Entropy is not conserved

Entropy

• Unlike energy functions, entropy is not conserved in natural process or in isolated systems.
• For example, consider heat transfer between two objects with different temperature T_I and T_II (T_I > T_I)in an isolated system. The two objects are connected by a thermal conductor such as a copper wire. The heat will flow from the high temperature side (I) to the low temperature side (II). Assuming no heat loss, the heat from I will go into II, thus $Q$_$I$=−$Q$_$II$

• The internal energy change of the whole system (I+II) is thus zero.
  ∆$U$_$I+II$=$∆U$_$I$+$∆U$_$II$=$Q$_$I$+$Q$_$II$=0
  The energy is conserved.
  
• Let’s consider the entropy change of the whole system (I+II).
  

Insert the relation $Q$_$I$=−$Q$_$II$

The entropy of this isolated system is not conserved. This heat flow is spontaneous and irreversible and the irreversibility cause entropy increase.