On what law is the first law of thermodynamics based? Explain the sign conventions in the equation \(\Delta U=q+w\)

Imagine you are playing with a set of building blocks. You can stack them up in different structures, knock them down, or move them from place to place, but unless you add or remove blocks, you end up with the same number of blocks that you started with. This is the essence of the law of conservation of energy. In the realm of physics, energy behaves much like your set of blocks. It can change form—think of electrical energy lighting up a bulb or chemical energy in food turning into kinetic energy when you walk—but the total amount of energy in an isolated system remains constant. This principle is the foundation for the first law of thermodynamics.

The first law, often expressed with the formula \(\Delta U=q+w\), succinctly captures this law but in the context of thermodynamic systems. Whether energy enters a system as heat (\(q\)) or work (\(w\)), it contributes to the internal energy (\(\Delta U\)) of the system. In practical terms, this means that if you're heating a pot of water, the energy from the stove isn't lost; rather, it's transferred to the water, increasing its internal energy and potentially changing its state, say from liquid to gas.

The term 'internal energy' refers to the total energy contained within a thermodynamic system, represented by the symbol \(U\). It encompasses all forms of energy within the system, such as kinetic energy of atoms, potential energy from molecular bonds, or even the energy due to the temperature of the system. When we describe the internal energy change (\(\Delta U\)), we talk about the difference in energy between two states of the system. A positive change implies that energy has increased (such as when a substance is heated up), whereas a negative change indicates a decrease in energy (like when a gas expands and cools down).

The subtlety here is that, while you can measure the change in internal energy, the total internal energy itself is a more elusive quantity. In the realm of education, understanding this change is crucial for predicting how a system will react to various inputs of heat and work. If students grasp this concept, they can make sense of why a refrigerator, for example, grows warm on the outside as it cools the inside: the internal energy of the refrigerated interior decreases as a result of work and heat transfer.

Dealing with positive and negative signs can sometimes feel like trying to remember which way to turn a stubborn screw—'righty-tighty, lefty-loosey.' In thermodynamics, the sign conventions are crucial for understanding the direction of energy transfer. The overarching rule is that the system is the focus of our attention.

For heat (\(q\)), a positive value means that energy in the form of heat is added to the system—the opposite of getting chilly when you step out into the cold. Conversely, a negative \(q\) means the system is losing heat, much like a melting ice cube. As for work (\(w\)), if a system expands against external pressure (think of a balloon inflating), it does work on its surroundings—indicated by a negative sign. In contrast, if you compress a piston, you're doing work on the system, which is shown as a positive \(w\).

If students remember that positive signs mean 'energy in' and negative signs 'energy out' with respect to the system, they'll have an easier time determining whether energy is gained or lost. A firm grasp of these conventions helps avoid confusion and leads to better intuition around energy processes.