The molecule of which gas have highest speed ?

Options

A. `H_2 " at " - 73^@C`
B. `CH_4 " at " 300K`
C. `N_2 " at " 1027^@C`
D. `O_2 " at " 0^@C`

Answer

To determine which gas has the highest speed, we can use the formula for the root mean square velocity (URMS) of a gas, which is given by:

\[ URMS = \sqrt{\frac{3RT}{M}} \]

Where:
- \( R \) is the universal gas constant,
- \( T \) is the temperature in Kelvin,
- \( M \) is the molar mass of the gas in kg/mol.

### Step 1: Convert Temperatures to Kelvin
We need to convert all given temperatures into Kelvin.

1. **Hydrogen (H2)**: Given at -73 °C
\[ T_{H2} = -73 + 273 = 200 \, K \]

2. **Methane (CH4)**: Given at 300 K (already in Kelvin)
\[ T_{CH4} = 300 \, K \]

3. **Nitrogen (N2)**: Given at 1027 °C
\[ T_{N2} = 1027 + 273 = 1300 \, K \]

4. **Oxygen (O2)**: Given at 0 °C
\[ T_{O2} = 0 + 273 = 273 \, K \]

### Step 2: Determine Molar Masses
Next, we need the molar masses of the gases:

- **Hydrogen (H2)**: \( M_{H2} = 2 \, g/mol = 0.002 \, kg/mol \)
- **Methane (CH4)**: \( M_{CH4} = 16 \, g/mol = 0.016 \, kg/mol \)
- **Nitrogen (N2)**: \( M_{N2} = 28 \, g/mol = 0.028 \, kg/mol \)
- **Oxygen (O2)**: \( M_{O2} = 32 \, g/mol = 0.032 \, kg/mol \)

### Step 3: Calculate URMS for Each Gas
Now we can calculate the URMS for each gas:

1. **Hydrogen (H2)**:
\[ URMS_{H2} = \sqrt{\frac{3 \times R \times 200}{0.002}} \]
(Since R is a constant, we can ignore it for comparison.)

2. **Methane (CH4)**:
\[ URMS_{CH4} = \sqrt{\frac{3 \times R \times 300}{0.016}} \]

3. **Nitrogen (N2)**:
\[ URMS_{N2} = \sqrt{\frac{3 \times R \times 1300}{0.028}} \]

4. **Oxygen (O2)**:
\[ URMS_{O2} = \sqrt{\frac{3 \times R \times 273}{0.032}} \]

### Step 4: Compare the Speeds
Since \( R \) is constant, we can compare the values inside the square roots to determine which gas has the highest speed:

1. **Hydrogen**:
\[ \frac{3 \times 200}{0.002} = 300000 \]

2. **Methane**:
\[ \frac{3 \times 300}{0.016} = 56250 \]

3. **Nitrogen**:
\[ \frac{3 \times 1300}{0.028} = 139285.71 \]

4. **Oxygen**:
\[ \frac{3 \times 273}{0.032} = 25687.5 \]

### Conclusion
Comparing the values:
- Hydrogen: 300000
- Methane: 56250
- Nitrogen: 139285.71
- Oxygen: 25687.5

The highest value is for Hydrogen, which indicates that hydrogen molecules have the highest speed.

### Final Answer
**The gas with the highest speed is Hydrogen (H2).**

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