At STP, 0.50 mole `H_(2)` gas and 1.0 mole He gas

Options

A. Occupy equal volumes
B. Have equal diffusion rates
C. Have equal molecular speeds
D. Have equal average kinetic energies

Answer

To solve the question regarding the behavior of 0.50 moles of H₂ gas and 1.0 mole of He gas at STP (Standard Temperature and Pressure), we need to analyze the properties of gases under these conditions.

### Step-by-Step Solution:

1. **Understanding STP**:
At STP, the temperature is 273.15 K (0°C) and the pressure is 1 atm. Under these conditions, one mole of any ideal gas occupies a volume of 22.4 liters.

2. **Volume Calculation**:
- For H₂ gas:
\[ \text{Volume of } H_2 = 0.50 \text{ moles} \times 22.4 \text{ L/mole} = 11.2 \text{ L} \]
- For He gas:
\[ \text{Volume of He} = 1.0 \text{ mole} \times 22.4 \text{ L/mole} = 22.4 \text{ L} \]

3. **Equal Volume**:
The question states that both gases occupy equal volume. However, based on our calculations, they do not occupy equal volumes. Therefore, we need to clarify that at STP, the volume occupied by each gas is directly proportional to the number of moles.

4. **Diffusion Rate**:
The rate of diffusion of gases can be compared using Graham's law, which states that the rate of diffusion is inversely proportional to the square root of the molar mass of the gas.
\[ \text{Rate of diffusion} \propto \frac{1}{\sqrt{M}} \]
- Molar mass of H₂ = 2 g/mol
- Molar mass of He = 4 g/mol
Therefore, H₂ will diffuse faster than He.

5. **Molecular Speed**:
The average molecular speed can be calculated using the formula:
\[ v = \sqrt{\frac{3RT}{M}} \]
Since both gases are at the same temperature (STP), the average molecular speed will depend on their molar masses. H₂ will have a higher average speed than He due to its lower molar mass.

6. **Average Kinetic Energy**:
The average kinetic energy of a gas is given by:
\[ KE = \frac{3}{2} k T \]
where \( k \) is the Boltzmann constant and \( T \) is the temperature. Since both gases are at the same temperature, they will have the same average kinetic energy regardless of the number of moles.

### Conclusion:
At STP, while the two gases do not occupy equal volumes and have different diffusion rates and molecular speeds, they do have equal average kinetic energy because kinetic energy depends only on temperature.