Four molecules of a gas have velocity 2, 4, 6 and 8 `kms^(-1)`, respectively. The ratio of their root mean velocity and average speed is :

Options

A. (a)`0.913`
B. (b)`1.059`
C. (c)`0.931`
D. (d)`1.095`

Explanation

To solve the problem, we need to find the ratio of the root mean square velocity (RMS velocity) and the average speed of the gas molecules with given velocities of 2, 4, 6, and 8 km/s.

### Step 1: Calculate the Root Mean Square Velocity (RMS Velocity)

The formula for the root mean square velocity is given by:

\[ v_{rms} = \sqrt{\frac{v_1^2 + v_2^2 + v_3^2 + v_4^2}{N}} \]

where \(N\) is the number of molecules.

Given the velocities:
- \(v_1 = 2 \, \text{km/s}\)
- \(v_2 = 4 \, \text{km/s}\)
- \(v_3 = 6 \, \text{km/s}\)
- \(v_4 = 8 \, \text{km/s}\)

We can calculate \(v_{rms}\):

\[ v_{rms} = \sqrt{\frac{2^2 + 4^2 + 6^2 + 8^2}{4}} = \sqrt{\frac{4 + 16 + 36 + 64}{4}} = \sqrt{\frac{120}{4}} = \sqrt{30} \]

### Step 2: Calculate the Average Speed

The formula for the average speed is given by:

\[ v_{avg} = \frac{v_1 + v_2 + v_3 + v_4}{N} \]

Calculating the average speed:

\[ v_{avg} = \frac{2 + 4 + 6 + 8}{4} = \frac{20}{4} = 5 \, \text{km/s} \]

### Step 3: Calculate the Ratio of RMS Velocity to Average Speed

Now, we need to find the ratio of the root mean square velocity to the average speed:

\[ \text{Ratio} = \frac{v_{rms}}{v_{avg}} = \frac{\sqrt{30}}{5} \]

### Step 4: Simplify the Ratio

To simplify the ratio:

\[ \text{Ratio} = \frac{\sqrt{30}}{5} = \frac{1}{5} \sqrt{30} \]

### Final Result

Thus, the ratio of the root mean square velocity to the average speed is:

\[ \text{Ratio} = \frac{\sqrt{30}}{5} \approx 1.095 \]