta V\)}, where \textbf{\(w\)} is the work done, \textbf{\(P\)} is the external pressure, and \textbf{\(\Delta V\)} is the change in volume. It is considered in situations where a change in the system's volume results in energy transfer, such as when a gas expands in a cylinder with a movable piston. It's an essential parameter in calculating the enthalpy change (\textbf{\(\Delta H\)}) and internal energy change (\textbf{\(\Delta E\)}) of a system under constant pressure.