The change in entropy when mixing gases can be calculated using statistical mechanics principles. In this case, we have a mixture of O₂ and N₂.
Understanding the System
- We have 5 moles of O₂ and 5 moles of N₂.
- The process occurs at constant temperature and pressure.
Entropy Change Formula
The entropy change (ΔS) for mixing two ideal gases can be calculated using the formula:
\[
\Delta S = -nR \left( x_1 \ln x_1 + x_2 \ln x_2 \right)
\]
Where:
- \( n \) = total moles of gas
- \( R \) = universal gas constant (8.314 J/mol·K)
- \( x_1 \) and \( x_2 \) = mole fractions of the gases
Calculating Mole Fractions
- Total moles (n) = 5 moles O₂ + 5 moles N₂ = 10 moles
- Mole fraction of O₂ (\( x_1 \)) = \( \frac{5}{10} = 0.5 \)
- Mole fraction of N₂ (\( x_2 \)) = \( \frac{5}{10} = 0.5 \)
Plugging Values into the Formula
Now substitute \( n = 10 \), \( R = 8.314 \, \text{J/mol·K} \), \( x_1 = 0.5 \), and \( x_2 = 0.5 \) into the entropy change formula:
\[
\Delta S = -10 \times 8.314 \left( 0.5 \ln(0.5) + 0.5 \ln(0.5) \right)
\]
\[
\Delta S = -10 \times 8.314 \times \left( 0.5 \times -0.693 + 0.5 \times -0.693 \right)
\]
\[
\Delta S = 10 \times 8.314 \times 0.693
\]
Final Calculation
Calculating the final value:
\[
\Delta S \approx 57.9 \, \text{J/K}
\]
Thus, the change in entropy when mixing 5 moles of O₂ and 5 moles of N₂ is approximately 57.9 J/K. This value indicates an increase in disorder as the gases mix.