The Kinetic Theory Of Gases

Step 1: Count degrees of freedom.
A polyatomic molecule has: • 3 translational DOF, • 3 rotational DOF (for nonlinear molecules), • Vibrational mode: each contributes 2 DOF (kinetic + potential). Given: only one vibrational mode is excited ⇒ 2 extra DOF.

Step 2: Total degrees of freedom.
DOF.

Step 3: Use equipartition theorem.
Each DOF contributes to . Thus, .

Step 4: But vibrational energy counts as full per mode.
A vibrational mode contributes (not ), so: .

Step 5: Conclusion.
Therefore, .

Step 1: Use the formula for mean free path.
Mean free path for a gas: , where is number density.
Since both boxes contain equal number of molecules, .

Step 2: Relate the mean free paths.
.
Thus .

Step 3: Compare with molecular spacing.
Mean molecular separation . Hence the ratio is constant for equal number of molecules.

Step 4: Conclusion.
Therefore, .

Step 1: Use the relationship between mean free path and molecular diameter.
The mean free path is inversely proportional to the square of the molecular diameter . Mathematically, this is expressed as:
Step 2: Relate the mean free paths of the two gases.
Given that , we can write the ratio of the mean free paths:
Step 3: Conclusion.
Thus, .

Step 1: Recall Maxwell-Boltzmann distribution form.
The Maxwell-Boltzmann (MB) distribution is given by:

Step 2: Approximation condition.
For both Fermi-Dirac and Bose-Einstein, if , then which corresponds to the Maxwell-Boltzmann limit.

Step 3: Conclusion.
Therefore, the MB distribution is valid when

Step 1: Boltzmann distribution ratio.
The relative population of two energy levels is given by

Step 2: Take reciprocal.

Step 3: Conclusion.
Hence, the population ratio between the ground and excited state is .

Step 1: RMS speed formula.
The root mean square (rms) speed of gas molecules is given by where is temperature and is molecular mass.

Step 2: Ratio of rms speeds.
Given , and :

Step 3: Conclusion.
Hence, .

Step 1: Understanding the Concept:
The mean free path ( ) is the average distance a gas molecule travels between successive collisions. It depends on the size of the molecules (diameter ) and their number density ( ). The number density, in turn, is related to pressure and temperature by the ideal gas law. To achieve a very long mean free path, the pressure must be very low (a high vacuum).
Step 2: Key Formula or Approach:
The mean free path is given by: where is the molecular diameter and is the number density. The number density is related to pressure and temperature by the ideal gas law in the form , which gives . Substituting into the mean free path formula allows us to solve for the pressure .
Step 3: Detailed Explanation:
Substitute into the formula: Rearrange to solve for the pressure : We are given the following values: Temperature, K. Molecular diameter, m. Mean free path, m. Boltzmann constant, J/K. Substitute these values into the expression for : The question asks for the answer in the form of Pa. Step 4: Final Answer:
The pressure needs to be evacuated to Pa.

Step 1: Understanding the Concept:
The problem relates the temperature and molecular weight of two different ideal gases using their average molecular speed. The average speed of gas molecules is determined by the gas's temperature and the mass of its constituent molecules, as described by the Maxwell-Boltzmann distribution.
Step 2: Key Formula or Approach:
The average speed of molecules in an ideal gas at temperature T is given by the formula derived from the Maxwell-Boltzmann distribution: where is the Boltzmann constant, is the mass of a single molecule, is the universal gas constant, and is the molar mass (or molecular weight) of the gas.
Step 3: Detailed Explanation:
Let the properties of gas G1 be denoted by subscript 1 and gas G2 by subscript 2. We are given: Molecular weight relation: Average speeds are equal: Using the formula for average speed for each gas: For G1: For G2: Since , we can set their expressions equal to each other: Squaring both sides of the equation: The constant factor cancels out: We need to find the ratio . Let's rearrange the equation: Now, we use the given relationship between the molecular weights, . This implies that . Substituting this into our ratio: Step 4: Final Answer:
The ratio is 2.