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

Options

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

Answer

To solve the question regarding the statements about an ideal gas at a given temperature T, we will analyze each statement one by one.

### Step-by-Step Solution:

1. **Statement A**: "The coefficient of the volume expansion at a constant pressure is the same for all ideal gases."
- **Analysis**: The coefficient of volume expansion (β) at constant pressure is defined as:
\[ \beta = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_P \]
For ideal gases, this coefficient is indeed the same for all ideal gases and is given by \( \frac{R}{P} \) where R is the gas constant and P is the pressure.
- **Conclusion**: This statement is **correct**.

2. **Statement B**: "The average translational kinetic energy per molecule of oxygen K gas is 3 kT."
- **Analysis**: The average translational kinetic energy (E) per molecule of an ideal gas is given by:
\[ E = \frac{3}{2} kT \]
Therefore, the statement that it is \( 3 kT \) is incorrect.
- **Conclusion**: This statement is **incorrect**.

3. **Statement C**: "The mean free path of molecules increases with increasing pressure."
- **Analysis**: The mean free path (λ) is given by:
\[ \lambda = \frac{kT}{\sqrt{2} \pi d^2 P} \]
where d is the diameter of the gas molecules. As pressure (P) increases, the mean free path decreases because they are inversely related.
- **Conclusion**: This statement is **incorrect**.

4. **Statement D**: "In a gaseous mixture, the average translational kinetic energy of the molecules of each component is different."
- **Analysis**: In a gaseous mixture, all components at the same temperature have the same average translational kinetic energy given by \( \frac{3}{2} kT \). Thus, the average kinetic energy is the same for all components.
- **Conclusion**: This statement is **incorrect**.

### Summary of Results:
- Statement A: Correct
- Statement B: Incorrect
- Statement C: Incorrect
- Statement D: Incorrect

### Final Answer:
The correct statement concerning ideal gas at any given temperature T is **Statement A**.