The ratio of the number of molecules in 4 g of hydrogen to the number of molecules in 5.6 `dm^3` of oxygen at standard temperature and pressure is [Given: Atomic mass of H = 1 u and O = 16 u]

Options

A. `2:1`
B. `3:4`
C. `1:4`
D. `8:1`

Answer

To solve the problem, we need to find the ratio of the number of molecules in 4 g of hydrogen to the number of molecules in 5.6 dm³ of oxygen at standard temperature and pressure (STP).

### Step 1: Calculate the number of moles of hydrogen
- The molecular formula for hydrogen gas is H₂.
- The molecular mass of H₂ = 2 × Atomic mass of H = 2 × 1 u = 2 g/mol.
- Given mass of hydrogen = 4 g.

Using the formula for moles:
\[ \text{Number of moles of H₂} = \frac{\text{Given mass}}{\text{Molecular mass}} = \frac{4 \text{ g}}{2 \text{ g/mol}} = 2 \text{ moles} \]

### Step 2: Calculate the number of molecules of hydrogen
- Using Avogadro's number (6.022 × 10²³ molecules/mol), we can find the number of molecules in 2 moles of hydrogen:
\[ \text{Number of molecules of H₂} = 2 \text{ moles} \times 6.022 \times 10^{23} \text{ molecules/mole} = 12.044 \times 10^{23} \text{ molecules} \]

### Step 3: Calculate the number of moles of oxygen
- The molecular formula for oxygen gas is O₂.
- The molecular mass of O₂ = 2 × Atomic mass of O = 2 × 16 u = 32 g/mol.
- At STP, 1 mole of any gas occupies 22.4 L (or 22.4 dm³).
- Given volume of oxygen = 5.6 dm³.

Using the formula for moles:
\[ \text{Number of moles of O₂} = \frac{\text{Volume}}{22.4 \text{ dm³/mol}} = \frac{5.6 \text{ dm³}}{22.4 \text{ dm³/mol}} = 0.25 \text{ moles} \]

### Step 4: Calculate the number of molecules of oxygen
- Using Avogadro's number:
\[ \text{Number of molecules of O₂} = 0.25 \text{ moles} \times 6.022 \times 10^{23} \text{ molecules/mole} = 1.5055 \times 10^{23} \text{ molecules} \]

### Step 5: Calculate the ratio of the number of molecules of hydrogen to oxygen
\[ \text{Ratio} = \frac{\text{Number of molecules of H₂}}{\text{Number of molecules of O₂}} = \frac{12.044 \times 10^{23}}{1.5055 \times 10^{23}} \approx 8 \]

### Final Answer
The ratio of the number of molecules in 4 g of hydrogen to the number of molecules in 5.6 dm³ of oxygen at standard temperature and pressure is approximately **8:1**.

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