PHYS3410 BIOPHYSICS II
Lecture Notes

Section I: Thermodynamics of Biological Systems

Lecture 2:Second Law of Thermodynamics: evolution of systems in time

Sadi Carnot (1796 – 1832) wanted to make the steam engine more efficient. His father Lazare Carnot was interested in efficiency of mechanical engines. He concluded that the most efficient engines should avoid contacts between bodies moving at different speeds.

Sadi Carnot applied similar principles to heat engines. Firstly, he observed that work is produced from the flow of heat from hot to cold reservoirs. Secondly, the most efficient heat engine needs to avoid contact between bodies of different temperatures. This is only possible in some parts of the operating cycle: for instance during volume expansion.

Reversibility

In a reversible process, the series of states that the engine goes through could be retraced  in the exact opposite order. A reversible engine can use same amount of mechanical work W to transfer same amount of heat Q from cold reservoir to hot reservoir.

Carnot’s cycle

Carnot realized that engine has to operate in a cycle: after producing work, the engine had to be reset to its initial state.

T_H, T_L are the operating temperatures of the engine, W is the work performed by the engine and Q_H and Q_L are amounts of heat taken from the high temperature reservoir and expelled into low temperature reservoir. The material, which is heated and cooled, is called working fluid.  The engine can run both ways: refrigerator uses work to move heat “uphill”.

Efficiency of heat engine, e, is the ratio of work performed to heat input from the hot reservoir:

(2.1)

(as tutorial exercise, show that W = Q_H – Q_L)

Carnot argued that reversible engine must produce maximum work compared to irreversible engine. More generally: all reversible engines must produce same amount of work from given amount of heat regardless of their construction.

Carnot showed that efficiency e can be rewritten as:

 (2.2)

(as tutorial exercise, prove this by computing total work W for one cycle)

If the efficiency of the reversible heat engine is independent of the physical and chemical nature of the engine, then the temperatures must be on an absolute temperature scale, as e cannot be greater than one. This fact was noted by Lord Kelvin (William Thomson, 1824 – 1907). 

	(2.3)
	(2.4)

Equation 2.4 is reminiscent of equation 1.5.

Entropy

Clausius (1822 – 1888) generalized eqn (2.4) for any closed cycle. He thought that the  invariance of Q/T might be useful and defined Entropy S  = Q/T (“Entropy” comes from Greek: “transformation”). Similarly to total energy for reversible process, S only depends on the initial and final states.

  (2.5)

The usefulness of this definition depends on the assumption that any two states can be connected by a reversible transformation. In a reversible process, system and reservoir have the same temperature when heat is exchanged and the entropy change of the reservoir is equal and opposite to the entropy change of the system.

In real, irreversible engines there is an energy dissipation in the piston as it performs work. Thus Q_L is greater than those in reversible cycle.

Q_Lirr > Q_L

   (2.6)

Since the irreversible engine returns to initial state in each cycle, there is no change in its entropy and to achieve this, the system expels more entropy (heat) to the exterior.

To gain better understanding of the real irreversible engine cycle we define change of entropy, dS, in each cycle as consisting of two parts:

dS = d_eS + d_iS   (2.7)

The d_eS term arises from exchange of heat between the engine and exterior and can be positive or negative. These processes involve “useful” energy, which can do work or reset the engine to be ready for the next cycle.

The d_iS term arises from dissipative processes, such as friction and turbulence and involve energy, which is “wasted”. This term is always positive.

There is no real system in nature that can go through a cycle of operations and return to its initial state without increasing the entropy of the exterior (universe). It is now thought that this degradation of energy gives the whole universe direction in time, time arrow: an evolution towards “heat death”, where no energy is available to do work. This will be accompanied by total disorder.

At present we do not have experimental evidence for the universe at large undergoing heat death: gravity, relativity and thermodynamics will have to be considered together to assess if this final doom is coming.