![]() Unless you also stirred vigorously, in which case you added energy making more motion states available, so entropy went up. water and water, entropy doesn’t change: the situation hasn’t changed. In the example of the coffee and milk, that mixture is complete and irreversible, so the entropy has gone up. Each bit can now be in more places, so there are more possible states of the final product. Generally, the entropy will go up when you mix materials. Add a surfactant to the oil and it can also be turned into an emulsion. Milk and water have far less surface energy so they mix easily. In the case of oil and water there is a surface energy along the interface between the substances that makes $\Delta U$ big and proportional to the interface surface area - hence converting oil and water into an even mixture is not favoured. ![]() Normally a big increase in entropy ($\Delta S \gg 0$) is enough to cause this and can even overcome an increase in internal energy (as is the case of dissolving ammonium nitrate in water, which is endothermic and cools the mixture).īut if the increase in internal energy would be big enough then even a big entropy gain may not be enough to outweigh it. However, the change in free energy of a system $\Delta H = \Delta U-T\Delta S$ matters for whether a change can happen spontaneously: this happens if $\Delta H<0$ ($U$ is the internal energy). So generally a mixture will have a higher entropy (and it will take energy to purify it into the component substances). The entropy of a mole of a mixture is $$S=RT \sum_i x_i \log(x_i)$$ where $R$ is the universal gas constant, $T$ is the temperature and $x_i$ is the molar fraction of substance $i$ $\sum_i x_i =1$.
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