The relation between rms velocity, `v_(rms)` and the most probable velocity, `v_(mp)`, of a gas is

Options

A. `v_(rms)=v_(mp)`
B. `v_(rms)=sqrt((3)/(2))mp`
C. `v_(rms)=(2)/(3)v_(mp)`
D. `v_(rms)=(2)/(3)v_(mp)`

Answer

To find the relation between the root mean square (RMS) velocity (\(v_{rms}\)) and the most probable velocity (\(v_{mp}\)) of a gas, we can use the definitions of these velocities in the context of the kinetic theory of gases.

### Step-by-Step Solution:

1. **Understanding RMS Velocity**:
The RMS velocity is defined as:
\[ v_{rms} = \sqrt{\frac{3kT}{m}} \]
where \(k\) is the Boltzmann constant, \(T\) is the absolute temperature, and \(m\) is the mass of a gas molecule.

2. **Understanding Most Probable Velocity**:
The most probable velocity is defined as:
\[ v_{mp} = \sqrt{\frac{2kT}{m}} \]

3. **Finding the Relation**:
To find the relation between \(v_{rms}\) and \(v_{mp}\), we can express both in terms of the same variables:
- From the equations above, we can rewrite \(v_{rms}\) and \(v_{mp}\):
\[ v_{rms} = \sqrt{3} \cdot \sqrt{\frac{kT}{m}} \quad \text{and} \quad v_{mp} = \sqrt{2} \cdot \sqrt{\frac{kT}{m}} \]

4. **Expressing \(v_{rms}\) in terms of \(v_{mp}\)**:
Now, we can express \(v_{rms}\) in terms of \(v_{mp}\):
\[ v_{rms} = \sqrt{3} \cdot \frac{v_{mp}}{\sqrt{2}} = \sqrt{\frac{3}{2}} \cdot v_{mp} \]

5. **Final Relation**:
Thus, the relation between RMS velocity and most probable velocity is:
\[ v_{rms} = \sqrt{\frac{3}{2}} \cdot v_{mp} \]