Five molecules of gas have velocities 10,20,30,40 and 50 m/s. The ratio of R.M.S speed to average speed is

Options

A. `1.105:1`
B. `1:1.105`
C. `11.05:1`
D. `1:11.05`

Answer

To find the ratio of the root mean square (R.M.S) speed to the average speed of the gas molecules with given velocities, we can follow these steps:

### Step 1: Calculate the Average Speed
The average speed \( V_{\text{avg}} \) is calculated using the formula:
\[ V_{\text{avg}} = \frac{V_1 + V_2 + V_3 + V_4 + V_5}{5} \]
Given velocities are \( V_1 = 10 \, \text{m/s}, V_2 = 20 \, \text{m/s}, V_3 = 30 \, \text{m/s}, V_4 = 40 \, \text{m/s}, V_5 = 50 \, \text{m/s} \).

Calculating the sum:
\[ V_{\text{avg}} = \frac{10 + 20 + 30 + 40 + 50}{5} = \frac{150}{5} = 30 \, \text{m/s} \]

### Step 2: Calculate the R.M.S Speed
The root mean square speed \( V_{\text{rms}} \) is calculated using the formula:
\[ V_{\text{rms}} = \sqrt{\frac{V_1^2 + V_2^2 + V_3^2 + V_4^2 + V_5^2}{5}} \]
Calculating the squares of the velocities:
\[ V_1^2 = 10^2 = 100, \quad V_2^2 = 20^2 = 400, \quad V_3^2 = 30^2 = 900, \quad V_4^2 = 40^2 = 1600, \quad V_5^2 = 50^2 = 2500 \]
Now, summing these values:
\[ V_1^2 + V_2^2 + V_3^2 + V_4^2 + V_5^2 = 100 + 400 + 900 + 1600 + 2500 = 4500 \]
Now, substituting back into the R.M.S formula:
\[ V_{\text{rms}} = \sqrt{\frac{4500}{5}} = \sqrt{900} = 30 \, \text{m/s} \]

### Step 3: Calculate the Ratio of R.M.S Speed to Average Speed
Now we can find the ratio:
\[ \text{Ratio} = \frac{V_{\text{rms}}}{V_{\text{avg}}} = \frac{30}{30} = 1 \]

### Final Answer
The ratio of R.M.S speed to average speed is \( 1 \).

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