Which of the following has possesses maximum rms velocity? All being at same temperature

Options

A. Oxygen
B. Carbon dioxide
C. Hydrogen
D. Helium

Answer

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

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

### Step 2: Analyze the Relationship
From the formula, we can see that \( V_{rms} \) 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 decreases, the RMS velocity increases.

### Step 3: Identify the Molar Masses of the Gases
Let's identify the molar masses of the gases mentioned:
- Hydrogen (\( H_2 \)): \( 2 \, \text{g/mol} = 0.002 \, \text{kg/mol} \)
- Helium (\( He \)): \( 4 \, \text{g/mol} = 0.004 \, \text{kg/mol} \)
- Oxygen (\( O_2 \)): \( 32 \, \text{g/mol} = 0.032 \, \text{kg/mol} \)
- Carbon Dioxide (\( CO_2 \)): \( 44 \, \text{g/mol} = 0.044 \, \text{kg/mol} \)

### Step 4: Compare the Molar Masses
Now, we can compare the molar masses:
- \( H_2 \): 0.002 kg/mol
- \( He \): 0.004 kg/mol
- \( O_2 \): 0.032 kg/mol
- \( CO_2 \): 0.044 kg/mol

### Step 5: Determine the Gas with the Lowest Molar Mass
The gas with the lowest molar mass is \( H_2 \) (Hydrogen), which has a molar mass of \( 0.002 \, \text{kg/mol} \).

### Step 6: Conclusion
Since \( V_{rms} \) is inversely proportional to the square root of the molar mass, the gas with the lowest molar mass will have the highest RMS velocity. Therefore, the gas that possesses the maximum RMS velocity at the same temperature is:
\[ \text{Hydrogen (H}_2\text{)} \]

### Summary
The answer to the question is Hydrogen, as it has the lowest molar mass, leading to the highest RMS velocity.

---