Four molecules of a gas have speeds `2, 4, 6` and `8 kms^(-1)` respectively. Calculate their root mean square speed.

Options

A. `sqrt(30)m//s`
B. `120m//s`
C. `2sqrt(30)m//s`
D. `60m//s`

Answer

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

### Step 1: Write down the formula for RMS speed.
The formula for the root mean square speed (RMS) is given by:

\[ v_{rms} = \sqrt{\frac{c_1^2 + c_2^2 + c_3^2 + \ldots + c_n^2}{n}} \]

where \(c_1, c_2, c_3, \ldots, c_n\) are the speeds of the gas molecules, and \(n\) is the total number of molecules.

### Step 2: Identify the speeds of the molecules.
From the question, the speeds of the four molecules are:
- \(c_1 = 2 \, \text{km/s}\)
- \(c_2 = 4 \, \text{km/s}\)
- \(c_3 = 6 \, \text{km/s}\)
- \(c_4 = 8 \, \text{km/s}\)

### Step 3: Calculate the squares of the speeds.
Now, we calculate the squares of each speed:
- \(c_1^2 = 2^2 = 4\)
- \(c_2^2 = 4^2 = 16\)
- \(c_3^2 = 6^2 = 36\)
- \(c_4^2 = 8^2 = 64\)

### Step 4: Sum the squares of the speeds.
Next, we sum these squared values:

\[ c_1^2 + c_2^2 + c_3^2 + c_4^2 = 4 + 16 + 36 + 64 = 120 \]

### Step 5: Divide by the number of molecules.
Now, we divide this sum by the number of molecules, which is \(n = 4\):

\[ \frac{120}{4} = 30 \]

### Step 6: Take the square root.
Finally, we take the square root of the result to find the RMS speed:

\[ v_{rms} = \sqrt{30} \, \text{km/s} \]

### Final Answer:
Thus, the root mean square speed of the gas molecules is:

\[ v_{rms} = \sqrt{30} \, \text{km/s} \]