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What is free energy?

Free energy is (Delta G = Delta H - T Delta S) — but where does it comes from and why does a negative (Delta G) mean a reaction will happen?
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Previously, we talked about spontaneity, which  is whether a chemical reaction will happen   or not. And also entropy. These are closely  related concepts. A reaction will only happen   if the total entropy of the universe increases,  that’s the second law of thermodynamics.  But sometimes entropy goes down locally. After  all, things freeze, and their entropy reduces.   That can only happen if the entropy  goes up somewhere else to compensation.  So now it’s time to look back at free energy,  and where that comes from.
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How can we even   begin to calculate the entropy change of the whole  universe to see if something is going to happen…  Actually, that’s easier than you might  think, because what we’re interested in   are entropy changes, and all we need  is the entropy change of a system,   and what that system then  does to its surroundings.  We can calculate entropy changes of systems by  looking at changes in standard molar entropies.   Remember, unlike enthalpy and internal energy,  it’s possible, in principle, to calculate absolute   values of entropy. It’s a defined thing. And  we can also do some experiments to determine   or estimate entropy changes of reactions, too. If we work that out, we have the   entropy change of a system.
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But what about surroundings?  We also know that energy, in the form  of heat, is going to be exchanged   with the surroundings. And, in principle,  if that process is approximately reversible,   we know how to calculate entropy. It’s that  heat transferred, q, divided by temperature.  And the heat transferred to the  surroundings, is the heat that’s lost   by the system! So that’s something we can find  from experiment. We’ve covered calorimetry,   so if you can measure an enthalpy change, you can  measure the entropy change of the surroundings. 
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So, let’s start to put this all together.  The entropy change of the surroundings is the  enthalpy released into it, divided by temperature.   That comes from one of our definitions of entropy.  It’s effectively a measure of how that heat is   diffused into the rest of the universe. Then we know the entropy change of the   system.
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This is something we can look up  and predict, or determine from experiment.  Finally, the entropy of the whole universe is  just the sum of these, so we add them together  We could leave this here, but to make it a little  more convenient, we can do some more manipulation.   So, we start with this equation that says  the entropy change of the whole universe is   a sum of the entropy change of a system, and the  entropy change of the surroundings, which comes   from the heat added, or sometimes taken from, it. We then multiply by temperature, which at least   tidies it up a little, and then we do  a little rearrangement.
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This gets us   the equation we saw at the very beginning of  this section the one for Gibb’s free energy.  
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This is a great equation because it tells us  the entropy change of the whole universe itself,   knowing only the enthalpy and entropy  changes of a system that we’re looking at. 
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If free energy of a reaction is negative,  the entropy of the universe will   increase. It’s spontaneous, and will happen. If  the free energy is positive, the entropy of the   universe will decrease, so it won’t happen – we  would have to put in some additional energy, which   would increase the entropy of the  universe through a process elsewhere. 
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That’s the basics of where free energy comes from,   but what about some applications? That free  energy equation has a temperature dependence,   which becomes important. So let’s look at it.   We want to look only at the sign of enthalpy  and entropy here, is it positive or negative?  If the enthalpy change is negative, meaning heat  is released into the system, and the entropy   change is positive, meaning that it increases  for the system, then free energy is always   negative. The reaction is spontaneous. That  should make sense.
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An increase in entropy   of the system is allowed, and dissipating heat  into the surroundings always increases entropy.  If, on the other hand, enthalpy is  positive, meaning that it’s absorbing   heat from the surroundings, and entropy  of our system or chemical reaction is   negative, meaning that it’s becoming less  disordered, then free energy is positive.  Again, this should make sense. Things don’t just  absorb heat and become more arranged and more   ordered over time spontaneously. But, what if they have the same   sign what if enthalpy and  entropy were both positive,   or both negative?
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Then the sign of the free  energy change has a temperature dependence.  If enthalpy is positive, then we need a higher  temperature to make the free energy negative.   If enthalpy is negative, and  entropy is also negative,   then a higher temperature will make the free  energy change positive, and stop the reaction.  This is a useful point to understand because  we can calculate the temperature that this will   happen at if we know enthalpy and entropy  changes, and set the free energy to zero. 
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That’s pretty much it for an introduction to  free energy. It’s a way of measuring the entropy   change of the whole universe, just by looking  at the system alone. And if it’s negative,   the reaction will happen,  if it’s positive, it won’t.  That doesn’t say anything  about the rate of reaction,   however – spontaneous reactions can still be  very slow, but it’s still the key determining  
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factor in controlling the most fundamental idea  of chemistry: can we make this reaction happen?

Free energy is (Delta G = Delta H – T Delta S) — but where does it comes from and why does a negative (Delta G) mean a reaction will happen?

Total entropy

The Second Law of Thermodynamics states that entropy (of everything) must increase for a reaction or process to happen. Since we’re thinking in terms of systems and surroundings, then the entire universe is just the sum of both of these:

[Delta S_{universe}=Delta S_{system:}+Delta S_{surroundings}]

Entropy of systems

The entropy change of a chemical reaction can be calculated from standard molar entropies. These can be looked up in a table, and then the difference between products and reactants found:

[Delta S_{system}=sum nS_{products}-sum nS_{reactants}]

It’s important that if you do this, you need to multiply by the number of moles of reactant involved.

Entropy of surroundings

The definition of entropy for a reversible process is the heat transferred divided by temperature:

[frac{q_{rev}}{T}=Delta S]

And the heat transferred for a reaction is the enthalpy change:

[-frac{Delta H_{system}}{T}=Delta S_{surroundings}]

Note: 1) This is the entropy change of surroundings, and not related to the entropy change of the system and (2) the negative sign, as we’re looking at surroundings, which have absorbed the heat released by a reaction, or provided the heat consumed by it. So the sign of the heat transfer is opposite.

Free energy

To get the overall change of entropy for the whole universe, these need to be added together:

[Delta S_{universe}=Delta S_{system}-frac{Delta H_{system}}{T}]

This rearranges to the equation for Gibb’s free energy:

[Delta G=-TDelta_{universe:}=Delta H_{system}-TDelta_{system}]

This gives us a function for the entropy change of the entire universe (system + surroundings) that depends only on the system and what it does to the surroundings.

Temperature dependence

Looking at the equation for free energy, there’s a clear temperature dependence:

[Delta G=Delta H-TDelta S]

Whether ΔG is positive or negative will depend on the sign (+ or -) of the enthalpy and entropy changes, and temperature:

ΔH ΔS ΔG
+ Always negative/spontaneous
+ Always positive/non-spontaneous
+ + Negative/spontaneous at higher temperature
Negative/spontaneous at lower temperature
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Introduction to Thermodynamics

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