Five molecules of a gas have speeds 1, 1, 3, 3, 2 km/s the value of the r.m.s spreed of the gas molecules is

Options

A. `sqrt(4/5)km//s`
B. `sqrt(23/5)km//s`
C. `sqrt(24/5)km//s`
D. `sqrt(87/5)km//s`

Answer

To find the root mean square (r.m.s) speed of the gas molecules, we will follow these steps:

### Step 1: List the speeds of the gas molecules
The speeds of the gas molecules are given as:
- v1 = 1 km/s
- v2 = 1 km/s
- v3 = 3 km/s
- v4 = 3 km/s
- v5 = 2 km/s

### Step 2: Square each speed
Next, we will square each of these speeds:
- v1² = (1 km/s)² = 1 km²/s²
- v2² = (1 km/s)² = 1 km²/s²
- v3² = (3 km/s)² = 9 km²/s²
- v4² = (3 km/s)² = 9 km²/s²
- v5² = (2 km/s)² = 4 km²/s²

### Step 3: Sum the squared speeds
Now, we will sum the squared speeds:
\[ \text{Sum} = v1² + v2² + v3² + v4² + v5² = 1 + 1 + 9 + 9 + 4 = 24 \text{ km²/s²} \]

### Step 4: Divide by the number of molecules
Since there are 5 molecules, we will divide the sum of the squared speeds by 5:
\[ \text{Mean of squares} = \frac{\text{Sum}}{N} = \frac{24 \text{ km²/s²}}{5} = 4.8 \text{ km²/s²} \]

### Step 5: Take the square root
Finally, we will take the square root of the mean of the squares to find the r.m.s speed:
\[ V_{rms} = \sqrt{4.8 \text{ km²/s²}} \approx 2.19 \text{ km/s} \]

### Final Answer
The root mean square speed of the gas molecules is approximately **2.19 km/s**.

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