Which of the following has maximum root mean square velocity at the same temperature ?

Options

A. `SO_(2)`
B. `CO_(2)`
C. `O_(2)`
D. `H_(2)`

Answer

To determine which gas has the maximum root mean square (RMS) velocity at the same temperature, we can follow these steps:

### Step 1: Understand the Formula for RMS Velocity
The formula for root mean square velocity (V_rms) is given by:
\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature in Kelvin,
- \( M \) is the molar mass of the gas in kg/mol.

### Step 2: Analyze the Relationship
From the formula, we can see that the RMS velocity is inversely proportional to the square root of the molar mass (\( M \)):
\[ V_{rms} \propto \frac{1}{\sqrt{M}} \]
This means that as the molar mass increases, the RMS velocity decreases, and vice versa.

### Step 3: Compare the Molar Masses of the Given Gases
We need to find the molar masses of the gases provided in the options:
1. **SO2 (Sulfur Dioxide)**:
- Sulfur (S) = 32 g/mol
- Oxygen (O) = 16 g/mol (2 atoms)
- Total = 32 + (2 × 16) = 64 g/mol

2. **CO2 (Carbon Dioxide)**:
- Carbon (C) = 12 g/mol
- Oxygen (O) = 16 g/mol (2 atoms)
- Total = 12 + (2 × 16) = 44 g/mol

3. **O2 (Oxygen)**:
- Oxygen (O) = 16 g/mol (2 atoms)
- Total = 2 × 16 = 32 g/mol

4. **H2 (Hydrogen)**:
- Hydrogen (H) = 1 g/mol (2 atoms)
- Total = 2 × 1 = 2 g/mol

### Step 4: Identify the Gas with the Lowest Molar Mass
Now, we compare the molar masses:
- SO2 = 64 g/mol
- CO2 = 44 g/mol
- O2 = 32 g/mol
- H2 = 2 g/mol

The gas with the lowest molar mass is **H2 (Hydrogen)**, which has a molar mass of 2 g/mol.

### Step 5: Conclusion
Since the RMS velocity is inversely proportional to the square root of the molar mass, H2 will have the maximum root mean square velocity among the given options.

Thus, the answer is:
**H2 (Hydrogen)** has the maximum root mean square velocity at the same temperature.

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