With what minimum pressure (in \(k P a\) ), a given volume of an ideal gas \(\left(C_{p, m}=7 / 2 R\right.\) ), originally at \(400 \mathrm{~K}\) and \(100 \mathrm{kPa}\) pressure can be compressed irreversibly adiabatically in order to raise its temperature to \(600 \mathrm{~K}\) : (a) \(362.5 \mathrm{kPa}\) (b) \(275 \mathrm{kPa}\) (c) \(437.5 \mathrm{kPa}\) (d) \(550 \mathrm{kPa}\)

The adiabatic index, also known as the heat capacity ratio, is a crucial concept in thermodynamics, specifically when dealing with adiabatic processes in ideal gases. But what exactly is an adiabatic index? It is a dimensionless number denoted by \( n \) and is defined as the ratio of the specific heat capacity at constant pressure \( C_p \) to the specific heat capacity at constant volume \( C_v \) for a given gas. \[ n = \frac{C_p}{C_v} \]
For an ideal gas, this index helps in describing how the gas will respond to compression without heat exchange with its environment. For instance, the greater the adiabatic index, the faster the increase in temperature of the gas when it is compressed adiabatically. In our exercise, the adiabatic index was calculated using the provided \( C_{p, m} \) and \( C_{v, m} \) values, leading to \( n \) equaling 1.4. This value is consistent with a diatomic gas, like nitrogen or oxygen, which is what most of our atmosphere consists of.

Moving on to specific heat capacities, these are properties that dictate how much heat energy is needed to raise the temperature of a substance. When it comes to gases, we must distinguish between the specific heat at constant volume \( C_v \) and the specific heat at constant pressure \( C_p \).

The former, \( C_v \) is the amount of heat required to raise the temperature of a unit mass of gas by one degree while keeping the volume constant. The latter, \( C_p \) is the energy needed under constant pressure conditions. Their relationship is fundamental in thermodynamics and is represented by the equation \( C_p - C_v = R \) for ideal gases, where \( R \) is the ideal gas constant. Understanding these capacities is essential in problems involving adiabatic processes, like in our exercise, where they were used to determine the adiabatic index.

The ideal gas law is a foundational equation in chemistry and physics, underpinning our understanding of gas behavior under various conditions. It relates the pressure \( P \) of a gas to its volume \( V \) and temperature \( T \) via \( PV=nRT \), where \( n \) is the amount of gas in moles and \( R \) is the universal gas constant. This law assumes that the gas particles are point particles that exhibit no intermolecular forces and occupy no volume.

Although actual gases do not perfectly adhere to these assumptions, the ideal gas law provides an excellent approximation under many conditions. It becomes particularly useful when dealing with processes that do not involve changes in the amounts of gas, such as the adiabatic process we encountered in the exercise, which assumes a constant quantity of gas throughout the compression.

Irreversible adiabatic compression is a particular type of adiabatic process that occurs without heat exchange between the system and its surroundings and isn't performed infinitesimally slowly. In the real world, most compressions are irreversible due to factors like friction and rapid change.

During such a process, the entropy of the system increases, indicating that it is not at equilibrium at all points in the process. In our exercise, by compressing the gas adiabatically and irreversibly, the gas is raised to a higher temperature and pressure. The relationship described by the formula \( PV^n = \text{constant} \) was integral to finding the final pressure after the temperature was increased, embodying how closely these concepts are intertwined in the study of thermodynamics.