Clausius–Duhem inequality

The Clausius-Duhem inequality tells us something important about how energy flows through things like engines and heat pumps. Imagine that you have a big bucket of water that's really hot on one side and really cold on the other. If you put a turbine inside, you could create electricity from the temperature difference. But there's a catch, because the process of turning the turbine will create friction and heat, which means that some of that energy is lost. The Clausius-Duhem equation sets a limit on how much energy can be converted into electricity in this way, and tells us that some energy will always be lost as heat. It's like saying that if you try to make a piggy bank fuller by taking out some coins, you can't end up with more coins than you started with. So scientists use the Clausius-Duhem inequality to understand the limits of energy conversion, and come up with ways to make processes more efficient.