Consider a box filled with an ideal gas. The box undergoes a sudden free expansion from \(V_{1}\) to \(V_{2}\). Which of the following correctly describes this process? a) Work done by the gas during the expansion is equal to \(n R T \ln \left(V_{2} / V_{1}\right)\) b) Heat is added to the box. c) Final temperature equals initial temperature times \(\left(V_{2} / V_{1}\right)\). d) The internal energy of the gas remains constant.

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In particular, it describes how thermal energy is converted to and from other forms of energy and how it affects matter. The behavior of an ideal gas during processes such as expansion is a classic study case in thermodynamics. It explores how variables like temperature, volume, and pressure are interrelated and how they influence the energy within a system.

For the free expansion of an ideal gas, thermodynamics tells us that since the system is isolated with no external forces acting on it, there isn't a transfer of heat or work being done on the surrounding environment. This has direct implications on the internal energy of the gas, which also ties into another crucial aspect of thermodynamics – the conservation of energy, as expressed by the first law of thermodynamics.

The ideal gas law is a cornerstone in the study of thermodynamics and provides a clear relationship between the pressure (P), volume (V), temperature (T), and number of moles (n) of a gas. Given by the equation \(PV = nRT\), where 'R' is the ideal gas constant, the law is a mathematical relationship that helps predict and explain the behavior of gases under various conditions.

In the context of a free expansion, such as the one described in the exercise, where a gas expands without external constraint and without performing work, the ideal gas law can be used to infer that if the temperature of the gas remains constant, the pressure may drop to zero since the gas is doing no work on its surroundings and no heat transfer occurs. This upholds that certain parameters, like the temperature in an adiabatic free expansion of an ideal gas, can remain unchanged despite a change in volume.

The first law of thermodynamics is essentially the law of conservation of energy applied to thermal processes. It states that the change in the internal energy of a system (\(\Delta U\)) is equal to the heat added to the system (Q) minus the work done by the system on its surroundings (W), mathematically expressed as \(\Delta U = Q - W\). When we talk about a free expansion of an ideal gas, both heat transfer and work done are zero, leading to no change in internal energy.

This principle helps us understand the exercise's solution. Since there is no heat transfer (Q=0) and no work is done (W=0) during a free expansion, we see that the internal energy (U) remains constant. This insight from the first law of thermodynamics validates our conclusion that the correct answer is the internal energy of the gas does not change during such an expansion.

An adiabatic process is a type of thermodynamic process in which there is no heat transfer into or out of the system. The system is effectively insulated from its surroundings. For an ideal gas undergoing an adiabatic process, this would mean that any change in the system's energy can only come from work done by or on the system.

In the specific case of an adiabatic free expansion, no external work is performed, and because the gas doesn't exchange heat with its environment, the process results in the temperature of the ideal gas remaining constant. This is a special type of adiabatic process because it defies the usual expectation that an adiabatic compression increases the gas temperature while an adiabatic expansion decreases it. But in a free expansion, the expansion happens without any external pressure; thus, this unique scenario maintains the temperature constant.