If three molecules have velocities `0.5kms^(-1),1kms^(-1)and 2kms^(-1),` the ratio of the rms speed and average speed is a) 2.15 b) 1.13 c) 0.53 d) 3.96

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 of three molecules with given velocities, we will follow these steps:

### Step 1: Identify the velocities of the molecules
The velocities of the three molecules are given as:
- \( v_1 = 0.5 \, \text{km/s} \)
- \( v_2 = 1 \, \text{km/s} \)
- \( v_3 = 2 \, \text{km/s} \)

### Step 2: Calculate the rms speed
The formula for the root mean square (rms) speed is:
\[ v_{\text{rms}} = \sqrt{\frac{v_1^2 + v_2^2 + v_3^2}{n}} \]
where \( n \) is the number of molecules (in this case, \( n = 3 \)).

Calculating \( v_{\text{rms}} \):
\[ v_{\text{rms}} = \sqrt{\frac{(0.5)^2 + (1)^2 + (2)^2}{3}} \]
Calculating the squares:
\[ (0.5)^2 = 0.25, \quad (1)^2 = 1, \quad (2)^2 = 4 \]
Adding these values:
\[ 0.25 + 1 + 4 = 5.25 \]
Now divide by 3:
\[ \frac{5.25}{3} = 1.75 \]
Taking the square root:
\[ v_{\text{rms}} = \sqrt{1.75} \approx 1.32 \, \text{km/s} \]

### Step 3: Calculate the average speed
The formula for the average speed is:
\[ v_{\text{average}} = \frac{v_1 + v_2 + v_3}{n} \]
Calculating \( v_{\text{average}} \):
\[ v_{\text{average}} = \frac{0.5 + 1 + 2}{3} \]
Adding the velocities:
\[ 0.5 + 1 + 2 = 3.5 \]
Now divide by 3:
\[ v_{\text{average}} = \frac{3.5}{3} \approx 1.17 \, \text{km/s} \]

### Step 4: Calculate the ratio of rms speed to average speed
Now we can find the ratio:
\[ \text{Ratio} = \frac{v_{\text{rms}}}{v_{\text{average}}} = \frac{1.32}{1.17} \approx 1.13 \]

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

### Options
The correct option is (b) 1.13.

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