State the second law of thermodynamics in words and express it mathematically.

Entropy is one of the most captivating and essential concepts in thermodynamics. It's a measure of disorder or randomness in a system. In a more accessible sense, imagine a room full of scattered toys. The messier it is, the higher the entropy. In the realm of physics, we consider entropy as a kind of 'spread' of energy or particles. When a system evolves, its energy tends to disperse more widely, which increases its entropy.

Mathematically, entropy is denoted as 'S' and is measured in units called joules per kelvin (J/K). According to the second law of thermodynamics, in an isolated system, entropy can either stay the same—in the case of a reversible process—or increase, but it can never decrease. This means that the universe tends toward greater disorder and randomness.

An isolated system is a theoretical concept where a system is completely shut off from its surroundings—no exchange of matter or energy occurs across its boundaries. Think of an insulated box with perfectly reflective walls that neither lose energy through heat nor allow energy to penetrate from outside. In the real world, truly isolated systems don't exist, but the notion is vital for understanding thermodynamics.

In the context of the second law, an isolated system is crucial because it is in this system that the nature of entropy is most clearly observed. Here, without external influence, the system will evolve towards a state of maximum entropy, or thermal equilibrium, driven solely by its internal processes.

Thermal equilibrium is the state achieved within a system or between different systems when no net heat transfer occurs between them. This means that all parts of the system or different systems in contact have the same temperature. At that point, no energy moves around because temperature differences drive energy transfer.

For example, if you pour some cream into your coffee and leave it undisturbed, after a while, you will find a uniform mixture—the cream and coffee are at thermal equilibrium. In terms of entropy, reaching thermal equilibrium is synonymous with reaching maximum entropy within an isolated system. It is the endgame for the second law of thermodynamics, signifying that the energy within the system is as spread out as it can be.

Reversible processes are idealized transformations in a system that occur in such a way that the system and its surroundings can be returned to their original states without any net change. It's important to understand that these processes are theoretical constructs; no real-world process is truly reversible because there are always some inherent inefficiencies. However, they provide a valuable benchmark for understanding the limitations of energy transformations.

During a reversible process, because the system can return to its original state, the entropy does not increase, which is a remarkable exception to the general rule that entropy tends to increase. These processes help define the limits of what is thermodynamically attainable and are exemplified by the careful control of temperature and the slow, infinitesimal changes in a system's variables to avoid generating any irreversibility.