Which one of the following statements is FALSE?

The correct option (the FALSE statement) is: For a real gas going through an adiabatic reversible process, the process equation is given by 𝑃𝑉^𝛾 = constant, where P is the pressure, V is the volume and 𝛾 is the ratio of the specific heats of the gas at constant pressure and constant volume.

Here is a step-by-step breakdown of why this statement is false, along with an explanation of why the other options are true:

1. Why the chosen statement is FALSE:
The classic adiabatic reversible process equation:
$$P{V}^{\gamma }=\text{constant}$$
is derived specifically using the assumptions of an ideal gas. In this derivation, the ideal gas law:
$$PV=nRT$$
and the relation for internal energy of an ideal gas (where internal energy depends solely on temperature, i.e., $dU=C_{v}dT$) are strictly applied.
For a real gas, intermolecular forces and the finite volume occupied by the gas molecules cannot be neglected. Real gases follow more complex equations of state (such as the van der Waals equation), and their internal energy depends on both temperature and volume. Consequently, the relationship $P{V}^{\gamma }=\text{constant}$ does not describe a real gas undergoing an adiabatic reversible process.

2. Verification of the other statements (which are TRUE):

• For an ideal gas, the enthalpy is independent of pressure:
For an ideal gas, internal energy $U$ and enthalpy $H$ are functions of temperature alone. By definition, enthalpy is:
$$H=U+PV$$
Substituting the ideal gas law ($PV=nRT$) yields:
$$H=U\left(T\right)+nRT$$
Since both terms on the right side depend only on temperature, the enthalpy $H$ of an ideal gas depends only on temperature and is completely independent of pressure or volume.

• For an ideal gas undergoing a polytropic process, the state relation holds:
No matter what path or process (polytropic, isobaric, isothermal, etc.) an ideal gas undergoes, it must satisfy its equation of state at every equilibrium point. The relation connecting pressure, volume, and temperature is always:
$$PV=mRT$$
Rearranging this formula gives:
$$\frac{P}{R}=\frac{mT}{V}$$
This equation is always valid at any point along the process for an ideal gas.

• Real gases behave ideally at low pressure or high temperature:
At sufficiently low pressures, the gas molecules are spaced very far apart, making intermolecular forces negligible. At sufficiently high temperatures, the thermal kinetic energy of the molecules is high enough to easily overcome intermolecular attractions. Under these conditions, any real gas closely obeys the ideal gas assumptions.