Reversible reservoirs; entropy change expressions

Reversible heat transfer reservoir

A reversible heat transfer reservoir (thermal energy reservoir) is a very large system that exchanges energy only as heat, and whose temperature stays constant however much heat is added or removed. It is internally reversible: all the heat transfer takes place without friction or other irreversibility inside it. Examples: the atmosphere, a lake, a furnace.

For a reservoir at temperature T supplying or receiving heat Q, the entropy change is ΔS = Q/T (Q positive when heat enters it).

Reversible work transfer reservoir

A reversible work transfer reservoir (mechanical energy reservoir) is a system that exchanges energy only as work, with no heat transfer and no dissipation. Examples: a frictionless weight-and-pulley arrangement, a spring, a flywheel, a frictionless piston raising a weight. Its entropy stays constant.

Entropy as a property

By Clausius, ∮ (δQ/T)_rev = 0 for a reversible cycle. The integral ∫(δQ/T)_rev between two states is therefore path independent. This defines the property entropy:

dS = (δQ/T)_rev

Entropy change for reversible heat transfer

dS = δQ_rev/T, so S₂ − S₁ = ∫₁² δQ_rev/T.

For a reservoir at constant T: ΔS_reservoir = Q/T. When heat Q leaves a reservoir at T_H, ΔS = −Q/T_H. When it enters a reservoir at T_L, ΔS = +Q/T_L.

Entropy change for reversible work transfer

For reversible work, δQ = 0 and no dissipation occurs, so

dS = 0, ΔS = 0

The work reservoir undergoes an isentropic process. Work therefore transfers no entropy, while heat does.

Result: heat transfer changes entropy by δQ_rev/T; reversible work transfer changes it by zero.