In Wikipedia, the definition of entropy goes like this: $ d S = \dfrac{\delta q_{\rm }}{T}$. The literal interpretation of this equation is that some amount of heat transferred into a system, if the temperature of the system is high, the entropy is low. Wikipedia also said that the entropy stands for the "waste energy" which means the amount of thermal energy that cannot be used to do work. So entropy is a function of the state of a system.
My question is, why does the equation above mean "waste energy"?
There are a few similar questions I found here (like
The definition of entropy), but I don't think they address this "waste energy" concept.
Best Answer
Yes, this is a nice way of thinking about energy.
Consider the Christmas toy shown below. The candles create hot air, which rises through the fan blades and spins the angels around. This is a simple heat engine. It only works because there's a temperature difference between the hot air below and the cold air above. For example, you could put the engine in a room where the air was very hot, but if it was uniformly hot, the engine wouldn't spin.
This shows that if we let hot and cold matter reach an equilibrium, we have less energy available to perform mechanical work. There is just as much energy as before (energy is conserved), but not in a form we can use, because now the temperature is all uniform.
Now look at the equation $dS=\delta Q/T$. Say we have some hot matter at temperature $T_H$, and some matter at a lower temperature $T_L$. We allow some heat $\delta Q$ to be conducted from hot to cold. The entropy of the hot matter has changed by $-\delta Q/T_H$, while the entropy of the cold matter has changed by $\delta Q/T_L$. The latter number is positive and greater in absolute value, so the total change in entropy is positive. This tells us that there is now less energy available to do work. The energy is still there, but it's wasted for our purposes.