The ratio of root mean square speed, average speed and most probable speed is

Options

A. `3pi : 8 :2pi`
B. `pi : 2pi : 3`
C. `3pi :2pi: 8`
D. `3pi:2pi:6`

Explanation

To find the ratio of root mean square speed (RMS), average speed, and most probable speed of gas particles, we can follow these steps:

### Step 1: Understand the Definitions
- **Root Mean Square Speed (RMS)**: It is defined as the square root of the average of the squares of the speeds of the gas particles.
- **Average Speed**: It is the mean speed of all the gas particles.
- **Most Probable Speed**: It is the speed at which the maximum number of gas particles are moving.

### Step 2: Write Down the Formulas
The formulas for these speeds in terms of temperature (T) and molar mass (m) are:
- RMS Speed (v_rms) = √(3RT/m)
- Average Speed (v_avg) = √(8RT/πm)
- Most Probable Speed (v_mp) = √(2RT/m)

Where:
- R = Universal gas constant
- T = Absolute temperature
- m = Molar mass of the gas

### Step 3: Set Up the Ratio
We need to find the ratio of RMS speed to Average speed to Most Probable speed:
\[ v_{rms} : v_{avg} : v_{mp} \]

### Step 4: Substitute the Formulas into the Ratio
Substituting the formulas into the ratio gives:
\[ \sqrt{\frac{3RT}{m}} : \sqrt{\frac{8RT}{\pi m}} : \sqrt{\frac{2RT}{m}} \]

### Step 5: Simplify the Ratio
1. Factor out common terms (√(RT/m)):
\[ \sqrt{3} : \sqrt{\frac{8}{\pi}} : \sqrt{2} \]

2. Now we can express the ratio in terms of numerical values:
- RMS Speed: \( \sqrt{3} \)
- Average Speed: \( \sqrt{\frac{8}{\pi}} \)
- Most Probable Speed: \( \sqrt{2} \)

### Step 6: Calculate the Numerical Values
Using approximate values:
- \( \sqrt{3} \approx 1.732 \)
- \( \sqrt{2} \approx 1.414 \)
- \( \sqrt{\frac{8}{\pi}} \approx \sqrt{\frac{8}{3.14}} \approx \sqrt{2.55} \approx 1.597 \)

### Step 7: Final Ratio
Now, we can express the ratio in a simplified form:
\[ 1.732 : 1.597 : 1.414 \]
To express it in whole numbers, we can multiply each term by a common factor (for simplicity, let's multiply by 1000):
\[ 1732 : 1597 : 1414 \]

However, to express in a simpler ratio, we can use the original square roots:
\[ \sqrt{3} : \sqrt{\frac{8}{\pi}} : \sqrt{2} \]

### Conclusion
The final ratio of root mean square speed, average speed, and most probable speed is:
\[ \sqrt{3} : \sqrt{\frac{8}{\pi}} : \sqrt{2} \]