Pick the correct statement (s) :

Options

A. The rms translational speed for all ideal gas molecules at the same temperature is not the same but it depends on the mass
B. Each particle in a gas has average transloational kinetic energy and the equation `1//2 mv_(max)^(2) = 3//2 kT` establisehes the relationship between the average translational kinetic energy per particle and temperature of an ideal gas.
C. If the temperature of an ideal gas is doubled form `100^(@)C` to `200^(@)C`, the average kinetic energy of each particle is also doubled.
D. It is possible for both pressure and volume of a monatomic ideal gas to change simultaneously without casuing the internal enregy of the gas to change

Explanation

To solve the question, we need to analyze each statement given and determine if they are correct or incorrect based on the principles of the kinetic theory of gases and thermodynamics.

### Step-by-Step Solution:

1. **Evaluate Statement 1**:
- The statement claims that the RMS (Root Mean Square) translational speed for an ideal gas at the same temperature is not the same and depends on mass.
- The formula for RMS speed is given by \( v_{RMS} = \sqrt{\frac{3RT}{M}} \), where \( R \) is the universal gas constant, \( T \) is the temperature, and \( M \) is the molar mass.
- Since \( M \) varies for different gases, the RMS speed will indeed vary even at the same temperature.
- **Conclusion**: Statement 1 is **correct**.

2. **Evaluate Statement 2**:
- The statement asserts that the average translational energy per particle is related to the equation \( \frac{1}{2} mv_{max}^2 = \frac{3}{2} kT \).
- The correct relationship for average translational kinetic energy is given by \( \frac{1}{2} mv_{RMS}^2 = \frac{3}{2} kT \), not \( v_{max} \).
- **Conclusion**: Statement 2 is **incorrect**.

3. **Evaluate Statement 3**:
- The statement claims that if the temperature of an ideal gas is doubled from 100°C to 200°C, the average kinetic energy of each particle is also doubled.
- First, convert the temperatures to Kelvin:
- \( T_1 = 100 + 273 = 373 \, K \)
- \( T_2 = 200 + 273 = 473 \, K \)
- The average kinetic energy is given by \( KE = \frac{3}{2} kT \). If we double the temperature in Kelvin, the kinetic energy will also double.
- However, since the temperatures are given in Celsius, we need to convert them to Kelvin to analyze the change correctly. The temperatures are not simply doubled in Kelvin.
- **Conclusion**: Statement 3 is **incorrect**.

4. **Evaluate Statement 4**:
- The statement claims that it is possible for both pressure and volume of a monoatomic ideal gas to change simultaneously without causing the internal energy of the gas to change.
- The internal energy \( U \) of an ideal gas depends on temperature, given by \( \Delta U = nC_v\Delta T \). If the temperature remains constant, then the change in internal energy is zero.
- According to the ideal gas law \( PV = nRT \), if the temperature is constant (isothermal process), any change in volume must result in a corresponding change in pressure to maintain constant temperature.
- **Conclusion**: Statement 4 is **correct**.

### Final Answer:
The correct statements are:
- Statement 1: Correct
- Statement 4: Correct