Question

Which one of the following statements is FALSE?

• A. For an ideal gas, the enthalpy is independent of pressure.
• B. For a real gas going through an adiabatic reversible process, the process equation is given by \(PV^\gamma = \text{constant}\), where P is the pressure, V is the volume and \(\gamma\) is the ratio of the specific heats of the gas at constant pressure and constant volume.
• C. For an ideal gas undergoing a reversible polytropic process \(PV^{1.5} = \text{constant}\), the equation connecting the pressure, volume and temperature of the gas at any point along the process is \(\frac{P}{R} = \frac{mT}{V}\), where R is the gas constant and m is the mass of the gas.
• D. Any real gas behaves as an ideal gas at sufficiently low pressure or sufficiently high temperature.

Hint:

Be very precise about the conditions under which common thermodynamic relations apply. Relations like \(PV=mRT\) and \(h=f(T)\) are for ideal gases. The process equation \(PV^\gamma = \text{constant}\) is for an isentropic process of an ideal gas with constant specific heats. Real gases require more complex equations of state and property relations.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The question asks to identify the false statement among four statements related to the thermodynamics of ideal and real gases.
Step 2: Detailed Explanation:
Let's analyze each statement: - (A) For an ideal gas, the enthalpy is independent of pressure. The enthalpy of an ideal gas is defined as \(H = U + PV\). Since for an ideal gas, internal energy \(U\) is a function of temperature only (\(U = f(T)\)) and \(PV = mRT\), the enthalpy can be written as \(H = f(T) + mRT\). This shows that the enthalpy of an ideal gas is also a function of temperature only. Therefore, it is independent of pressure. This statement is TRUE.
- (B) For a real gas going through an adiabatic reversible process, the process equation is given by \(PV^\gamma = \text{constant}\)...
The relation \(PV^\gamma = \text{constant}\) is derived specifically for an ideal gas undergoing a reversible adiabatic (isentropic) process, under the assumption that the specific heats are constant. This equation does not hold for a real gas, as real gases do not follow the ideal gas law and their specific heats can vary with temperature and pressure. This statement is FALSE.
- (C) For an ideal gas undergoing a reversible polytropic process... the equation connecting... is \(\frac{P}{R} = \frac{mT}{V}\).
The equation given is \(\frac{P}{R} = \frac{mT}{V}\). Rearranging this gives \(PV = mRT\). This is the ideal gas equation of state. The ideal gas equation of state holds for an ideal gas at any state point, regardless of the process connecting the states. Since the gas is ideal, this equation is valid at any point along the polytropic process. This statement is TRUE.
- (D) Any real gas behaves as an ideal gas at sufficiently low pressure or sufficiently high temperature.
The ideal gas model assumes that intermolecular forces are negligible and the volume of the gas molecules is zero compared to the container volume. At very low pressures, molecules are far apart, so intermolecular forces are negligible. At very high temperatures, the kinetic energy of the molecules is much greater than the potential energy of intermolecular attraction, so the forces become insignificant. Therefore, real gases approach ideal gas behavior at these conditions. This statement is TRUE.
Step 3: Final Answer:
The false statement is (B).
Step 4: Why This is Correct:
Statement (B) incorrectly applies the isentropic process equation for an ideal gas to a real gas. The other three statements are fundamental principles of thermodynamics for ideal and real gases.