Understanding Helmholtz Free Energy: Get Heat for Free?

[aaaa202]

I'm having some trouble understanding this concept. Why is it that you sometimes can get heat for free from the environment? Like suppose you have a system, on which you make an energyconsuming proces which creates entropy. Then you subtract TΔS because apparently heat can enter when the entropy increases - why is that? Does the temperature drop, when we increase the entropy?
Can you guys please explain in terms of microstates too, as it is that way I feel I understand it the best.
Also, how do you know that the temperatures between the environment and the system is the same AFTER it has absorbed exactly TΔS?

 

[Valerie Park]

Heat can enter a system from the environment because of the Second Law of Thermodynamics. This law states that entropy always increases over time, and that the entropy of an isolated system can never decrease. Entropy is related to the number of microstates available for a given macrostate. As the entropy of a system increases, so does its number of microstates. The increase in microstates allows energy to enter the system from the environment, resulting in an increase in temperature. When a system absorbs exactly TΔS of heat from the environment, the temperatures between the environment and the system are the same AFTER the transfer of heat. This is because the amount of heat transferred is equal to the temperature difference multiplied by the entropy change. Since the temperatures are now equal, there is no longer a temperature difference between the two and therefore no more heat can be transferred.