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."

To understand why this statement is false, let us examine the thermodynamic principles governing reversible adiabatic processes for both ideal and real gases.

1. Derivation for an Ideal Gas:
For any reversible process, the first law of thermodynamics can be written as:
$$dQ=dU+PdV$$
For an adiabatic process, there is no heat exchange with the surroundings, meaning:
$$dQ=0$$
Thus, the equation simplifies to:
$$dU+PdV=0$$
For an ideal gas, the internal energy is a function of temperature only, which gives:
$$dU=m{C}_v$$
Additionally, an ideal gas strictly obeys the ideal gas equation of state:
$$PV=mRT$$
Combining these relations and integrating leads to the well-known relation:
$$P{V}^{\gamma }=\text{constant}$$
where $\gamma =C_p/C_v$ is the ratio of specific heats.

2. Why it Fails for a Real Gas:
For a real gas, the assumptions of an ideal gas do not hold:
• The internal energy $U$ depends on both temperature and volume due to intermolecular attraction forces. That is, $dU$ contains an additional volume-dependent term and is not simply equal to $m{C}_{v}dT$.
• Real gases do not obey the simple ideal gas equation $PV=mRT$. Instead, they follow more complex equations of state (such as the van der Waals equation or Redlich-Kwong equation).

Consequently, substituting the properties of a real gas into the first law of thermodynamics does not yield the relation $P{V}^{\gamma }=\text{constant}$. Thus, the statement claiming this relation holds for a real gas is false.