The speeds of 5 molecules of a gas ( in arbitrary units) are as follows: 2, 3, 4, 5, 6. The root mean square speed for these molecules is ……..

Answer

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

### Step 1: Identify the speeds of the molecules
The speeds of the 5 molecules are given as:
- Speed 1 (V1) = 2
- Speed 2 (V2) = 3
- Speed 3 (V3) = 4
- Speed 4 (V4) = 5
- Speed 5 (V5) = 6

### Step 2: Square each of the speeds
Next, we will square each of the speeds:
- V1² = 2² = 4
- V2² = 3² = 9
- V3² = 4² = 16
- V4² = 5² = 25
- V5² = 6² = 36

### Step 3: Sum the squared speeds
Now, we will sum these squared values:
\[ \text{Sum} = V1² + V2² + V3² + V4² + V5² = 4 + 9 + 16 + 25 + 36 \]
Calculating the sum:
\[ \text{Sum} = 4 + 9 + 16 + 25 + 36 = 90 \]

### Step 4: Divide by the number of molecules
Now, we divide the sum of the squared speeds by the number of molecules (n = 5):
\[ \text{Mean of squares} = \frac{\text{Sum}}{n} = \frac{90}{5} = 18 \]

### Step 5: Take the square root
Finally, we take the square root of the mean of the squares to get the root mean square speed:
\[ V_{rms} = \sqrt{18} \]
Calculating the square root:
\[ V_{rms} \approx 4.24 \]

### Final Answer
The root mean square speed for these molecules is approximately **4.24**.

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