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 (RMS) 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:
- \( v_1 = 1 \) km/s
- \( v_2 = 1 \) km/s
- \( v_3 = 3 \) km/s
- \( v_4 = 3 \) km/s
- \( v_5 = 2 \) km/s

### Step 2: Square each speed
We will square each of the speeds:
- \( v_1^2 = 1^2 = 1 \)
- \( v_2^2 = 1^2 = 1 \)
- \( v_3^2 = 3^2 = 9 \)
- \( v_4^2 = 3^2 = 9 \)
- \( v_5^2 = 2^2 = 4 \)

### Step 3: Sum the squared speeds
Now we will sum the squared speeds:
\[ \text{Sum} = v_1^2 + v_2^2 + v_3^2 + v_4^2 + v_5^2 = 1 + 1 + 9 + 9 + 4 = 24 \]

### Step 4: Calculate the mean of the squared speeds
To find the mean of the squared speeds, we divide the sum by the number of molecules (which is 5):
\[ \text{Mean} = \frac{\text{Sum}}{N} = \frac{24}{5} = 4.8 \]

### Step 5: Take the square root of the mean
Finally, we take the square root of the mean to find the RMS speed:
\[ v_{\text{RMS}} = \sqrt{4.8} \]

### Step 6: Calculate the value
Calculating the square root:
\[ v_{\text{RMS}} \approx 2.19 \text{ km/s} \]

### Final Answer
The value of the RMS speed of the gas molecules is approximately \( \sqrt{\frac{24}{5}} \) km/s or about 2.19 km/s.

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