From the following statements concerning ideal gas at any given temperature T, select the correct one (s)

Options

A. A) The coefficient of volume expansion at constant pressure is the same for all ideal gases
B. B) The average translational kinetic energy per molecule of oxygen gas is 3 kT, k being Boltzmann constant
C. C) The mean-free path of molecules increases with decrease in the pressure
D. D) In a gaseous mixture, the average tanslational kinetic energy of the molecules of each component is different

Answer

To solve the question regarding the statements concerning an ideal gas at a given temperature \( T \), we will evaluate each statement based on the principles of ideal gas behavior.

### Step-by-Step Solution:

1. **Understanding Ideal Gas Law**:
The ideal gas law is given by the equation:
\[ PV = nRT \]
where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

**Hint**: Recall the ideal gas law and its components.

2. **Analyzing the Statement about Average Translational Kinetic Energy**:
The average translational kinetic energy per molecule of an ideal gas is given by:
\[ KE = \frac{3}{2} kT \]
where \( k \) is the Boltzmann constant. This indicates that the average kinetic energy is proportional to the temperature and does not depend on the type of gas.

**Hint**: Remember that kinetic energy is related to temperature and is independent of the gas type.

3. **Considering the Effect of Pressure on Volume**:
According to Boyle's Law, at constant temperature, if the pressure decreases, the volume of the gas increases. This is a direct consequence of the ideal gas law.

**Hint**: Think about how pressure and volume are inversely related in an ideal gas.

4. **Coefficient of Volume Expansion**:
The coefficient of volume expansion at constant pressure is the same for all ideal gases. This means that when the temperature of an ideal gas increases, its volume expands uniformly regardless of the type of gas.

**Hint**: Consider how gases behave under temperature changes and their uniform expansion properties.

5. **Identifying Correct Statements**:
Based on the analysis:
- **Statement A**: The average translational kinetic energy is the same for all ideal gases at the same temperature. (Correct)
- **Statement C**: The volume of the gas increases with a decrease in pressure. (Correct)

Therefore, both statements A and C are correct.

**Hint**: Review each statement carefully and relate them to the principles of gas behavior.

### Final Answer:
The correct statements concerning the ideal gas at any given temperature \( T \) are:
- Statement A: The average translational kinetic energy of the molecules is independent of their nature.
- Statement C: With the decrease in pressure, the volume of the gas increases.