Second Law of Thermodynamics

Second Law for isolated systems.
For any process in an isolated system the entropy will tend to increase

$$\Delta S ~\geq~ 0 ~~~~~~~(2nd~Law, ~isolated~system)$$

We will differentiate the two cases

$$\Delta S =  0       (reversible process, isolated system)$$

$$\Delta S >  0       (irreversible process, isolated system)$$

Second Law for closed systems.
For any process in a closed system the total entropy will tend to increase

$$\Delta S_{tot} \geq  0     (2nd Law, closed system)$$

where the total entropy change is defined as

$$\Delta S_{tot} =  \Delta S +  \Delta S'$$

$\Delta S$ - entropy change in the system
$\Delta S'$ - entropy change in the surroundings
Again, we will differentiate the two cases

$$\Delta S_{tot} ~=~ 0 ~~~~~~~(reversible~process, ~closed~system)$$

$$\Delta S_{tot} >  0       (irreversible process, closed system)$$

Comments

• The Second Law predicts the change in entropy for reversible processes. For irreversible processes, it only predicts the direction of change.
• There is a third kind of process we can conceive of, for which
  
  $$\Delta S <  0       (isolated sys)$$
  
  
  or
  
  $$\Delta S_{tot} <  0       (closed sys)$$
  
  
  According to the Second Law, such processes wil not be observed in nature. These are called "impossible" or "non-spontaneous" processes. In view of our statistical analysis above, we know that such processes will practically never occur for macroscopic systems.
• Note the philosophical difference between the First and Second Law. The First Law is exact, energy conservation is expected to hold always ndependent of system size. The Second Law only predicts the direction of change and processes violating it will not be observed because they are practically impossible for macroscopic bodies.

The two forms of the law are equivalent. Let us start with the Second Law for closed systems

$$\Delta S_{tot} =  \Delta S +  \Delta S' \geq  0$$

and apply it to an isolated system. We can think of the isolated system as a special kind of open system for which

$$\Delta S' =  0$$

This leads to

$$\Delta S_{tot} ~=~ \Delta S ~\geq~ 0$$

where the final inequality is exactly our 2nd Law for isolated systems.

Alternatively, let us start with the Second Law for isolated systems and appply it the the isolated "supersystem" consisting of our system plus its surroundings. The isolated system form of the 2nd Law tells us that the entropy change of the isolated supersysem is

$$\Delta S_{tot} \geq  0$$

but the total entropy change may be broken down into a sum of changes in system and surroundings:

$$\Delta S_{tot} =  \Delta S +  \Delta S'$$

which leads to the open system form of the 2nd Law

$$\Delta S_{tot} =  \Delta S +  \Delta S' \geq  0$$

Second Law at p,T=constThe Second Law of thermodynamics at p,T=const may be expressed as

$$\Delta G \leq  0     (closed sys; pV work; p,T=const)$$

i.e. "for any process in a closed system with pV work only and at p,T=const the Gibbs free energy tends to decrease".

The two possibilities are

$$\Delta G ~=~ 0 ~~~~~~~(reversible~process;~closed~sys;~pV~work;~p,T=const)$$

$$\Delta G ~<~ 0 ~~~~~~~(irreversible~process;~closed~sys;~pV~work;~p,T=const)$$

For justification see Gibbs free energy section.

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