A combustion cxperiment is performed by burning a mixture of fuel and oxygen in a constant-volume container surrounded by a water bath. During the experiment, the temperature of the water rises. If the system is the mixture of fuel and oxygen: (a) Has heat been transferred? (b) Has work been done? (c) What is the sign of \(\Delta U\) ?

When we talk about heat transfer, we are referring to the movement of thermal energy from one place to another. In the example provided, we can see that the temperature of the water bath increases during the combustion experiment. This indicates that thermal energy is moving from the mixture of fuel and oxygen (the system) to the water bath (the surroundings).

Heat transfer can happen in three ways: conduction, convection, and radiation. In this specific case, it's likely happening through conduction, where thermal energy moves through a solid (the container) from the hot gas inside to the cooler water outside.

This is important in the first law of thermodynamics because it tells us that energy is conserved. If the system loses energy through heat transfer, that energy must go somewhere, in this case to the water bath.

Internal energy, represented as \(\text{U}\), is the total energy contained within a system. It includes the kinetic energy of all the molecules due to their motion and the potential energy due to interactions between molecules.

In our example, when the fuel and oxygen burn, the chemical energy stored in their bonds is converted into thermal energy, raising the temperature of the system and the surrounding water bath.

According to the first law of thermodynamics, any change in a system’s internal energy (\(\text{\Delta U}\)) is the result of heat transfer (\(q\)) and work done (\(w\)). In equation form: \(\text{\Delta U = q - w}\).

Since heat is transferred out of the system (negative \(q\)) and no work is done (\(w = 0\)), the internal energy change \(\Delta U\) for this system is negative, indicating a loss of energy.

Work is done by a system when it causes a displacement through a force. For thermodynamics, this most commonly refers to a change in volume.

In the exercise, the combustion occurs in a constant-volume container, meaning the volume does not change at all. Because of this, no work is done by or on the system (\(w = 0\)).

This is why, while determining the change in internal energy, the formula simplifies to \(\text{\Delta U = q}\), focusing only on the heat transfer aspect. The absence of work makes the calculation straightforward and underscores an important point: even without volume change, energy transformations and transfers still happen and must be accounted for.