Problem 15 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

According to kinetic theory of gases, for a diatomic molecule (a) the pressure exerted by the gas is proportional to mean velocity of the molecule (b) the pressure exerted by the gas is proportional to the root mean velocity of the molecule (c) the root mean square velocity of the molecule is inversely proportional to the temperature (d) the mean translational kinetic energy of the molecule is proportional to the absolute temperature.

Step-by-Step Solution

Answer

Option (d) is correct.

Step 1: Understanding the Kinetic Theory of Gases

The kinetic theory of gases states that the behavior of gas molecules can be explained by their constant motion, collisions, and interactions. This theory assumes that gas molecules occupy negligible volume and their collisions are perfectly elastic, which means there is no loss of kinetic energy.

Step 2: Analyzing Each Option

Let's consider each option:(a) The pressure exerted by the gas is not directly proportional to the mean velocity of the molecules.(b) Pressure is not directly related to the root mean square velocity of gas molecules.(c) The root mean square velocity is actually proportional to the square root of temperature \( (v_{rms} \propto \sqrt{T}) \), not inversely.(d) The mean translational kinetic energy \( (KE) \) of a molecule is given by \( KE = \frac{3}{2}kT \), where \( k \) is the Boltzmann constant and \( T \) is the absolute temperature. Thus, \( KE \) is proportional to \( T \).

Step 3: Evaluate Option (d)

Option (d) states that the mean translational kinetic energy of the molecule is proportional to the absolute temperature. Given the formula \( KE = \frac{3}{2}kT \), this option correctly describes a fundamental aspect of the kinetic theory of gases.

Step 4: Drawing a Conclusion

From our analysis, option (d) is consistent with the properties of gas molecules as described by kinetic theory. The mean translational kinetic energy of gas molecules is indeed directly proportional to the absolute temperature.

Key Concepts

Diatomic molecule

In the world of chemistry, a diatomic molecule is a molecule composed of only two atoms. These atoms can either be of the same chemical element, like oxygen (\( \text{O}_2 \)), or of different elements, like carbon monoxide (\( \text{CO} \)). Understanding diatomic molecules is crucial for grasping concepts in kinetic theory of gases:

• Polar vs. Nonpolar: Diatomic molecules can be polar, like hydrogen chloride (\( \text{HCl} \)), or nonpolar, like nitrogen (\( \text{N}_2 \)).
• Bonding: These molecules are bonded through covalent bonds, where the atoms share electrons.
• Kinetic Theory: In kinetic theory, diatomic gases behave like rigid dumbbells rotating and translating through space. This affects their kinetic energy and specific heat capacities.

Understanding these molecules aids in predicting the properties and behavior of gases, especially how they move and exert pressure.

Mean translational kinetic energy

Mean translational kinetic energy is a key concept in understanding gas behavior through kinetic theory. This energy refers to the average kinetic energy due to the motion of molecules in a gas. For a molecule:

• It's calculated using the formula \( KE = \frac{3}{2} k T \), where \( k \) is the Boltzmann constant and \( T \) is the absolute temperature.
• The formula indicates that the translational kinetic energy is directly proportional to the absolute temperature. This means, as the temperature increases, the kinetic energy increases, causing more vigorous movement of gas molecules.

Because kinetic energy is related to the temperature, it affects how gases respond to changes in their environment, impacting pressure and volume.

Root mean square velocity

The root mean square (RMS) velocity is an important measure in kinetic theory used to derive the speed of particles in a gas.
It represents the square root of the average of the squares of individual molecular velocities. The equation is:\[ v_{\text{rms}} = \sqrt{\frac{3kT}{m}} \]Where:

• \( v_{\text{rms}} \) is the root mean square velocity.
• \( k \) is the Boltzmann constant.
• \( T \) is the absolute temperature.
• \( m \) is the mass of a gas molecule.

This velocity helps determine the gas pressure and temperature relationship, showing that pressure rises as temperature rises, due to increased molecular motion. Overall, it’s a useful way to analyze gaseous behaviors.

Absolute temperature

Absolute temperature is a vital component in physics and chemistry, defining a zero-point at which a system's thermal energy is at its minimum, referred to in Kelvin. The Kelvin scale is used because it starts at absolute zero (\( 0 \, \text{K} \)), the point where particles have minimum thermal motion.
This concept is crucial for kinetic theory as:

• Temperature in Kelvin is directly used in equations like \( KE = \frac{3}{2}kT \), influencing kinetic energy and molecular velocity computations.
• It makes calculations simpler since it avoids negative values, that would arise in Celsius or Fahrenheit when extrapolating backwards.
• Understanding absolute temperature allows scientists to predict real-world gas behaviors under varying temperature conditions.

This deep understanding of temperature helps describe how molecular energies change, impacting pressure and volume in gas laws based on kinetic theory.