What is the increase in entropy when `11.2 L of O_(2)` are mixed with `11.2 L of H_(2)` at STP?

Options

A. (a) `0.576 J/K`
B. (b) `5.76 J/K`
C. (c) `7.56 J/K`
D. (d) `2.76 J/K`

Answer

To find the increase in entropy when 11.2 L of O₂ are mixed with 11.2 L of H₂ at standard temperature and pressure (STP), we can follow these steps:

### Step 1: Calculate the number of moles of O₂ and H₂
At STP, 1 mole of any gas occupies 22.4 L. Therefore, we can find the number of moles for both gases.

For O₂:
\[ \text{Moles of O₂} = \frac{11.2 \, \text{L}}{22.4 \, \text{L/mol}} = 0.5 \, \text{moles} \]

For H₂:
\[ \text{Moles of H₂} = \frac{11.2 \, \text{L}}{22.4 \, \text{L/mol}} = 0.5 \, \text{moles} \]

### Step 2: Calculate the mole fractions of O₂ and H₂
The total number of moles when both gases are mixed is:
\[ \text{Total moles} = \text{Moles of O₂} + \text{Moles of H₂} = 0.5 + 0.5 = 1 \, \text{mole} \]

Now, we can calculate the mole fractions:
\[ \text{Mole fraction of O₂} (X_{O₂}) = \frac{\text{Moles of O₂}}{\text{Total moles}} = \frac{0.5}{1} = 0.5 \]
\[ \text{Mole fraction of H₂} (X_{H₂}) = \frac{\text{Moles of H₂}}{\text{Total moles}} = \frac{0.5}{1} = 0.5 \]

### Step 3: Use the formula for entropy of mixing
The formula for the change in entropy of mixing (\( \Delta S_{mix} \)) for two gases is given by:
\[ \Delta S_{mix} = -R \left( n_{O₂} \ln X_{O₂} + n_{H₂} \ln X_{H₂} \right) \]
Where \( R \) is the gas constant (8.314 J/(mol·K)), and \( n_{O₂} \) and \( n_{H₂} \) are the number of moles of O₂ and H₂, respectively.

Substituting the values:
\[ \Delta S_{mix} = -8.314 \left( 0.5 \ln(0.5) + 0.5 \ln(0.5) \right) \]
\[ = -8.314 \left( 0.5 \ln(0.5) + 0.5 \ln(0.5) \right) = -8.314 \left( \ln(0.5) \right) \]

### Step 4: Calculate \( \ln(0.5) \)
Using the value \( \ln(0.5) \approx -0.693 \):
\[ \Delta S_{mix} = -8.314 \times (0.5 \times -0.693 + 0.5 \times -0.693) \]
\[ = -8.314 \times (-0.693) = 5.76 \, \text{J/K} \]

### Final Answer
The increase in entropy when 11.2 L of O₂ are mixed with 11.2 L of H₂ at STP is:
\[ \Delta S_{mix} = 5.76 \, \text{J/K} \]

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