keeping the number of moles, volume and temperature the same, which of the following are the same for all ideal gases?

Options

A. (a)Rms speed of a molecule
B. (b)Density
C. (c)pressure
D. (d)Average magnitude of momentum.

Answer

To solve the question, we need to analyze each option given the conditions that the number of moles, volume, and temperature are the same for all ideal gases.

### Step-by-Step Solution:

1. **Understanding the 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 universal gas constant, and \( T \) is the temperature.

2. **Analyzing Pressure**:
Given that the number of moles (\( n \)), volume (\( V \)), and temperature (\( T \)) are constant, we can rearrange the ideal gas law to find pressure:
\[ P = \frac{nRT}{V} \]
Since \( n \), \( R \), \( T \), and \( V \) are constants, it follows that pressure \( P \) must also be constant for all ideal gases under these conditions.
- **Conclusion**: Pressure is the same for all ideal gases.

3. **Analyzing RMS Speed**:
The root mean square (RMS) speed of gas molecules is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
Where \( M \) is the molar mass of the gas. Since different gases have different molar masses, the RMS speed will vary for different gases even if \( R \) and \( T \) are constant.
- **Conclusion**: RMS speed is not the same for all ideal gases.

4. **Analyzing Density**:
Density (\( \rho \)) is defined as mass per unit volume:
\[ \rho = \frac{m}{V} \]
The mass \( m \) can be expressed as:
\[ m = n \cdot M \]
Therefore, density can be rewritten as:
\[ \rho = \frac{nM}{V} \]
Since \( n \) and \( V \) are constant but \( M \) varies for different gases, the density will also vary.
- **Conclusion**: Density is not the same for all ideal gases.

5. **Analyzing Average Magnitude of Momentum**:
The momentum \( p \) of a molecule is given by:
\[ p = mv \]
The average momentum can be calculated as:
\[ \text{Average momentum} = \frac{\sum p}{N} \]
Since the mass \( m \) and speed \( v \) of the molecules will differ for different gases, the average momentum will also differ.
- **Conclusion**: Average magnitude of momentum is not the same for all ideal gases.

### Final Answer:
The only property that remains the same for all ideal gases under the given conditions is **Pressure**.