If different ideal gases are at the same temperature, pressure and have same volume, then all gases have same

Options

A. Density
B. Number of molecules
C. Most probable speed
D. Internal energy per mole

Answer

To solve the question, we need to analyze the properties of ideal gases under the conditions given: same temperature, pressure, and volume.

### Step-by-Step Solution:

1. **Understanding the Conditions**:
- We have different ideal gases at the same temperature (T), pressure (P), and volume (V).
- According to the ideal gas law, \( PV = nRT \), where \( n \) is the number of moles and \( R \) is the ideal gas constant.

2. **Using Avogadro's Law**:
- Avogadro's law states that equal volumes of gases at the same temperature and pressure contain an equal number of molecules.
- Since the volume is the same for all gases and the temperature and pressure are constant, the number of moles (n) of each gas must also be the same.

3. **Calculating Density**:
- Density (\( \rho \)) is defined as mass per unit volume (\( \rho = \frac{m}{V} \)).
- Since the volume is the same and the number of moles is the same, the mass of the gases will be the same if they are the same type of gas. Thus, the density will also be the same.

4. **Number of Molecules**:
- The number of molecules (N) can be calculated using Avogadro's number (\( N = n \times N_A \), where \( N_A \) is Avogadro's number).
- Since the number of moles is the same for all gases, the number of molecules will also be the same.

5. **Most Probable Speed**:
- The most probable speed (\( v_{mp} \)) of gas molecules is given by the formula:
\[ v_{mp} = \sqrt{\frac{2kT}{m}} \]
where \( k \) is Boltzmann's constant and \( m \) is the mass of a gas molecule.
- Since different gases have different molecular masses, the most probable speed will not be the same for all gases.

6. **Internal Energy per Mole**:
- The internal energy (U) of an ideal gas is given by:
\[ U = \frac{3}{2} nRT \]
- Since \( n \) (number of moles) is the same for all gases and \( R \) and \( T \) are constants, the internal energy per mole will also be the same.

### Conclusion:
From the analysis, we can conclude that if different ideal gases are at the same temperature, pressure, and volume, then:
- All gases have the same **number of molecules** and **internal energy per mole**.
- They do not have the same **most probable speed**.

### Final Answer:
All gases have the same number of molecules and internal energy per mole.