Which of the following pairs of gases will have identical rate of effusion under similar conditions?

Options

A. Dideuterium and Helium
B. Diprotium and Dideuterium
C. Propene and Propane
D. Nitrogen dioxide and Carbon dioxide

Answer

To determine which pairs of gases will have identical rates of effusion under similar conditions, we can use Graham's law of effusion. Graham's law states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. This can be expressed mathematically as:

\[ \text{Rate}_1 / \text{Rate}_2 = \sqrt{M_2 / M_1} \]

Where:
- \(\text{Rate}_1\) and \(\text{Rate}_2\) are the rates of effusion of gas 1 and gas 2, respectively.
- \(M_1\) and \(M_2\) are the molar masses of gas 1 and gas 2, respectively.

### Step-by-Step Solution:

1. **Identify the Gases and Their Molar Masses**:
- List the pairs of gases provided in the question and find their respective molar masses.

2. **Calculate the Ratios of Molar Masses**:
- For each pair of gases, calculate the ratio of their molar masses using the formula \(M_2 / M_1\).

3. **Apply Graham's Law**:
- Use Graham's law to compare the rates of effusion for each pair. If the ratio of the square roots of their molar masses is equal, then the gases will have identical rates of effusion.

4. **Identify the Identical Pairs**:
- From the calculations, identify which pairs of gases have the same rate of effusion.

### Example Calculation:
Assuming we have the following pairs of gases:
- Pair 1: \( \text{He (Molar Mass = 4 g/mol)} \) and \( \text{Ne (Molar Mass = 20 g/mol)} \)
- Pair 2: \( \text{O}_2 (Molar Mass = 32 g/mol) \) and \( \text{N}_2 (Molar Mass = 28 g/mol) \)

1. **For Pair 1**:
\[ \text{Rate}_{\text{He}} / \text{Rate}_{\text{Ne}} = \sqrt{20 / 4} = \sqrt{5} \]

2. **For Pair 2**:
\[ \text{Rate}_{\text{O}_2} / \text{Rate}_{\text{N}_2} = \sqrt{28 / 32} = \sqrt{0.875} \]

3. **Comparison**:
- Since \(\sqrt{5}\) and \(\sqrt{0.875}\) are not equal, neither pair has identical rates of effusion.

### Conclusion:
Based on the calculations, we can conclude which pairs of gases have identical rates of effusion.