The molecular velocity of any gas is

Options

A. inversely proportional to the square root of temperature
B. inversely proportional to absolute temperature
C. directly proportional to square of temperature
D. directly proportional to square root of temperature.

Explanation

To solve the question regarding the molecular velocity of any gas, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding Molecular Velocity**:
- Molecular velocity refers to the speed of gas particles as they move in random motion. According to the kinetic molecular theory, gas particles are in constant random motion and collide with each other and the walls of their container.

2. **Definition of Root Mean Square Speed (Vrms)**:
- The molecular velocity can be described using the root mean square speed (Vrms), which is a measure of the average velocity of gas particles. It accounts for the different speeds and directions of the particles.

3. **Kinetic Energy Relation**:
- The average kinetic energy of gas particles is given by the equation:
\[ KE = \frac{3}{2} k_B T \]
where \( k_B \) is the Boltzmann constant and \( T \) is the absolute temperature.

4. **Relating Kinetic Energy to Velocity**:
- Kinetic energy can also be expressed in terms of mass and velocity:
\[ KE = \frac{1}{2} mv^2 \]
- Setting these two expressions for kinetic energy equal gives:
\[ \frac{3}{2} k_B T = \frac{1}{2} mv^2 \]

5. **Deriving the Expression for Vrms**:
- Rearranging the equation gives:
\[ v^2 = \frac{3 k_B T}{m} \]
- Taking the square root, we find the root mean square speed:
\[ V_{rms} = \sqrt{\frac{3 k_B T}{m}} \]

6. **Using Ideal Gas Constant**:
- We can relate the Boltzmann constant to the ideal gas constant \( R \) and Avogadro's number \( N_A \):
\[ k_B = \frac{R}{N_A} \]
- Substituting this into the equation for \( V_{rms} \):
\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]
- Here, \( M \) is the molar mass of the gas.

7. **Analyzing the Options**:
- The derived equation shows that \( V_{rms} \) is directly proportional to the square root of the temperature \( T \). Thus, we can conclude that:
- It is **not** inversely proportional to temperature.
- It is **not** proportional to the square of temperature.
- It is **correct** that it is directly proportional to the square root of temperature.

8. **Conclusion**:
- The correct answer is that the molecular velocity of any gas is directly proportional to the square root of the temperature.

### Final Answer:
The molecular velocity of any gas is directly proportional to the square root of the temperature.

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