Chemistry - Why does entropy increase when the difference in temperatures is decreased?
Consider two tanks of water. One hot, one cold.
You open a valve between the two. The temperature between the two bleeds, and the hot tank loses entropy, and the cold tank gains entropy. However, entropy is also gained just by this process happening, as it would be impossible to force the 'cold' tank to retransfer the heat back to the 'hot' tank.
Also, you must consider something else. The two conjoined systems now have more molecules than either system alone, increasing the 'randomness' as a whole. Considering dice, it's like having two sets of two, then one set of four. The set of four has more possibilities than just the set of two.
Because the same amount of heat has different effects on the two systems in terms of entropy. The difference in temperature is just an arrow for the transformation, but for entropy, it's the individual temperatures and the total exchanged heat that matters.
The situation is equivalent to having a large room filled with people chatting, taking a small group out of it and moving their conversation to a public library. The room will pretty much stay noisy, but the library will have a huge increment in noise. Same with heat. You moved some heat from one entity to another, but one had a lower temperature. That heat has a huge impact, increasing the entropy of the low temperature body more than the decrease in entropy of the higher temperature body.
Temperature is a tricky concept, because a "uniform" temperature does not mean the system has a lot of order. Uniform temperature merely means the statistical distribution of particle velocities is homogenous.
Before the systems are combined, you know a fair amount about the system. For instance, you know particles on the "cold" side are going to stay on the cold side, and particles on the "warm" side are going to stay on the warm side. Informally, this means you know something about every particle in the system. Once you combine the systems, you no longer know where the particles are going to be, so there is more disorder to the system.
During the time that there are temperature gradients present, the combined entropy of the system is increasing with time. If the initial state is two blocks of material at different temperatures, and the final state is the same two blocks both at the same temperature, then there are a hugely greater number of quantum mechanical energy states available in the final state than in the initial state. The final state includes all the energy states that were available in the initial state (since these are still available to the final state within the framework of conservation of energy) plus many more states that were not available in the initial state.