A gaseous mixture contains 4 molecules with a velocity of 6 cm `sec^(-1)`,5 molecules with a velocity of 2 cm `sec^(-1)` and 10 molecules with a velocity of 3 cm `sec^(-1)`. What is the rms speed of the gas:

Options

A. 2.5 cm `sec^(-1)`
B. 1.9 cm `sec^(-1)`
C. 3.6 cm `sec^(-1)`
D. 4.6 cm `sec^(-1)`

Explanation

To find the root mean square (RMS) speed of the gaseous mixture, we can follow these steps:

### Step 1: Identify the number of molecules and their velocities
We have:
- 4 molecules with a velocity of 6 cm/s
- 5 molecules with a velocity of 2 cm/s
- 10 molecules with a velocity of 3 cm/s

### Step 2: Calculate the total number of molecules
Total number of molecules (N) = 4 + 5 + 10 = 19 molecules

### Step 3: Calculate the square of each velocity multiplied by the number of molecules
We need to calculate:
- For 6 cm/s: \( n_1 \cdot c_1^2 = 4 \cdot (6^2) = 4 \cdot 36 = 144 \)
- For 2 cm/s: \( n_2 \cdot c_2^2 = 5 \cdot (2^2) = 5 \cdot 4 = 20 \)
- For 3 cm/s: \( n_3 \cdot c_3^2 = 10 \cdot (3^2) = 10 \cdot 9 = 90 \)

### Step 4: Sum these values
Total = \( 144 + 20 + 90 = 254 \)

### Step 5: Divide by the total number of molecules
Now, we divide the total by the number of molecules:
\[ \text{Mean of squares} = \frac{254}{19} \approx 13.3684 \]

### Step 6: Take the square root to find the RMS speed
Now, we take the square root of the mean of squares:
\[ \text{RMS speed} = \sqrt{13.3684} \approx 3.65 \text{ cm/s} \]

### Final Answer
The RMS speed of the gas is approximately **3.65 cm/s**.

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