The Kinetic Theory Of Gases

We are given that , , and are the mean speed, root mean square speed, and most probable speed of molecules in an ideal monoatomic gas. These quantities are related to the temperature and mass of the molecules. The expressions for these speeds are as follows: 1. The most probable speed is given by: 2. The root mean square speed is given by: 3. The mean speed is given by: where: - is the Boltzmann constant, - is the absolute temperature, - is the mass of a molecule. Step 1: Analyzing Option (A) - Speed Greater than From the Maxwell-Boltzmann distribution, the speeds of molecules follow a probability distribution. Although some molecules can have speeds much higher than the most probable speed , the probability density rapidly decreases as speed increases. Therefore, no molecule can have a speed greater than , making this statement true. Thus, Option (A) is correct. Step 2: Analyzing Option (B) - Speed Less than The speed corresponds to the most probable speed, which is the speed at which the maximum number of molecules are moving. The distribution of speeds has a wide spread, and some molecules will have speeds less than , so this statement is not correct. Thus, Option (B) is incorrect. Step 3: Analyzing Option (C) - From the equations for the speeds: we know that , as: Thus, the relationship is correct. Hence, Option (C) is correct. Step 4: Analyzing Option (D) - Kinetic Energy Formula The average kinetic energy per molecule is given by: The statement in Option (D) suggests the kinetic energy is proportional to , which is not correct. The correct relationship for the kinetic energy is based on the temperature of the gas and not directly on the speed. Therefore, this statement is incorrect. Thus, Option (D) is incorrect. Conclusion The correct statement is:

Concept:
Use ideal gas law:


Step 1: Analyse segments in P–T graph
- AB: horizontal line ⇒ constant pressure (isobaric).
- BC: vertical line ⇒ constant temperature (isothermal).
- CA: slanted line ⇒ linear P–T relation ⇒ neither iso-P nor iso-T.

Step 2: Convert to P–V plot
- Isobaric (AB) ⇒ horizontal line in P–V.
- Isothermal (BC) ⇒ hyperbola in P–V.
- Remaining segment (CA) closes loop.

Step 3: Match graph
Only option with:
- Straight horizontal AB
- Curved BC
is (C).