Relationship between free energy, enthalpy, entropy?

Yes, the typical relation we use is the one at constant temperature for chemical reactions at standard state (#25^@ "C"# and #"1 bar"#):

#DeltaG_(rxn)^@ = DeltaH_(rxn)^@ - TDeltaS_(rxn)^@#

And this comes from the Maxwell Relations for the Gibbs' free energy and enthalpy in a thermodynamically-closed system (closed to mass transfer but not energy transfer), in terms of differentials:

#dG = -SdT + VdP#
#dH = TdS + VdP#

where #G# is Gibbs' free energy, #S# is entropy, #H# is enthalpy, #T# is temperature, #V# is volume, and #P# is pressure.

We can see that #VdP = dH - TdS#, so we obtain a relation for varying temperature:

#dG = -SdT + dH - TdS#

#~~ dH - SdT - TdS - overbrace(dSdT)^"small"#

#=> color(blue)(dG = dH - d(TS))#

and integration from state 1 to 2 would give:

#color(blue)(DeltaG) = DeltaH - Delta(TS)#

#= color(blue)(DeltaH - TDeltaS - SDeltaT - DeltaSDeltaT)#

where we have used similar steps to what is shown here.

As mentioned, the one we usually use is at constant temperature, so #-SDeltaT = -DeltaSDeltaT = 0# and:

#color(blue)barul|stackrel(" ")(" "DeltaG = DeltaH - TDeltaS" ")|#