# Gibbs Free Energy Equation: why is the entropy change of the system not q/T ?

Homework Statement:
The way we stipulate the conditions for the Gibbs Free Energy Equation (GFEE) - it seems to me that ΔS(system) must always equal negative ΔS(surroundings) ? Obviously this is incorrect but I can't see why. The GFEE says the magnitude of the entropy change of the surroundings is given by

ΔS(surr) = ΔH(syst)/T

Since the only exchange between the system-surroundings is the heat of ΔH(syst), then the change of heat in the system is -tautologically- also the same ΔH that the surroundings experience. So then surely the magnitude of the entropy change of the system is also the same quantity as the above expression for that of the surroundings. Because both system and surroundings have the same magnitude of heat change at the same temperature with no PV work or matter exchange i.e. it erroneously appears to me that

ΔS(syst) = ΔH(syst)/T = ΔS(surr)

But of course if this were true it would render the GFEE useless since ΔG would always be zero.
So why does the normal entropy equation ΔS=q/T not work for the entropy change of the SYSTEM in the GFEE?
What instead is the expression (or way to consider) the entropy change of the system in the GFEE ?
I have never seen this discussed in my Thermodynamics text books or anywhere online.
Relevant Equations:
ΔS(univ) = ΔS(surr) + ΔS(syst)
ΔS = q/T
q=ΔH
ΔS(surr) = ΔH(syst)/T
leading towards Gibbs:
ΔS(univ) = ΔH(syst)/T + ΔS(syst)
multiply by T to give Gibbs Free Energy Equation
T.ΔS(univ) = ΔG(univ) = ΔH(syst) - T.ΔS(syst)
What is the entropy change of the system in the Gibbs Free Energy Equation?
The general expression for entropy change is ΔS=q/T
The only exchange between the system and the surroundings is ΔH done reversibly, with no PV work and no matter transfer, therefore
q(syst) = ΔH(syst)
therefore surely the entropy change of the system is given by
ΔS(syst) = q(syst)/T
therefore
ΔS(syst) = ΔH(syst)/T
but I know this isn't correct, i just can't see why this expression for entropy change of the system is incorrect

## Answers and Replies

Chestermiller
Mentor
Problem Statement: The way we stipulate the conditions for the Gibbs Free Energy Equation (GFEE) - it seems to me that ΔS(system) must always equal negative ΔS(surroundings) ? Obviously this is incorrect but I can't see why. The GFEE says the magnitude of the entropy change of the surroundings is given by

ΔS(surr) = ΔH(syst)/T

Since the only exchange between the system-surroundings is the heat of ΔH(syst), then the change of heat in the system is -tautologically- also the same ΔH that the surroundings experience. So then surely the magnitude of the entropy change of the system is also the same quantity as the above expression for that of the surroundings. Because both system and surroundings have the same magnitude of heat change at the same temperature with no PV work or matter exchange i.e. it erroneously appears to me that

ΔS(syst) = ΔH(syst)/T = ΔS(surr)

But of course if this were true it would render the GFEE useless since ΔG would always be zero.
So why does the normal entropy equation ΔS=q/T not work for the entropy change of the SYSTEM in the GFEE?
What instead is the expression (or way to consider) the entropy change of the system in the GFEE ?
I have never seen this discussed in my Thermodynamics text books or anywhere online.
Relevant Equations: ΔS(univ) = ΔS(surr) + ΔS(syst)
ΔS = q/T
q=ΔH
ΔS(surr) = ΔH(syst)/T
leading towards Gibbs:
ΔS(univ) = ΔH(syst)/T + ΔS(syst)
multiply by T to give Gibbs Free Energy Equation
T.ΔS(univ) = ΔG(univ) = ΔH(syst) - T.ΔS(syst)

What is the entropy change of the system in the Gibbs Free Energy Equation?
The general expression for entropy change is ΔS=q/T
The only exchange between the system and the surroundings is ΔH done reversibly, with no PV work and no matter transfer, therefore
q(syst) = ΔH(syst)
therefore surely the entropy change of the system is given by
ΔS(syst) = q(syst)/T
therefore
ΔS(syst) = ΔH(syst)/T
but I know this isn't correct, i just can't see why this expression for entropy change of the system is incorrect
If you are talking about a system where a phase change is involved, then it is correct. If you are talking about a system involving a chemical reaction, then G changes as a result of changes in the amounts of the various species present. Which are you referring to?

See the following thread: https://www.physicsforums.com/threads/thermochemistry-challenge-problem-chets-paradox.913567/

DrDu
Science Advisor
The equation S=q/T holds true only for reversible reactions, i.e. for a reaction which moves through a sequence of equilibrium states. And in deed in equilibrium ##\Delta G=0##. When you see some ##\Delta G## values, they typically refer to some standard state, where all reactants are at a concentration (more precisely activity) of 1 mole/L, which is but rarely an equilibrium state.