# 5: Entropy and The Second Law of Thermodynamics


• 5.1: Energy Does not Determine Spontaneity
There are many spontaneous events in nature. If you open the valve in both cases a spontaneous event occurs. In the first case the gas fills the evacuated chamber, in the second the gases will mix. The state functions $$U$$ and $$H$$ do not give us a clue what will happen. You might think that only those events are spontaneous that produce heat.  The development of the new state function entropy has brought us much closer to a complete understanding of how heat and work are related.
• 5.2: Nonequilibrium Isolated Systems Evolve in a Direction That Increases Their Energy Dispersal
An isolated system, one that does not exchange heat with its surroundings, not at equilibrium will evolve in a direction that increases the overall energy dispersion of the state. The system will reach equilibrium when the energy dispersal, or entropy ($$S$$), is maximum.
• 5.3: Unlike heat, Entropy is a State Function
Entropy, $$S$$, is a state function, so it does not depend on the thermodynamic path. We can take any path we want to calculate the entropy of a thermodynamic system.
• 5.4: The Second Law of Thermodynamics
An isolated system is a little more than just adiabatic. In the latter heat cannot get in or out. In an isolated system nothing gets in or out, neither heat nor mass nor even any radiation, such as light. The isolated system is like a little universe all to itself.
• 5.5: The Famous Equation of Statistical Thermodynamics is S=k ln W
Entropy of can be calculated from the molecular viewpoint when considering the number of microstates that exist in a corresponding macrostate.
• 5.6: We Must Always Devise a Reversible Process to Calculate Entropy Changes
The second law of thermodynamics can be formulated in many ways, but in one way or another, they are all related to the fact that the state function entropy, $$S$$, tends to increase over time in isolated systems. The second law has important consequences for the question of how we can use heat to do useful work.
• 5.7: Thermodynamics Provides Insight into the Conversion of Heat into Work
Heat and work are both forms of transferring energy, and under the right circumstance, one form may be transformed into the other. However, the second law of thermodynamics puts a limitation on this. To go from work to heat is called dissipation and there is no limitation on this at all. In fact it was through dissipation (by friction) that we discovered that heat and work were both forms of energy. There is, however, a limitation on converting heat to work.
• 5.8: Entropy Can Be Expressed in Terms of a Partition Function
We have seen that the partition function of a system gives us the key to calculate thermodynamic functions like energy or pressure as a moment of the energy distribution. We can extend this formulism to calculate the entropy of a system once its Q is known. The derivation is shown on page 840 and involves the use of the Stirling approximation. The end result is
• 5.9: The Statistical Definition of Entropy is Analogous to the Thermodynamic Definition
The molecular formula for calculating entropy is directly related to the macroscopic, thermodynamic formula for calculating changes in entropy.
• 5.E: Entropy and The Second Law of Thermodynamics (Exercises)

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