Which of the following contains the same number of molecules ?

Options

A. `1g` of `O_(2)`,`2g` of `SO_(2)`
B. `1g` of `CO_(2)` , `1g` of `N_(2)O`
C. `112ml` of `O_(2)` at `STP,` `224 ml` of `He` at `0.5atm` and `273K`
D. All of these

Explanation

To determine which of the given options contains the same number of molecules, we need to calculate the number of moles for each substance in the pairs provided. Here’s a step-by-step solution:

### Step 1: Calculate the number of moles for 1 gram of Oxygen (O2)
- **Molecular mass of O2**: Oxygen has an atomic mass of approximately 16 g/mol, so for O2:
\[ \text{Molecular mass of O2} = 2 \times 16 = 32 \text{ g/mol} \]
- **Number of moles of O2**:
\[ \text{Number of moles} = \frac{\text{mass}}{\text{molecular mass}} = \frac{1 \text{ g}}{32 \text{ g/mol}} = \frac{1}{32} \text{ moles} \]

### Step 2: Calculate the number of moles for 2 grams of Sulfur Dioxide (SO2)
- **Molecular mass of SO2**: Sulfur has an atomic mass of approximately 32 g/mol and Oxygen is 16 g/mol:
\[ \text{Molecular mass of SO2} = 32 + 2 \times 16 = 32 + 32 = 64 \text{ g/mol} \]
- **Number of moles of SO2**:
\[ \text{Number of moles} = \frac{2 \text{ g}}{64 \text{ g/mol}} = \frac{1}{32} \text{ moles} \]

### Conclusion for Pair 1:
Since both O2 and SO2 have the same number of moles (\(\frac{1}{32}\)), they contain the same number of molecules.

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### Step 3: Calculate the number of moles for 1 gram of Carbon Dioxide (CO2)
- **Molecular mass of CO2**: Carbon has an atomic mass of approximately 12 g/mol:
\[ \text{Molecular mass of CO2} = 12 + 2 \times 16 = 12 + 32 = 44 \text{ g/mol} \]
- **Number of moles of CO2**:
\[ \text{Number of moles} = \frac{1 \text{ g}}{44 \text{ g/mol}} = \frac{1}{44} \text{ moles} \]

### Step 4: Calculate the number of moles for 1 gram of Nitrous Oxide (N2O)
- **Molecular mass of N2O**: Nitrogen has an atomic mass of approximately 14 g/mol:
\[ \text{Molecular mass of N2O} = 2 \times 14 + 16 = 28 + 16 = 44 \text{ g/mol} \]
- **Number of moles of N2O**:
\[ \text{Number of moles} = \frac{1 \text{ g}}{44 \text{ g/mol}} = \frac{1}{44} \text{ moles} \]

### Conclusion for Pair 2:
Since both CO2 and N2O have the same number of moles (\(\frac{1}{44}\)), they contain the same number of molecules.

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### Step 5: Calculate the number of moles for 112 ml of Oxygen at STP
- At STP (Standard Temperature and Pressure), 1 mole of gas occupies 22,400 ml.
- **Number of moles of O2**:
\[ \text{Number of moles} = \frac{112 \text{ ml}}{22400 \text{ ml/mol}} = \frac{112}{22400} = \frac{1}{200} \text{ moles} \]

### Step 6: Calculate the number of moles for 224 ml of Helium at 0.5 atm
- To find the volume at 1 atm, we use the formula:
\[ V_1 = \frac{P_1 \times V_2}{P_2} = \frac{1 \text{ atm} \times 224 \text{ ml}}{0.5 \text{ atm}} = 448 \text{ ml} \]
- **Number of moles of He**:
\[ \text{Number of moles} = \frac{448 \text{ ml}}{22400 \text{ ml/mol}} = \frac{448}{22400} = \frac{1}{50} \text{ moles} \]

### Conclusion for Pair 3:
Since the number of moles for O2 is \(\frac{1}{200}\) and for He is \(\frac{1}{50}\), they do not contain the same number of molecules.

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### Final Conclusion:
From the calculations:
- Pair 1 (O2 and SO2) contains the same number of molecules.
- Pair 2 (CO2 and N2O) contains the same number of molecules.
- Pair 3 (O2 and He) does not contain the same number of molecules.

Thus, the answer is that the first two pairs contain the same number of molecules, while the third does not.