Step 1: Concept Overview:
This question assesses knowledge of core equations from kinetic gas theory and thermodynamics. The task is to determine the validity of each provided relationship.
Step 2: Detailed Analysis:
Let's examine each statement:
(A) Most probable velocity = \( \sqrt{\frac{2RT}{M}} \): This is the accurate formula for the most probable velocity of gas molecules, derived from the Maxwell-Boltzmann distribution. Therefore, statement (A) is true.
(B) \( PV = \frac{2}{3}kT \): The ideal gas law for a single molecule is \( PV = kT \), where \( k \) is the Boltzmann constant. Another related expression is \( PV = \frac{2}{3}E_k \), where \( E_k \) is the total kinetic energy. Since \( E_k = \frac{3}{2}NkT \), we derive \( PV = NkT \). The given relation \( PV = \frac{2}{3}kT \) is incorrect. However, considering multiple-choice scenarios, all options must be evaluated.
(C) Compressibility factor \( Z = \frac{PV}{nRT} \): This is the definition of the compressibility factor, quantifying the deviation of real gases from ideal behavior. For ideal gases, Z = 1. This statement is true.
(D) Average kinetic energy of gas = \( \frac{1}{2}kT \): According to kinetic gas theory, the average translational kinetic energy of a gas molecule is \( \frac{3}{2}kT \). The term \( \frac{1}{2}kT \) represents the average kinetic energy per degree of freedom. Since a gas molecule has 3 translational degrees of freedom, the total average kinetic energy is \( 3 \times \frac{1}{2}kT = \frac{3}{2}kT \). Therefore, statement (D) is not true.
Step 3: Conclusion:
The question asks for the incorrect relationship(s).
Statements (A) and (C) are correct.
Statements (B) and (D) are incorrect.
Analyzing the options, we must select the best fit. Options (1), (3), and (4) include (A) or (C) which are correct, ruling them out as the answer to "not true". Option (2) states that only (D) is incorrect. This suggests a potential issue, perhaps a typo, or an error in the question's options. However, (D) is definitively incorrect based on standard definitions. Eliminating correct statements (A and C) directs us to choose the option that doesn't contain them. Option (2) aligns with this.