The Kinetic Theory Of Gases

Step 1: Recall the equipartition of energy theorem.
The equipartition of energy theorem states that for a system in thermal equilibrium, the total energy is shared equally among all its degrees of freedom. Specifically, the average kinetic energy associated with each translational degree of freedom is , where is the Boltzmann constant and is the absolute temperature.

Step 2: Determine the average translational kinetic energy of a molecule.
A gas molecule in three-dimensional space has three translational degrees of freedom (corresponding to motion along the x, y, and z axes). According to the equipartition theorem, the total average translational kinetic energy of a single molecule is:

Step 3: Analyze the dependence of translational kinetic energy.
The formula shows that the average translational kinetic energy depends only on the absolute temperature . It does not depend on the mass of the molecule, the type of gas (monatomic, diatomic, etc.), or any other property of the gas.

Step 4: Calculate the ratio for hydrogen and oxygen.
We are given that the hydrogen ( ) and oxygen ( ) gases are at the same temperature. Since the average translational kinetic energy depends only on temperature, the molecules of both gases will have the same average translational kinetic energy. The ratio is therefore: The ratio is 1:1.

The root mean square (rms) speed of gas molecules is given by the formula:
, where T is the absolute temperature in Kelvin.
From this formula, we can see that the rms speed is proportional to the square root of the absolute temperature: .
We are given two states of the gas. Let's denote them by subscripts 1 and 2.
State 1:
Temperature . We must convert this to Kelvin: K.
RMS speed m/s.
State 2:
Temperature . Convert to Kelvin: K.
RMS speed is , which we need to find.
Using the proportionality, we can set up a ratio:
.
.
.
Solving for :
m/s.

Step 1: Understanding the Concept:
According to the ideal gas law ( ) and Dalton's law of partial pressures, the total pressure of a gas mixture in a fixed volume at a constant temperature is directly proportional to the total number of moles of gas present. If some moles of gas are removed, the pressure will decrease proportionally.
Step 2: Key Formula or Approach:
Since Volume (V) and Temperature (T) are constant, , where is the total number of moles.
This means .
We need to calculate the number of moles ( ) for each gas initially and finally.
Step 3: Detailed Explanation:
First, we need the molar masses of the gases:
- Molar mass of Oxygen (O₂): g/mol.
- Molar mass of Nitrogen (N₂): g/mol.
- Molar mass of Carbon Dioxide (CO₂): g/mol.
Calculate the initial number of moles ( ):
- Moles of O₂: mol.
- Moles of N₂: mol.
- Moles of CO₂: mol.
- Initial total moles: mol.
Calculate the final number of moles ( ):
The entire oxygen is removed. The remaining gases are Nitrogen and Carbon Dioxide.
- Final total moles: mol.
Calculate the final pressure ( ):
Using the proportionality :
Step 4: Final Answer:
The pressure of the remaining mixture is . Therefore, option (D) is correct.