The ratio of most probable speed, average speed and rms speed of gas molecules is………………………………

Options

A. `1 : 1.128 : 1.224`
B. `1 : 1.128 : 1.424`
C. `1 : 2.128 : 1.224`
D. `1 : 1.428 : 1.442`

Explanation

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

### Step 1: Define the Speeds
1. **Most Probable Speed (v_mp)**: This is the speed at which the maximum number of gas molecules are moving. It is given by the formula:
\[ v_{mp} = \sqrt{\frac{2RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the absolute temperature, and \( M \) is the molar mass of the gas.

2. **Average Speed (v_avg)**: This is the arithmetic mean of the speeds of gas molecules. It is given by the formula:
\[ v_{avg} = \sqrt{\frac{8RT}{\pi M}} \]

3. **RMS Speed (v_rms)**: This is the square root of the mean of the squares of the speeds of gas molecules. It is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]

### Step 2: Set Up the Ratio
Now, we will set up the ratio of these three speeds:
\[ v_{mp} : v_{avg} : v_{rms} = \sqrt{\frac{2RT}{M}} : \sqrt{\frac{8RT}{\pi M}} : \sqrt{\frac{3RT}{M}} \]

### Step 3: Simplify the Ratio
In the ratio, \( RT/M \) is common in all three terms, so we can cancel it out:
\[ = \sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3} \]

### Step 4: Calculate the Numerical Values
Now we will calculate the numerical values of each term:
- \( \sqrt{2} \approx 1.414 \)
- \( \sqrt{\frac{8}{\pi}} = \sqrt{8} / \sqrt{\pi} \approx 2.828 / 1.772 \approx 1.596 \)
- \( \sqrt{3} \approx 1.732 \)

### Step 5: Normalize the Ratio
To express the ratio in a simpler form, we can take \( \sqrt{2} \) as the common factor:
\[ 1 : \frac{1.596}{1.414} : \frac{1.732}{1.414} \]
Calculating these gives:
\[ 1 : 1.128 : 1.224 \]

### Final Ratio
Thus, the ratio of the most probable speed, average speed, and RMS speed of gas molecules is:
\[ 1 : 1.128 : 1.224 \]

### Conclusion
The correct answer is \( 1 : 1.128 : 1.224 \).

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