Five gas molecules chosen at random are found to have speed of `500 , 600, 700, 800 and 900 m//s`. Find the rms speed. Is it the same as the average speed ?

Explanation

To solve the problem, we need to calculate the root mean square (RMS) speed of the gas molecules and then compare it with the average speed. Here are the steps to find the solution:

### Step 1: List the speeds of the gas molecules
The speeds of the five gas molecules are:
- \( v_1 = 500 \, \text{m/s} \)
- \( v_2 = 600 \, \text{m/s} \)
- \( v_3 = 700 \, \text{m/s} \)
- \( v_4 = 800 \, \text{m/s} \)
- \( v_5 = 900 \, \text{m/s} \)

### Step 2: Calculate the squares of the speeds
We need to square each of the speeds:
- \( v_1^2 = 500^2 = 250000 \)
- \( v_2^2 = 600^2 = 360000 \)
- \( v_3^2 = 700^2 = 490000 \)
- \( v_4^2 = 800^2 = 640000 \)
- \( v_5^2 = 900^2 = 810000 \)

### Step 3: Sum the squares of the speeds
Now we add these squared values together:
\[ \text{Sum} = 250000 + 360000 + 490000 + 640000 + 810000 = 2550000 \]

### Step 4: Calculate the mean of the squares
To find the mean of the squares, divide the sum by the number of molecules (which is 5):
\[ \text{Mean of squares} = \frac{2550000}{5} = 510000 \]

### Step 5: Calculate the RMS speed
The RMS speed is the square root of the mean of the squares:
\[ \text{RMS speed} = \sqrt{510000} \approx 714.14 \, \text{m/s} \]

### Step 6: Calculate the average speed
The average speed is calculated by summing the speeds and dividing by the number of molecules:
\[ \text{Average speed} = \frac{500 + 600 + 700 + 800 + 900}{5} = \frac{3500}{5} = 700 \, \text{m/s} \]

### Step 7: Compare RMS speed and average speed
From our calculations:
- RMS speed \( \approx 714.14 \, \text{m/s} \)
- Average speed \( = 700 \, \text{m/s} \)

### Conclusion
The RMS speed is not the same as the average speed.