# Why can a system extract heat from an environment at constant temperature for free?

I was reading through Schroeder's Thermal Physics textbook, where I came across the following (in the section on Helmholtz free energy): "If the environment is one of constant temperature, the system can extract heat from this environment for free, so all we need to provide, to create the system from nothing, is any additional work needed."

My question is now: Why can a system extract heat for free when the environment is at constant temperature? Or is this the same as saying dW=-dQ for an isothermal process?

• You can extract some energy if the system's temperature is less than the environment simply due to the fact that heat flows from hot to cold. The quote is explaining Helmholtz free energy as something which tracks changes in energy which occur due to things other than heat exchange with the environment. Mar 30, 2019 at 16:26
• Octonion is correct. Helmholtz free energy is one of four so called thermodynamic potentials. For a compact description of what these mean, check out the following:hyperphysics.phy-astr.gsu.edu/hbase/thermo/helmholtz.html Mar 30, 2019 at 17:30
• I can't help you as I don't understand a single word of your quotation. What kind of book is that? "to create the system from nothing". I didn't know that thermodynamics was about creating things. Mar 30, 2019 at 20:28

Assume that the surroundings are infinitely large and at temperature $$T_\infty$$, and that the system and surroundings are in thermal contact
• If the system is at $$T > T_\infty$$ then there will be spontaneous heat transfer from the system to the surroundings until the system reaches $$T_\infty$$
• If the system is at $$T < T_\infty$$ then there will be spontaneous heat transfer from the surroundings to the system until the system reaches $$T_\infty$$
Note that it's not possible to heat the system above $$T_\infty$$ (or cool it below $$T_\infty$$) "for free." The only change that is "free" is bringing the system to $$T_\infty$$ and keeping it there.