The ratio of most probable velocity, average velocity and root mran square velocity is

Options

A. `sqrt2 : sqrt(8/pi) : sqrt3 `
B. `1 : sqrt2 : sqrt3`
C. `sqrt2 : sqrt3 : sqrt8`
D. `sqrt1 : sqrt(8pi) : sqrt3`

Answer

To find the ratio of the most probable velocity (MPS), average velocity (AV), and root mean square velocity (RMS), we can follow these steps:

### Step 1: Write the formulas for each velocity type
1. **Most Probable Speed (MPS)**:
\[ v_{mp} = \sqrt{\frac{2RT}{M}} \]
2. **Average Speed (AV)**:
\[ v_{avg} = \sqrt{\frac{8RT}{\pi M}} \]
3. **Root Mean Square Speed (RMS)**:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]

### Step 2: Set up the ratio
We want to find the ratio of MPS, AV, and RMS:
\[ \text{Ratio} = v_{mp} : v_{avg} : v_{rms} \]

### Step 3: Substitute the formulas into the ratio
Substituting the expressions for each velocity into the ratio gives:
\[ \text{Ratio} = \sqrt{\frac{2RT}{M}} : \sqrt{\frac{8RT}{\pi M}} : \sqrt{\frac{3RT}{M}} \]

### Step 4: Simplify the ratio
To simplify, we can factor out the common terms:
\[ \text{Ratio} = \sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3} \]

### Step 5: Calculate the numerical values
1. **Calculate \(\sqrt{2}\)**:
\[ \sqrt{2} \approx 1.414 \]
2. **Calculate \(\sqrt{\frac{8}{\pi}}\)**:
\[ \sqrt{\frac{8}{\pi}} \approx \sqrt{\frac{8}{3.14}} \approx \sqrt{2.546} \approx 1.596 \]
3. **Calculate \(\sqrt{3}\)**:
\[ \sqrt{3} \approx 1.732 \]

### Step 6: Write the final ratio
Now we can write the final ratio:
\[ \text{Ratio} \approx 1.414 : 1.596 : 1.732 \]

### Step 7: Express the ratio in simplest form
To express this in a simpler form, we can divide each term by the smallest value (1.414):
\[ \text{Ratio} \approx 1 : 1.127 : 1.225 \]

Thus, the final ratio of most probable velocity, average velocity, and root mean square velocity is approximately:
\[ 1 : 1.127 : 1.225 \]