Entropy Change in the Mixing of Ideal Gases

When two ideal gases are allowed to mix in an isolated system, the entropy change can be determined by considering the factors that contribute to the overall change in entropy. The sign of the entropy change depends on the nature of the mixing process and the initial conditions of the gases.

1. Entropy and Mixing of Ideal Gases
Entropy is a measure of the randomness or disorder in a system. In the context of gases, entropy is related to the distribution of molecular velocities and positions. When two ideal gases are mixed, their molecules spread out and occupy a larger volume, increasing the overall disorder of the system.

2. Entropy Change for Mixing Process
The entropy change for the mixing of ideal gases can be calculated using the formula:

ΔS = nR ln(Vf/Vi)

where ΔS is the change in entropy, n is the total number of moles of gas, R is the gas constant, Vf is the final volume, and Vi is the initial volume.

3. Effect of Pressure and Temperature
If the two gases have the same pressure and temperature, their initial volumes will be different. When the gases are allowed to mix, their volumes will equalize, resulting in an increase in the overall volume. This increase in volume leads to an increase in entropy.

4. Sign of Entropy Change
Since the entropy change is determined by the natural logarithm of the ratio of final and initial volumes, the sign of the entropy change depends on the relative volumes of the gases.

If the final volume (Vf) is greater than the initial volume (Vi), the natural logarithm of the ratio will be positive, leading to a positive entropy change. Conversely, if the final volume is smaller than the initial volume, the natural logarithm of the ratio will be negative, resulting in a negative entropy change.

5. Conclusion
In summary, when two ideal gases under the same pressure and temperature are allowed to mix in an isolated system, the entropy change will be positive if the final volume is greater than the initial volume. This is because the mixing process leads to an increase in the overall disorder or randomness of the system. Conversely, if the final volume is smaller than the initial volume, the entropy change will be negative, indicating a decrease in the system's disorder.