RMS velocity of a gas does not vary with________at a given temperature.

Options

A. pressure
B. volume
C. density
D. shape of the container

Answer

To solve the question regarding the RMS (Root Mean Square) velocity of a gas and what it does not vary with at a given temperature, we can follow these steps:

### Step 1: Understand the RMS Velocity Formula
The RMS velocity (c_rms) of a gas is given by the formula:
\[ c_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( M \) is the molar mass of the gas.

### Step 2: Analyze the Variables
From the formula, we can see that the RMS velocity depends on:
- Temperature (T)
- Molar mass (M)

### Step 3: Consider the Ideal Gas Law
According to the ideal gas law:
\[ PV = nRT \]
where:
- \( P \) is the pressure,
- \( V \) is the volume,
- \( n \) is the number of moles.

We can express \( R \) and \( T \) in terms of \( P \) and \( V \).

### Step 4: Substitute into the RMS Velocity Equation
We can express the RMS velocity in terms of pressure and density:
- Density (\( d \)) can be expressed as \( d = \frac{m}{V} \), where \( m \) is the total mass of the gas.
- Thus, we can relate the RMS velocity to pressure and density:
\[ c_{rms} = \sqrt{\frac{3P}{d}} \]

### Step 5: Identify What RMS Velocity Does Not Depend On
From the derived equations, we see that the RMS velocity depends on:
- Pressure (P)
- Density (d)
- Temperature (T)
- Molar mass (M)

However, it does not depend on the shape of the container. The shape of the container does not influence the kinetic energy of the gas particles or their average speed.

### Conclusion
Therefore, the RMS velocity of a gas does not vary with the shape of the container at a given temperature.

### Final Answer
The RMS velocity of a gas does not vary with **the shape of the container** at a given temperature.

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