One litre of a gas at 300 atm and \(473 \mathrm{~K}\) is compressed to a pressure of 600 atm and \(273 \mathrm{~K}\). The compressibility factors found to be \(1.072\) and \(1.375\) respectively at the initial and finally conditions. Calculate the final volume.

Understanding the concept of non-ideal gas behavior is crucial in the realm of physical chemistry, particularly when dealing with real gases under high pressures or low temperatures. Unlike ideal gases, which are described by the Ideal Gas Law, real gases do not always follow the law perfectly due to interactions between gas molecules and the volume occupied by the gas molecules themselves.

As pressure increases or temperature decreases, these interactions become more significant, causing deviations from the Ideal Gas Law. The compressibility factor, Z, is introduced to quantify this deviation. A Z value of 1 indicates ideal behavior, while a Z value different from 1 reflects the degree to which a gas is non-ideal. For real gases, corrections for intermolecular forces and molecular volume are made using the Van der Waals equation or other equations of state designed to more accurately predict the behavior of a gas under a given set of conditions.

The PV/nRT equation represents the Ideal Gas Law, which is a cornerstone of gas laws and is essential in understanding the relationship between the pressure (P), volume (V), mole number (n), gas constant (R), and temperature (T) of a gas. The equation is often stated as PV = nRT.

However, when dealing with real gases, the equation is modified to include the compressibility factor (Z), becoming Z = PV/nRT. This alteration accounts for non-ideal behavior by including the deviation from ideal conditions. The compressibility factor acts as a reality check, allowing scientists and engineers to make more accurate predictions and calculations regarding the behavior of gases under various conditions.

Gas compression calculations are integral to many scientific and industrial processes. Whether you're considering the compression of gases in cylinders or the operation of a refrigerator, understanding how to calculate changes in volume and pressure during compression is vital.

When compressing a gas, both pressure and temperature typically increase. However, if the temperature is controlled or allowed to change in a specific way—for example, if the process is adiabatic (no heat transfer) or isothermal (constant temperature)—the calculations adjust accordingly. The properties of the gas before and after compression are crucial data points; hence, the compressibility factor is a key element of these calculations. It adjusts the ideal gas equation to provide realistic results for the final volume or pressure after compression.

The Indian Institutes of Technology Joint Entrance Examination (IIT-JEE) is acclaimed for its challenging and conceptual Physical Chemistry problems. Mastery over concepts such as gas laws, thermodynamics, and chemical equilibrium become essential.

IIT-JEE aspirants often face questions that test their understanding of real versus ideal gas behavior and require them to perform gas compression calculations. A key aspect of solving these problems is the comprehension and manipulation of the compressibility factor within the context of various equations of state. Students must apply both conceptual knowledge and problem-solving skills to excel in this competitive exam, dealing with multifaceted problems like the one described in the original exercise, which intertwines the concepts of non-ideal gas behavior and the PV/nRT equation in a practical situation.