If three molecules have velocities `0.5kms^(-1),1kms^(-1)and 2kms^(-1),` the ratio of the rms speed and average speed is

Options

A. `2.15`
B. `1.13`
C. `0.53`
D. `3.96`

Explanation

To solve the problem of finding the ratio of the root mean square (rms) speed to the average speed for three molecules with velocities \(0.5 \, \text{km/s}\), \(1 \, \text{km/s}\), and \(2 \, \text{km/s}\), we will follow these steps:

### Step 1: Calculate the RMS Speed

The formula for the root mean square speed (\(v_{\text{rms}}\)) for \(n\) particles is given by:

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

For our case, we have three velocities:
- \(v_1 = 0.5 \, \text{km/s}\)
- \(v_2 = 1 \, \text{km/s}\)
- \(v_3 = 2 \, \text{km/s}\)

Calculating the squares of the velocities:

\[ v_1^2 = (0.5)^2 = 0.25 \]
\[ v_2^2 = (1)^2 = 1 \]
\[ v_3^2 = (2)^2 = 4 \]

Now, substituting these values into the rms formula:

\[ v_{\text{rms}} = \sqrt{\frac{0.25 + 1 + 4}{3}} = \sqrt{\frac{5.25}{3}} = \sqrt{1.75} \approx 1.32 \, \text{km/s} \]

### Step 2: Calculate the Average Speed

The formula for the average speed (\(v_{\text{avg}}\)) is given by:

\[ v_{\text{avg}} = \frac{v_1 + v_2 + v_3}{n} \]

Substituting the values:

\[ v_{\text{avg}} = \frac{0.5 + 1 + 2}{3} = \frac{3.5}{3} \approx 1.17 \, \text{km/s} \]

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

Now, we can find the ratio of the rms speed to the average speed:

\[ \text{Ratio} = \frac{v_{\text{rms}}}{v_{\text{avg}}} = \frac{1.32}{1.17} \approx 1.13 \]

### Final Answer

The ratio of the rms speed to the average speed is approximately \(1.13\).

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