The internal energy of a system increased by \(400 \mathrm{~J}\) when it absorbed \(600 \mathrm{~J}\) of heat. (a) Was work done by or on the system? (b) How much work was done?

Internal energy is a key concept in thermodynamics representing the total energy stored within a system. It is composed of various forms of microscopic energy including kinetic energy, due to the motion of the molecules, and potential energy, resulting from intermolecular forces. An important thing to understand is that changes in internal energy, denoted by \(\triangle U\), can occur without knowing the specific amounts of kinetic and potential energy involved.

In the context of the exercise, the internal energy of the system increased by \(400 \text{ J}\), indicating that the overall energy of the molecules has increased. This change could be due to an increase in temperature or a phase change within the system, among other factors. Internal energy is a state function, meaning its change is independent of the path taken, relying only on the initial and final states of the system.

Thermodynamic work is an energy transfer that occurs when a force acts through a distance, as in the expansion or compression of a gas within a system. Work done by the system, denoted by \(W\), usually involves moving against an external pressure.

In our exercise, the positive sign of \(W = 200 \text{ J}\) indicates that the system has done work on the surroundings. This might involve the system's volume increasing, such as gas pushing a piston outward. It's fundamental in thermodynamics to distinguish between work done by the system, which is conventionally positive, and work done on the system, considered as negative.

Heat transfer, represented by the symbol \(Q\), involves the movement of energy due to a temperature difference between the system and its surroundings. Heat can move into or out of a system in several ways, including conduction, convection, and radiation.

The exercise mentions that the system absorbed \(600 \text{ J}\) of heat. This transfer of energy as heat indicates that the surrounding environment was at a higher temperature than the system, prompting energy to flow into the system to reach thermal equilibrium. Heat transfer is not conserved in itself and it can affect the system’s internal energy and facilitate thermodynamic work.

The principle of energy conservation is a fundamental concept of physics and thermodynamics. It asserts that the total energy of a closed system remains constant—it can neither be created nor destroyed, only transformed from one form to another or transferred between systems.

In the context of the given exercise, the First Law of Thermodynamics encapsulates this principle of energy conservation. The equation \(\Delta U = Q - W\) shows that the change in internal energy \(\Delta U\) of a system is equal to the heat added to the system \(Q\) minus the work done by the system \(W\). The exercise shows an application of this law, with energy being transferred as both heat and work, but the overall energy within the system plus its surroundings remaining constant.