Which one of the following will have greatest average speed of its molecules ?

Options

A. 0.5 mol of `O_(2)` at 500 K
B. 0.2 mol of `CO_(2)` at 400 K
C. 1.0 mol of He at 200 K
D. 0.4 mol of `NH_(3)` at 300 K

Explanation

To determine which gas has the greatest average speed of its molecules, we can use the formula for the average speed of gas molecules, which is given by:

\[ u = \sqrt{\frac{8RT}{\pi m}} \]

Where:
- \( u \) = average speed of the gas molecules
- \( R \) = universal gas constant
- \( T \) = absolute temperature in Kelvin
- \( m \) = molar mass of the gas in kg/mol

Since \( R \) and \( \pi \) are constants, we can simplify our analysis by focusing on the term:

\[ \frac{T}{m} \]

This means that the average speed is directly proportional to the square root of the ratio of temperature to molar mass. We will calculate \( \sqrt{\frac{T}{m}} \) for each gas provided in the options.

### Step-by-Step Solution:

1. **Identify the Gases and Their Properties**:
- **Gas A**: Oxygen (O2) at 500 K, molar mass = 32 g/mol
- **Gas B**: Carbon Dioxide (CO2) at 400 K, molar mass = 44 g/mol
- **Gas C**: Helium (He) at 200 K, molar mass = 4 g/mol
- **Gas D**: Ammonia (NH3) at 300 K, molar mass = 17 g/mol

2. **Convert Molar Mass to kg/mol**:
- Oxygen: 32 g/mol = 0.032 kg/mol
- Carbon Dioxide: 44 g/mol = 0.044 kg/mol
- Helium: 4 g/mol = 0.004 kg/mol
- Ammonia: 17 g/mol = 0.017 kg/mol

3. **Calculate \( \sqrt{\frac{T}{m}} \) for Each Gas**:
- For **Oxygen**:
\[ \sqrt{\frac{500}{0.032}} \approx \sqrt{15625} \approx 125 \]
- For **Carbon Dioxide**:
\[ \sqrt{\frac{400}{0.044}} \approx \sqrt{9090.91} \approx 95.4 \]
- For **Helium**:
\[ \sqrt{\frac{200}{0.004}} \approx \sqrt{50000} \approx 223.6 \]
- For **Ammonia**:
\[ \sqrt{\frac{300}{0.017}} \approx \sqrt{17647.06} \approx 132.5 \]

4. **Compare the Values**:
- Oxygen: 125
- Carbon Dioxide: 95.4
- Helium: 223.6
- Ammonia: 132.5

5. **Determine the Greatest Average Speed**:
- The highest value of \( \sqrt{\frac{T}{m}} \) is for Helium (223.6), indicating that Helium has the greatest average speed of its molecules.

### Conclusion:
The gas with the greatest average speed of its molecules is **Helium (C)**.