If gas has a RMS velocity `5.2 xx 10^4 cms^(-1)` , find the average and most probable velocities of the gas 

Answer

To find the average and most probable velocities of a gas given its RMS (Root Mean Square) velocity, we can follow these steps:

### Step 1: Understand the Relationships
The relationships between the RMS velocity (v_rms), average velocity (v_avg), and most probable velocity (v_mp) of a gas are given by the following equations:
- \( v_{rms} = \sqrt{\frac{3RT}{M}} \)
- \( v_{avg} = \sqrt{\frac{8RT}{\pi M}} \)
- \( v_{mp} = \sqrt{\frac{2RT}{M}} \)

Where:
- R = Universal gas constant
- T = Temperature in Kelvin
- M = Molar mass of the gas

### Step 2: Calculate the Average Velocity
Given the RMS velocity \( v_{rms} = 5.2 \times 10^4 \, \text{cm/s} \), we can derive the average velocity using the ratio of the two velocities:

\[ \frac{v_{avg}}{v_{rms}} = \sqrt{\frac{8}{3\pi}} \]

Rearranging gives us:

\[ v_{avg} = v_{rms} \cdot \sqrt{\frac{8}{3\pi}} \]

### Step 3: Substitute the RMS Velocity
Now, substituting the value of \( v_{rms} \):

\[ v_{avg} = 5.2 \times 10^4 \cdot \sqrt{\frac{8}{3\pi}} \]

### Step 4: Calculate the Numerical Value
Calculating the numerical value:

1. Calculate \( \frac{8}{3\pi} \):
- \( \pi \approx 3.14 \)
- \( \frac{8}{3\pi} \approx \frac{8}{9.42} \approx 0.848 \)

2. Take the square root:
- \( \sqrt{0.848} \approx 0.921 \)

3. Now multiply by \( 5.2 \times 10^4 \):
- \( v_{avg} \approx 5.2 \times 10^4 \cdot 0.921 \approx 4.79 \times 10^4 \, \text{cm/s} \)

### Step 5: Calculate the Most Probable Velocity
Using the RMS velocity to find the most probable velocity:

\[ \frac{v_{mp}}{v_{rms}} = \sqrt{\frac{2}{3}} \]

Rearranging gives us:

\[ v_{mp} = v_{rms} \cdot \sqrt{\frac{2}{3}} \]

### Step 6: Substitute the RMS Velocity
Substituting the value of \( v_{rms} \):

\[ v_{mp} = 5.2 \times 10^4 \cdot \sqrt{\frac{2}{3}} \]

### Step 7: Calculate the Numerical Value
Calculating the numerical value:

1. Calculate \( \frac{2}{3} \):
- \( \frac{2}{3} \approx 0.667 \)

2. Take the square root:
- \( \sqrt{0.667} \approx 0.816 \)

3. Now multiply by \( 5.2 \times 10^4 \):
- \( v_{mp} \approx 5.2 \times 10^4 \cdot 0.816 \approx 4.25 \times 10^4 \, \text{cm/s} \)

### Final Answers
- Average Velocity \( v_{avg} \approx 4.79 \times 10^4 \, \text{cm/s} \)
- Most Probable Velocity \( v_{mp} \approx 4.25 \times 10^4 \, \text{cm/s} \)