Three gas molecules have velocities 0.3 km`s^(-1)` ,`0.6 kms^(-1), 1.5 kms^(-1)`. Calculate rms velocity and average velocity?

Explanation

To solve the problem of calculating the RMS (Root Mean Square) velocity and the average velocity of the three gas molecules with given velocities, we can follow these steps:

### Step 1: Identify the velocities
The velocities of the gas molecules are given as:
- \( v_1 = 0.3 \, \text{km/s} \)
- \( v_2 = 0.6 \, \text{km/s} \)
- \( v_3 = 1.5 \, \text{km/s} \)

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

Substituting the values:
\[ v_{\text{rms}} = \sqrt{\frac{(0.3)^2 + (0.6)^2 + (1.5)^2}{3}} \]

### Step 3: Calculate the squares of the velocities
Calculating the squares:
- \( (0.3)^2 = 0.09 \)
- \( (0.6)^2 = 0.36 \)
- \( (1.5)^2 = 2.25 \)

### Step 4: Sum the squares
Now, sum the squares:
\[ 0.09 + 0.36 + 2.25 = 2.70 \]

### Step 5: Divide by the number of molecules
Now divide by the number of molecules:
\[ \frac{2.70}{3} = 0.90 \]

### Step 6: Take the square root
Now take the square root to find the RMS velocity:
\[ v_{\text{rms}} = \sqrt{0.90} \approx 0.9487 \, \text{km/s} \]
Rounding this gives approximately:
\[ v_{\text{rms}} \approx 0.95 \, \text{km/s} \]

### Step 7: Calculate the average velocity
The formula for the average velocity is:
\[ v_{\text{average}} = \frac{v_1 + v_2 + v_3}{n} \]

Substituting the values:
\[ v_{\text{average}} = \frac{0.3 + 0.6 + 1.5}{3} \]

### Step 8: Sum the velocities
Now sum the velocities:
\[ 0.3 + 0.6 + 1.5 = 2.4 \]

### Step 9: Divide by the number of molecules
Now divide by the number of molecules:
\[ \frac{2.4}{3} = 0.8 \, \text{km/s} \]

### Final Results
- RMS velocity \( v_{\text{rms}} \approx 0.95 \, \text{km/s} \)
- Average velocity \( v_{\text{average}} = 0.8 \, \text{km/s} \)