A gas with \(\gamma=1.40\) is at 100 kPa pressure and occupies \(5.00 \mathrm{L}\) (a) How much work does it take to compress the gas adiabatically to \(2.50 \mathrm{L} ?\) (b) What's its final pressure?

Thermodynamics is a branch of physics that deals with the relationships between heat and other forms of energy. In the context of thermodynamics, systems are often described by their properties such as pressure, volume, temperature, and internal energy. The behavior of a gas during processes such as expansion or compression can be analyzed using the laws of thermodynamics.

There are four main laws of thermodynamics, but in the context of the exercise provided, the first law—which is the principle of conservation of energy—applies directly. This law states that energy cannot be created or destroyed, only transformed from one form to another. In an adiabatic process, the system does work on the surroundings or has work done on it without the transfer of heat, meaning all energy transfer is in the form of work.

In an adiabatic process, the transfer of energy between a system and its surroundings is done entirely through work; there is no heat exchange. For a gas in an adiabatic process, the work done during compression or expansion can be understood using the concept of pressure-volume work.

The formula for this work done \( W \) on a gas by external forces during an adiabatic process is:
\[ W = \frac{P1 \times V1 - P2 \times V2}{\text{(gamma - 1)}} \] where \( P1 \) and \( P2 \) are the initial and final pressures, \( V1 \) and \( V2 \) are the initial and final volumes, and \( \text{gamma} \) is the adiabatic index or heat capacity ratio. This particular formula is a result of integrating the equation for an adiabatic process which relates pressure and volume.

Gas laws are equations that relate the pressure, volume, and temperature of a gas. These include Boyle’s Law, Charles’s Law, Gay-Lussac's Law, and the Combined Gas Law, which merges the three into a single equation. However, for adiabatic processes, the key relationship is given by the adiabatic equation:
\[ PV^\text{gamma} = \text{constant} \] where \( P \) is the pressure, \( V \) is the volume, and \( \text{gamma} \) is the adiabatic index. This equation describes how pressure and volume change during an adiabatic process where no heat is transferred into or out of the system. In the exercise, the initial and final pressures and volumes must adhere to this relationship for a given adiabatic index \( \text{gamma} = 1.40 \) used for diatomic gases, which includes air typically.

The pressure-volume (PV) relationship in an adiabatic process is quite distinct from that in an isothermal process (where temperature remains constant). In an isothermal process, the PV relationship is described by Boyle's Law, stating that pressure inversely varies with volume. Conversely, in an adiabatic process, the relationship is described by \( PV^\text{gamma} = \text{constant} \).

This relationship implies that pressure and volume do not change in a simple inverse proportionality. Instead, volume changes raise the pressure more than they would in an isothermal process due to the adiabatic index \( \text{gamma} \) that appears as an exponent on the volume. This power, greater than 1 for real gases, shows that the pressure increases rapidly with decreasing volume during compression, and vice versa during expansion. The exercise showcase this relationship, where compressing the gas to half its original volume significantly increases its pressure due to the work done on the gas.