Kinetic Theory

Molar specific heat of water = molar mass of water specific heat of water [where, ]

Mean free path, where, n = Number of molecules per unit volume d = Diameter of the molecule As where, Boltzmann constant Pressure of gas Absolute temperature of the gas

Step 1: The root mean square velocity of a gas is given by: where:
- is the gas constant,
- is the temperature,
- is the molar mass of the gas.
Step 2: The ratio of the root mean square velocities of and molecules is: Step 3: The molar masses of and are approximately 32 and 48 g/mol, respectively. So,

The mean free path ( ) of a gas molecule is given by the following relation: where: - is the number of molecules per unit volume,
- is the effective collision cross-section, which depends on the molecular diameter ( ).
The effective cross-section is proportional to the square of the molecular diameter: Since both gases have the same number of molecules per unit volume, the ratio of their mean free paths will be inversely proportional to the ratio of their molecular diameters squared: Given that the molecular diameters are in the ratio 1 : 3, we have: Thus, the ratio of their mean free paths is:

Concept:

Step 1: Equate velocities.


Step 2: Substitute values.


Step 3: Solve.

The mean free path is the average distance a particle can travel before colliding with another particle. The mean free path is inversely proportional to the number density (the number of particles per unit volume) and the cross-sectional area of the particles. The formula for the mean free path is given by: where is the effective collision cross-section of the particles, which is proportional to the square of the particle diameter .
Thus, the mean free path is inversely proportional to , the diameter of the particles, as given in option (C).

Step 1: Understanding the Concept:
The molar heat capacities for an ideal gas depend on the degrees of freedom. For a diatomic molecule at moderate temperatures, and .

Step 2: Detailed Explanation:
Given . For a diatomic gas, .

Step 3: Final Answer:
The value of is .

Step 1: Understanding the Concept:
Mean free path , where = molecular diameter, = number density.

Step 2: Detailed Explanation:
. So mean free path is inversely proportional to the square of the diameter.

Step 3: Final Answer:
Inversely proportional to square of diameter.

Step 1: Understanding the Concept:
For an ideal gas, translational kinetic energy and .

Step 2: Detailed Explanation:


Therefore,

Step 3: Final Answer:
.

Concept:
• Mass of one molecule:

Step 1: Molar mass of water


Step 2: Use Avogadro number


Step 3: Compute
Final Conclusion:
Option (E)

Concept:
• Extensive variables depend on amount of substance
• Intensive variables do not

Step 1: Check options
Mass, energy, volume, work → depend on system size (extensive)
Density → independent of size (intensive) Final Conclusion:
Option (D)

Step 1: Understanding the Concept:
According to the kinetic theory of gases, the translational kinetic energy of an ideal gas is directly proportional to its absolute temperature. A monatomic gas (like Helium or Neon) has only 3 translational degrees of freedom.

Step 2: Key Formula or Approach:
The total translational kinetic energy ( ) for moles of a monatomic ideal gas is given by:

Where:
= number of moles
= universal gas constant
= absolute temperature in Kelvin

Step 3: Detailed Explanation:
Given values:
Kinetic energy, J
Number of moles, mole
Gas constant, J/(mol K)
Rearrange the formula to solve for temperature :

Substitute the known values:



Rounding to a reasonable number of significant figures gives approximately 1.96 K.

Step 4: Final Answer:
The temperature is approximately K.

Step 1: Understanding the Concept:
The Kinetic Theory of Gases relates macroscopic properties like pressure ( ) and volume ( ) to microscopic properties like the root-mean-square (rms) speed ( ) of gas molecules.
Step 2: Key Formula or Approach:
The pressure exerted by an ideal gas is given by:

Since mass and volume are constant, pressure is proportional to the square of rms speed:

Step 3: Detailed Explanation:
Let initial rms speed be , then increased speed:


So, .
Percentage increase:

Step 4: Final Answer:
The increase in pressure is 44%.

Concept:
Physics - Kinetic Theory of Gases.
Step 1: State the formula for rms speed.
The root mean square speed ( ) of gas molecules is given by the formula: $ R T M R 3 M $
Step 3: Conclusion.
The rms speed of the molecules is directly proportional to the square root of the absolute temperature of the gas.