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 translational kinetic energy and the equation `(1)/(2)mv_(max)^(2) = (3)/(2)kT` establishes the relatiship between the average translational kinetic energy per particle and temperature of an ideal gas. It can be concluded that single particle has a temperature
C. Temperature of an ideal gas is doubled from `100^(@)C to 200^(@)C`. The average kinetic energy of each particle is also doubled.
D. It is possible for both the pressure and volume of a monoatomic ideal gas to change simultaneously without causing the internal energy of the gas to change.

Explanation

To solve the question of picking the correct statements regarding the kinetic theory of gases, we will analyze each statement step by step.

### Step 1: Analyze the first statement
**Statement**: The RMS translational speed for all ideal gas molecules at the same temperature is not the same but it depends on the mass.

**Analysis**: The formula for root mean square velocity (VRMS) 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 \( V_{RMS} \) depends on \( M \), different gases (with different molar masses) at the same temperature will have different RMS speeds. Thus, this statement is **correct**.

### Step 2: Analyze the second statement
**Statement**: Each particle in a gas has average translational kinetic energy and the equation for average kinetic translational kinetic energy is given by \( \frac{1}{2}mv^2_{max} = \frac{3}{2}kT \).

**Analysis**: The average translational kinetic energy of a gas particle is given by:
\[ \frac{1}{2} m V_{RMS}^2 = \frac{3}{2} k T \]
However, the statement incorrectly uses \( V_{max} \) instead of \( V_{RMS} \). Since \( V_{max} \) does not represent the average speed, this statement is **incorrect**.

### Step 3: Analyze the third statement
**Statement**: The temperature of an ideal gas is doubled from 100 degrees Celsius to 200 degrees Celsius, so the average kinetic energy of each particle is also doubled.

**Analysis**: The average kinetic energy is directly proportional to the absolute temperature (in Kelvin). Converting the temperatures:
- 100°C = 373 K
- 200°C = 473 K

Since \( 473 K \) is not double \( 373 K \), the average kinetic energy does not double. Therefore, this statement is **incorrect**.

### Step 4: Analyze the fourth statement
**Statement**: It is possible for both the pressure and volume of a monatomic ideal gas to change simultaneously without causing the internal energy of the gas to change.

**Analysis**: The internal energy of an ideal gas is a function of temperature. If the internal energy does not change, the temperature must remain constant. According to the ideal gas law \( PV = nRT \), if \( T \) is constant, then \( PV \) must also remain constant, which means pressure and volume can change inversely. Thus, this statement is **correct**.

### Conclusion
The correct statements are:
1. The first statement is correct.
2. The second statement is incorrect.
3. The third statement is incorrect.
4. The fourth statement is correct.

### Final Answer
The correct statements are: **1 and 4**.

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