Is it possible to compress air without causing its internal energy to increase? If so, how?

When we discuss internal energy, we are talking about the energy contained within a substance due to the motion and interactions among its molecules. For an ideal gas, this energy is directly related to temperature; higher temperatures correlate with higher internal energy because the molecules move more vigorously. This relationship is crucial in understanding thermal processes.

To put it simply, if you keep the temperature of an ideal gas constant, its internal energy doesn't change, regardless of how you alter its pressure or volume. This principle underpins the concept of isothermal compression, where even though the gas is being compressed (and potentially does work on the surroundings or has work done on it), its internal energy stays the same because its temperature remains constant.

The ideal gas law is expressed as \(PV = nRT\), where \(P\) is the pressure of the gas, \(V\) is the volume, \(n\) is the number of moles of gas, \(R\) is the universal gas constant, and \(T\) is the absolute temperature. This equation is foundational in understanding how different physical properties of a gas are interrelated.

Often, gases are not ideal and real-life deviations occur; that's when the van der Waals equation comes into play, adding terms to account for molecular interactions and volumes. Nevertheless, the ideal gas law provides a solid baseline for understanding gaseous behavior, especially in educational examples and simplified situations. It plays a particularly important role in defining isothermal processes, as it implies that when temperature (\(T\)) is constant, the product of pressure (\(P\)) and volume (\(V\)) is also constant, even as their individual values might change.

An adiabatic process occurs when a system (such as a volume of gas) is compressed or expanded in such a way that no heat is transferred between the system and its surroundings. In an adiabatic process involving an ideal gas, if the system does work on the surroundings or has work done on it, this results in a temperature change since energy is not gained or lost through heat.

Contrastingly, during an isothermal process, temperature remains stable, meaning internal energy does not change. A perfect adiabatic process is an idealization because, in reality, some heat transfer always occurs, but it serves as a valuable concept in thermodynamics. In practice, adiabatic conditions can be approximated by insulating a system so that the heat exchange is minimized and the process happens quickly.