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Entropy is a central concept in thermodynamics that quantifies the degree of disorder or randomness in a system. In this exercise, we look at the change in entropy when a barrier is removed between two gases, allowing them to mix freely. When the partition separating the two gases is punctured, the gases expand into the entire available space, resulting in a new distribution of molecules.The entropy change due to the mixing of gases can be calculated using the formula:\[ \Delta S = -nR(x_1\ln x_1 + x_2\ln x_2) \]where:

• \( n \) is the total number of moles of gas.
• \( R \) is the ideal gas constant.
• \( x_1 \) and \( x_2 \) are the mole fractions of the gases involved.

After mixing, each type of gas molecule is equally likely to be found in each half of the box, so both \( x_1 \) and \( x_2 \) are 0.5. This results in an increase in entropy, which aligns with the second law of thermodynamics stating that the total entropy of an isolated system can only increase.

When the two gases in the exercise are allowed to mix by removing the barrier, they spread out evenly across the available volume. This is because the gases are at the same temperature, meaning they have the same average kinetic energy. As they mix, each molecule of gas has an equal probability of occupying any given section of the combined volume. Before the barrier is removed, the gases are separated, with nitrogen on one side and oxygen on the other. Upon reaching equilibrium after mixing, the gas molecules are randomly distributed, forming a homogenous mixture. In our specific exercise, this leads to 250 nitrogen molecules and 50 oxygen molecules on each side, on average. This reflects the natural tendency of systems to move towards a state of higher entropy, where molecules are more spread out evenly.

The process of gas mixing also involves understanding probability distributions, especially when evaluating the likelihood of a particular molecular arrangement. Initially, 500 nitrogen molecules are confined to one side, while 100 oxygen molecules are on the other. After the partition is removed, the gases randomize.The probability that the molecules will revert precisely to the initial configuration is calculated using binomial probability. For any single nitrogen molecule, the probability of being in its original half after mixing is 0.5, and similarly for oxygen. Thus, the probability for all nitrogen to remain on one side is \( (0.5)^{500} \), and for oxygen, it is \( (0.5)^{100} \). These probabilities are each incredibly low due to the large number of particles involved, highlighting the improbability of spontaneously returning to the original high-order state after random mixing.