How does an increase in entropy increase stability, despite an increase in energy My understanding of the basics of entropy are as follows: there are more ways for a system to be arranged in a disordered state, than an ordered state.
I am happy with that explanation. When it comes to defining entropy in terms of energy, though, I am slightly confused. My notes (I am taking A-Level Chemistry) say the following things:

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*A system is more energetically stable when at a higher entropy

*A reaction is (more) feasible if it involves an increase in entropy

My last point was going to be: an increase in entropy is an increase in energy, although this isn’t explicitly stated. That was my understanding. Perhaps this is where I’m wrong?
When I see that ∆S is positive for an increase in entropy, that confuses me. When ∆S is positive, we are increasing the energy of the system, but apparently also making it more stable. I am unfamiliar with systems that are more stable at higher energy levels. If I think of a gas, if would seem to me that a more stable arrangement is becoming a liquid i.e. losing energy, however, this contradicts the entropy conclusion.
Perhaps I am visualising this problem in the wrong way. I am seeing the changes of state (in this example) to be like a potential well, where energy needs to be added to move the substance out of the well and change its state. Then, the lower energy state would be at the bottom of the well, but then, apparently, with higher entropy. (It’s not a perfect analogy but it serves to show my thinking).
 A: You need to clear the difference in entropy of the system and the universe. A phenomenon will happen spontaneously if the entropy of the universe is positive and won't happen if it is negative or zero (again, spontaneously). Now you may object by saying that why does water freezes spontaneously at STP? After all the entropy of ice is less than that of water hence there is a decrease in entropy of the system, hence this should not be spontaneous. I reply that you didn't read the statement correctly; it is written the entropy of the universe and not of the system. Yes, I agree that entropy of the system (water) decreases but it happens that at STP the entropy of the universe (the water and its immediate surroundings) have a net increase in entropy. Net entropy change of a universe is given by $\Delta S= \Delta S_{sour}+\Delta S_{sys}$. Calculating the net change in entropy is very tedious and hence we devise a new function, called Gibbs Free energy, which is a property of the system and easy to calculate. By finding the sign of it alone we are able to predict whether a phenomenon will happen spontaneously or not. In this case, it is the opposite of that of entropy, the negative sign represents a spontaneous process. I recommend you to read this book.
