At moderate pressure, the compressibility factor for a gas is given as: \(Z=1+0.35 P\) \(-\frac{168}{T} \cdot P\), where \(P\) is in bar and \(T\) is in Kelvin. What is the Boyle's temperature of the gas? (a) \(168 \mathrm{~K}\) (b) \(480 \mathrm{~K}\) (c) \(58.8 \mathrm{~K}\) (d) \(575 \mathrm{~K}\)

The compressibility factor, symbolized as \( Z \), is a dimensionless quantity that describes how a real gas deviates from ideal gas behavior. For an ideal gas, the compressibility factor is equal to 1 across a range of temperatures and pressures. This simplifies to the Ideal Gas Law, \( PV=nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is temperature.

In the case of real gases, the value of \( Z \) differs from 1 due to intermolecular forces and the actual volume occupied by the gas molecules, which are not considered in the ideal gas model. Therefore, the equation \( PV=ZnRT \) is used, which adjusts the Ideal Gas Law to account for real gas behavior.

As an example, if we look at the equation provided in the exercise, \( Z=1+0.35 P - \frac{168}{T} \cdot P \), it suggests the compressibility factor changes with pressure \( P \) and temperature \( T \). To understand the gas behavior at different conditions, we can analyze how variations in \( P \) and \( T \) affect \( Z \), thus reflecting the non-ideal nature of the gas in question.

Real gases are gases that do not perfectly adhere to the assumptions of the Ideal Gas Law, especially under high pressure or low temperature conditions. Unlike ideal gases, real gases have intermolecular attractions or repulsions and occupy physical space, which must be accounted for.

In reality, as the molecules of a gas are brought closer together (at higher pressures) or cooled (at lower temperatures), their behavior deviates from ideality, due to the increased effect of intermolecular forces. For instance, real gases can condense to form a liquid, or solidify, which is a behavior that ideal gases, by definition, never exhibit.

Engineers and scientists use various equations of state, like the Van der Waals equation, to predict and describe the behavior of real gases by incorporating terms to account for intermolecular forces and the volume occupied by gas molecules. These equations are critical for designing equipment and processes in industries that deal with gases under a wide range of temperatures and pressures.

Ideal gas behavior is a theoretical model that simplifies the study of gases by assuming that the gas molecules do not interact with each other (no intermolecular forces) and occupy no volume. These assumptions allow for the derivation of the Ideal Gas Law, which works reliably within the limits of low pressure and high temperature.

When gases behave ideally, their compressibility factor \( Z \) is equal to 1. This aligns with the conditions set for an ideal gas in the Ideal Gas Law. However, in practice, no gas is truly ideal; real gases only approximate ideal behavior under certain conditions, and those conditions are typically characterized by low pressures and high temperatures, where the particles are far apart, and intermolecular forces are negligible.

The concept of Boyle's temperature, encountered in the given exercise, is closely connected to ideal gas behavior. It is defined as the temperature at which the gas exhibits ideal behavior over a range of pressures. At this temperature, any pressure-based deviations from ideality (represented by changes in \( Z \)) are minimized, and the gas best conforms to Boyle's Law, which is a special case of the Ideal Gas Law for a fixed amount of gas at constant temperature.