I am struggling with the concept of entropy but might be on the verge of understanding it. The Clausius inequality, $$\mathrm{d}S \geq \mathrm{d}q/T,$$ is true for all spontaneous changes with the equality for reversible changes. But what if a change is not spontaneous or is that impossible i.e a change cannot be forced. Any help with answering this question or to help conceptualize and understand the entropy would be greatly appreciated.
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As you perfectly say, Clausius inequality states that $$\mathrm{d}s\geq \mathrm{d}q/T.$$ This has to be true for any process.
For reversible processes, what happens is that $\mathrm{d}s= \mathrm{d}q/T$, and for non-reversible processes, $\mathrm{d}s > \mathrm{d}q/T$. If for a given process $\mathrm{d}s< \mathrm{d}q/T$, it would not fulfill Clausius inequality.
Consider the situation where the system is isolated from its surroundings. In this case, $\mathrm{d}q_\mathrm{(sys)} = 0$ , and thus the Clausius relation implies $\mathrm{d}S\geq0$, so this tells us that the entropy of an isolated system cannot decrease in the course of a spontaneous change.
Since the universe is itself an isolated system, this result shows that we can use entropy as the signpost of spontaneous change. Processes are only spontaneous if they cause an increase in the total entropy of the universe.
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One way of thinking about this is to consider what happens when you clean a messy room. Your room is an example of an open system (i.e. heat, work and mass can move across its boundary). If you put a lot of work into your room, in the form of good old fashioned elbow grease, you end up with a nice, clean room that has a lower entropy than what it started with. However, it is important to note that the entropy of the universe, which is an isolated system that happens to contain the open system of your room, will have increased as a result of your work. This is because you could not do the work in a reversible fashion. Your body heat, friction from moving things around, etc. all produced irreversible changes in the heat flow to your surroundings. This is where the Clausius equation holds. To describe the entropy decrease in your room you have to know the equation of state that can provide the initial and final entropies. Not really practical for an abstract concept for a room, but for a fluid such as a gas or liquid it's usually clear-cut.
