The pressure exerted by an ideal gas at a particular temperature is directly proportional to

Options

A. the mean speed of the gas molecules
B. the mean of the square of the speed of the gas molecules
C. the square of the mean speed of the gas molecules
D. the root mean square speed of the gas molecules

Explanation

To solve the question regarding the pressure exerted by an ideal gas at a particular temperature and its direct proportionality, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Kinetic Theory of Gases**:
The kinetic theory of gases explains the behavior of ideal gases in terms of molecular motion. According to this theory, gas pressure is a result of collisions between gas molecules and the walls of the container.

2. **Using the Ideal Gas Equation**:
The ideal gas equation is given by:
\[ PV = nRT \]
where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature.

3. **Relating Pressure to Molecular Speed**:
From the kinetic theory, we can derive that:
\[ P = \frac{1}{3} \frac{mN}{V} \langle c^2 \rangle \]
where:
- \( P \) is the pressure,
- \( m \) is the mass of a single gas molecule,
- \( N \) is the total number of molecules,
- \( V \) is the volume,
- \( \langle c^2 \rangle \) is the mean of the square of the molecular speeds.

4. **Identifying Proportionality**:
From the equation \( P = \frac{1}{3} \frac{mN}{V} \langle c^2 \rangle \), we can see that at a constant temperature, the pressure \( P \) is directly proportional to \( \langle c^2 \rangle \) (the mean of the square of the speeds of the gas molecules).

5. **Conclusion**:
Therefore, the pressure exerted by an ideal gas at a particular temperature is directly proportional to the mean of the square of the speed of the gas molecules.

### Final Answer:
The pressure exerted by an ideal gas at a particular temperature is directly proportional to the mean of the square of the speed of the gas molecules.

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