Which of the following gases has maximum rms speed at a given temperature?

Options

A. (a)hydrogen
B. (b)nitrogen
C. (c)oxygen
D. (d)carbon dioxide

Answer

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

### Step 1: Understand the RMS Speed Formula
The formula for the root mean square speed (Vrms) of a gas 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 Given Information
We are given four gases: hydrogen (H₂), nitrogen (N₂), oxygen (O₂), and carbon dioxide (CO₂). We need to compare their molar masses to find out which gas has the minimum molar mass since the RMS speed is inversely proportional to the square root of the molar mass.

### Step 3: Determine the Molar Mass of Each Gas
- **Hydrogen (H₂)**: The molar mass is approximately \( 2 \, \text{g/mol} \) or \( 0.002 \, \text{kg/mol} \).
- **Nitrogen (N₂)**: The molar mass is approximately \( 28 \, \text{g/mol} \) or \( 0.028 \, \text{kg/mol} \).
- **Oxygen (O₂)**: The molar mass is approximately \( 32 \, \text{g/mol} \) or \( 0.032 \, \text{kg/mol} \).
- **Carbon Dioxide (CO₂)**: The molar mass is approximately \( 44 \, \text{g/mol} \) or \( 0.044 \, \text{kg/mol} \).

### Step 4: Compare the Molar Masses
From the above values, we can summarize the molar masses:
- Hydrogen: \( 0.002 \, \text{kg/mol} \)
- Nitrogen: \( 0.028 \, \text{kg/mol} \)
- Oxygen: \( 0.032 \, \text{kg/mol} \)
- Carbon Dioxide: \( 0.044 \, \text{kg/mol} \)

### Step 5: Identify the Gas with Minimum Molar Mass
Among the gases listed, hydrogen has the lowest molar mass.

### Step 6: Conclusion
Since the RMS speed is inversely proportional to the square root of the molar mass, hydrogen will have the maximum RMS speed at a given temperature.

Thus, the gas with the maximum RMS speed at a given temperature is **hydrogen (H₂)**.

### Final Answer
The correct option is **hydrogen (H₂)**.

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