5.1.4.c. The general case

Let us consider the general case in which a change of state is brought about both by doing work and by transfer of heat. We write change in internal energy for this case as:

  ΔU = q + w

(5.1)

In thermodynamics, the change in internal energy (ΔU) of a system can be influenced by two main processes: the heat exchanged with the surroundings (q) and the work done on or by the system (w). The simplified equation ΔU = q + w shows that internal energy change is the sum of heat and work. This indicates that regardless of how these processes happen, the total change in internal energy depends only on the system's initial and final states, not the path taken to get there.

Think of this like filling a bathtub with water. The total amount of water in the tub is affected by both how long you leave the tap running (heat added) and whether you are scooping water out with a bucket (work done). The end result — the total water in the tub — is what matters, not how you got there.

For a given change in state, q and w can vary depending on how the change is carried out. However, q +w = ΔU will depend only on initial and final state. If there is no transfer of energy as heat or as work (isolated system), i.e., if w = 0 and q = 0, then ΔU = 0.

This part emphasizes that energy changes, whether heat or work, are not just arbitrary; they are interconnected. In an isolated system where no energy is exchanged with the surroundings, both heat transfer and work done are zero, thus leading to a net change in internal energy (ΔU) of zero. This means that the total energy in the system remains unchanged, highlighting the conservation of energy principle.

Imagine a sealed, insulated cooler containing ice. If kept untouched, the ice will remain the same over time; no heat has entered or left the cooler, and no work has been done on the cooler. The internal energy of the ice remains constant, exemplifying an isolated system.

The equation 5.1, i.e., ΔU = q + w is a mathematical statement of the first law of thermodynamics, which states that

The energy of an isolated system is constant. It is commonly stated as the law of conservation of energy i.e., energy can neither be created nor be destroyed.

The first law of thermodynamics, also known as the conservation of energy principle, asserts that energy within a closed system cannot change. It can only be transformed from one form to another, such as heat converting into work or vice versa. Consequently, as we analyze energy exchanges in thermodynamic cycles (like heat engines), we see that energy is merely redistributed and conserved within the system.

Think of energy like money in a bank account; it doesn’t just appear out of nowhere or disappear — it simply transfers between sources. Earning interest represents energy conversion, while withdrawals can be seen as work done with that energy. As with the first law, the total balance (or total energy) remains constant unless you add or remove funds.

note: There is considerable difference between the character of the thermodynamic property energy and that of a mechanical property such as volume. We can specify an unambiguous (absolute) value for volume of a system in a particular state, but not the absolute value of the internal energy. However, we can measure only the changes in the internal energy, ΔU of the system.

In thermodynamics, certain properties like volume can often be measured precisely at any given state. In contrast, internal energy is not assigned a specific value but can only be known in terms of changes (ΔU) from one state to another. This means that while the volume of a gas can be directly observed, the absolute energy within a system can't, only its variation when energy is added or extracted can be measured.

Consider this like measuring weight versus understanding the fuel level in a car. You can weigh something precisely; however, you might only know that your fuel tank is lower than it was before, but you can’t always measure the exact amount left without a gauge.

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