Calculate the entropy change associated with the isothermal expansion of \(5.25 \mathrm{~mol}\) of ideal gas atoms from \(24.252 \mathrm{~L}\) to \(34.058 \mathrm{~L}\).

Thermodynamics is a branch of physics that deals with the relationships between heat, work, temperature, and energy. The fundamental principles of thermodynamics are encapsulated in its four laws, which describe how energy moves and transforms within a system.

At the heart of thermodynamics is the concept of entropy, symbolically represented as 'S'. Entropy is a measure of the randomness or disorder within a system and is a key concept when analyzing the efficiency of thermal systems. It can be understood as a natural tendency towards dissipation of energy and an increase in chaos.

An important point to note is that while energy can neither be created nor destroyed (the first law of thermodynamics), the quality of energy can change, leading to an increase in entropy (the second law of thermodynamics). These laws set the stage for understanding processes like isothermal expansion, which involves changes both in the energy and the entropy of a system.

Isothermal expansion refers to the increase in volume of a gas while maintaining a constant temperature. This thermodynamic process is a common exercise in understanding entropy because it manifests how entropy increases even without a change in temperature.

During isothermal expansion, the gas does work on its surroundings because the external pressure must be gradually decreased to maintain the constant temperature. This aligns with the intuitive notion that as a gas expands, its particles spread out, becoming more disordered. The formula for calculating the change in entropy during this process helps us quantify this increase in disorder.

As underlined in thermodynamics, we also need to consider the conservation of energy. In an isothermal process, even though the internal energy of the ideal gas does not change, the transfer of heat is crucial to sustain the constant temperature. This transfer of heat, at a microscopic level, is the reason behind entropy change.

The ideal gas law is an equation of state for a hypothetical gas that allows us to predict the behavior of real gases under certain conditions. The ideal gas law is usually formulated as PV=nRT, where 'P' stands for pressure, 'V' is volume, 'n' is the number of moles, 'R' is the universal gas constant, and 'T' is the absolute temperature in Kelvins.

This law helps us link together four important physical quantities: pressure, volume, temperature, and the amount of substance. When dealing with problems in thermodynamics, particularly those involving isothermal processes and entropy changes, the ideal gas law provides a simplification that makes calculations manageable.

To illustrate, understanding the relationship between volume and temperature under constant pressure lets us predict how a gas will expand or compress. This forms a basis when calculating changes in entropy for a system undergoing isothermal expansion, as the ideal gas law can be modified to describe the conditions for this specific process.