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. `0.913`
B. `1.059`
C. `0.931`
D. `1.095`

Explanation

To find the ratio of the root mean square velocity (Vrms) and the average speed (Vaverage) of the gas molecules with given velocities, we can follow these steps:

### Step 1: Calculate Vrms
The formula for root mean square velocity (Vrms) is given by:

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

where \(v_1, v_2, v_3, v_4\) are the velocities of the molecules and \(N\) is the number of molecules.

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

Calculating the squares of the velocities:
- \(v_1^2 = 2^2 = 4\)
- \(v_2^2 = 4^2 = 16\)
- \(v_3^2 = 6^2 = 36\)
- \(v_4^2 = 8^2 = 64\)

Now, sum these squares:
\[ v_1^2 + v_2^2 + v_3^2 + v_4^2 = 4 + 16 + 36 + 64 = 120 \]

Now, substitute into the Vrms formula:
\[ V_{rms} = \sqrt{\frac{120}{4}} = \sqrt{30} \]

### Step 2: Calculate Vaverage
The formula for average speed (Vaverage) is given by:

\[ V_{average} = \frac{v_1 + v_2 + v_3 + v_4}{N} \]

Calculating the sum of the velocities:
\[ v_1 + v_2 + v_3 + v_4 = 2 + 4 + 6 + 8 = 20 \]

Now, substitute into the Vaverage formula:
\[ V_{average} = \frac{20}{4} = 5 \, \text{km/s} \]

### Step 3: Calculate the Ratio
Now that we have both Vrms and Vaverage, we can find the ratio:

\[ \text{Ratio} = \frac{V_{rms}}{V_{average}} = \frac{\sqrt{30}}{5} \]

### Step 4: Simplify the Ratio
To simplify:
\[ \text{Ratio} = \frac{\sqrt{30}}{5} \approx 1.095 \]

### Conclusion
Thus, the ratio of the root mean square velocity to the average speed is approximately \(1.095\).