Entropy
For any spontaneous process, total entropy change (DS_T) must be positive:
DS_T > 0    SPONTANEOUS
DS_T < 0    NOT SPONTANEOUS
DS_T = 0    EQUILIBRIUM

Free Energy
DS_T = DS_SYS  +  DS_SUR
DS_SUR  = -DH_SYS/T
DS_T = DS_SYS  - DH_SYS/T (multiply this equation by -T)
-TDS_T = -TDS_SYS  +  DH_SYS
Define DG  as -TDS_T(DG called Gibbs free energy)
DG =  DH - TDS   (since only using SYS subscript, omit SYS/SUR subscripts)
DG < 0   SPONTANEOUS
DG > 0   NOT SPONTANEOUS
DG = 0   EQUILIBRIUM

Explain:H2O(s)  H2O(l)(spontaneous at 20°C)
Solid Liquid  DS positive (favorable)
      Gains Heat    DH positive (unfavorable)
spontaneous only if TDS > DH
TDS = DH   at 0°C (equilibrium)
ice melts above 0°C
 

Explain:   H2O(l)  H2O(s)(spontaneous at -10°C)
Liquid Solid DS negative (unfavorable)
      loses Heat   DH negative (favorable)
spontaneous only if DH > TDS
TDS = DH   at 0°C (equilibrium)
water freezes below 0°C
 

Calculate temperature range when reaction can be spontaneous:
N2 + 3H2 2NH3
DH = 2DH(NH3)= 2(-46kJ) = -92kJ
DS = S2 - S1 = 2S(NH3) - 3S(H2)- S(N2) =
         2(192 J/K) - 3(131 J/K) - 192 J/K = -200 J/K

T = DH/DS = -92000J/(-200J/K) = 460K = 187°C
SPONTANEOUS FOR  T<187°C