Four molecules of gas have speeds 1,2,3 and 4 `km//s`.The value of the root mean square speed of the gas molecules is

Options

A. `1/2sqrt15kms^(-1)`
B. `1/2sqrt10kms^(-1)`
C. `2.5kms^(-1)`
D. `sqrt(15//2)kms^(-1)`

Explanation

To find the root mean square speed of the gas molecules, we can follow these steps:

### Step 1: Identify the speeds of the gas molecules
The speeds of the four gas molecules are given as:
- \( V_1 = 1 \, \text{km/s} \)
- \( V_2 = 2 \, \text{km/s} \)
- \( V_3 = 3 \, \text{km/s} \)
- \( V_4 = 4 \, \text{km/s} \)

### Step 2: Write down the formula for root mean square speed
The formula for the root mean square speed (\( V_{\text{rms}} \)) is given by:
\[ V_{\text{rms}} = \sqrt{\frac{V_1^2 + V_2^2 + V_3^2 + V_4^2}{n}} \]
where \( n \) is the number of molecules.

### Step 3: Calculate the number of molecules
In this case, we have:
\[ n = 4 \]

### Step 4: Calculate the squares of the speeds
Now, we will calculate the squares of each speed:
- \( V_1^2 = 1^2 = 1 \)
- \( V_2^2 = 2^2 = 4 \)
- \( V_3^2 = 3^2 = 9 \)
- \( V_4^2 = 4^2 = 16 \)

### Step 5: Sum the squares of the speeds
Now, we will sum these squares:
\[ V_1^2 + V_2^2 + V_3^2 + V_4^2 = 1 + 4 + 9 + 16 = 30 \]

### Step 6: Divide by the number of molecules
Now, we will divide the sum by the number of molecules:
\[ \frac{30}{4} = 7.5 \]

### Step 7: Take the square root
Finally, we take the square root of the result:
\[ V_{\text{rms}} = \sqrt{7.5} \]

### Step 8: Simplify the square root
We can express \( 7.5 \) as \( \frac{15}{2} \):
\[ V_{\text{rms}} = \sqrt{\frac{15}{2}} = \frac{\sqrt{15}}{\sqrt{2}} = \frac{\sqrt{15}}{2\sqrt{2}} \text{ km/s} \]

### Final Answer
Thus, the root mean square speed of the gas molecules is:
\[ V_{\text{rms}} = \frac{\sqrt{15}}{2} \text{ km/s} \]

### Conclusion
The correct option is:
**Option 4: \( \frac{\sqrt{15}}{2} \text{ km/s} \)**

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