Question

Which of the following relationships is/are not true?
(A). Most probable velocity = \( \sqrt{\frac{2RT}{M}} \)
(B). \( PV = \frac{2}{3}kT \)
(C). Compressibility factor \( Z = \frac{PV}{nRT} \)
(D). Average kinetic energy of gas = \( \frac{1}{2}kT \)
Choose the correct answer from the options given below

• (A) only.
• (D) only.
• (B) and (C) only.
• (A) and (C) only.

Hint:

In questions asking for incorrect statements, first identify the certainly correct ones. This allows you to eliminate options quickly. Here, knowing that the formulas for most probable velocity (A) and compressibility factor (C) are standard definitions immediately rules out options 1, 3, and 4.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question tests the knowledge of fundamental equations from the kinetic theory of gases and thermodynamics. We need to evaluate the correctness of each given relationship.
Step 2: Detailed Explanation:
Let's analyze each statement:
(A) Most probable velocity = \( \sqrt{\frac{2RT}{M}} \): This is the standard and correct formula for the most probable velocity of gas molecules, derived from the Maxwell-Boltzmann distribution. So, 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 relation from kinetic theory is \( PV = \frac{2}{3}E_k \), where \( E_k \) is the total kinetic energy of the gas. Since \( E_k = \frac{3}{2}NkT \), we get \( PV = NkT \). The given relation \( PV = \frac{2}{3}kT \) is incorrect. However, in the context of multiple-choice questions with potential flaws, we must evaluate all options.
(C) Compressibility factor \( Z = \frac{PV}{nRT} \): This is the definition of the compressibility factor, which measures the deviation of a real gas from ideal gas behavior. For an ideal gas, Z = 1. This statement is true.
(D) Average kinetic energy of gas = \( \frac{1}{2}kT \): According to the kinetic theory of gases, the average translational kinetic energy of a gas molecule is given by \( \frac{3}{2}kT \). The term \( \frac{1}{2}kT \) represents the average kinetic energy per degree of freedom. As 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, the statement (D) is not true.
Step 3: Final Answer:
The question asks which relationship(s) is/are not true.
Statements (A) and (C) are true.
Statements (B) and (D) are not true.
Looking at the given options, we must choose the best fit. Options (1), (3), and (4) include (A) or (C) which are true statements, so they cannot be the answer for "not true". Option (2) states that only (D) is not true. This implies that the question setter may have considered (B) to be true (possibly due to a typo in the intended formula, like confusing it with \(PV = \frac{2}{3}E_k\)) or that there is an error in the question's options. However, among the choices, (D) is definitively incorrect based on the standard definition of average kinetic energy. Eliminating the clearly true statements (A and C) forces us to select the option that does not contain them. Option (2) is the only one that fits this criterion.