Under similar conditions of temperature and pressure gas having highest most probable velocity is

Options

A. Methane
B. Nitrogen
C. Oxygen
D. Helium

Answer

To solve the question of which gas has the highest most probable velocity under similar conditions of temperature and pressure, we will follow these steps:

### Step 1: Understand the Formula
The most probable velocity (v_mp) of a gas can be calculated using the formula:
\[ v_{mp} = \sqrt{\frac{2RT}{M}} \]
where:
- \( R \) is the gas constant,
- \( T \) is the temperature in Kelvin,
- \( M \) is the molar mass of the gas in kg.

### Step 2: Identify the Relationship
From the formula, we can see that the most probable velocity is inversely proportional to the square root of the molar mass:
\[ v_{mp} \propto \frac{1}{\sqrt{M}} \]
This means that the gas with the lowest molar mass will have the highest most probable velocity.

### Step 3: Calculate the Molar Mass of Each Gas
Now, we will calculate the molar mass of the given gases:
1. **Methane (CH₄)**:
- Carbon (C): 12 g/mol
- Hydrogen (H): 1 g/mol × 4 = 4 g/mol
- Total: 12 + 4 = 16 g/mol

2. **Nitrogen (N₂)**:
- Nitrogen (N): 14 g/mol × 2 = 28 g/mol

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

4. **Helium (He)**:
- Helium: 4 g/mol

### Step 4: Compare the Molar Masses
Now we have the molar masses:
- Methane: 16 g/mol
- Nitrogen: 28 g/mol
- Oxygen: 32 g/mol
- Helium: 4 g/mol

### Step 5: Determine the Gas with the Lowest Molar Mass
From the calculated molar masses, we can see that:
- Helium has the lowest molar mass (4 g/mol).

### Step 6: Conclusion
Since the most probable velocity is inversely proportional to the square root of the molar mass, Helium will have the highest most probable velocity among the given gases.

### Final Answer
The gas with the highest most probable velocity is **Helium (Option D)**.

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