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What is entropy?

Entropy is a thermodynamic measure of 'disorder', which must always increase — but how can we measure it? 
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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. How can we even begin   to calculate the entropy change of the whole  universe to see if something is going to happen… 
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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.  But what about surroundings? We also know that energy,   in the form of heat, is going to be exchanged  with the surroundings.
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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. If you can measure an  enthalpy change through calorimetry, you can   measure the entropy change of the surroundings. So, let’s start to put this all together. It’s   a bit of brutal no-nonsense algebra from this  point, but the final equation is the key point.  The entropy change of the surroundings is the  enthalpy released into it, divided by temperature.   That comes from one of our definitions of  entropy.
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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.   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 
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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.   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. 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,  
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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. 
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But, what if they have the same sign  what if enthalpy and entropy were both   positive, or both negative? 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  
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key determining factor in controlling  the most fundamental idea of chemistry:   can we make this reaction happen?

Entropy is a thermodynamic measure of ‘disorder’, which must always increase — but how can we measure it?

Reversibility

A reversible process is one that can occur in infinitesimal steps – that is, steps that are increasingly small, until they are at a limit where the changes to the system is negligible.

If change is negligible, then the change can be reversed easily, and the system can be restored. Any energy exchanged in the forward reaction is perfectly restore by the backward reaction.

Examples in gases include most expansions and compressions. Examples in chemical reactions include anything happening at an equilibrium position. For example, a solid at its melting/freezing point is in equilibrium with its liquid.

At the freezing/melting point, solid and liquid forms can happily coexist, and moving between them is a reversible process (melting is only irreversible above the melting point, freezing is irreversible below it).

The heat change for a reversible process has an entropy associated with it:

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

The entropy change is the heat exchanged in the reversible process divided by temperature. This leads to entropy units of kJ K-1 (or molar quantities of kJ K-1 mol-1).

Absolute values of entropy

Unlike enthalpy and internal energy, absolute values of entropy can be known. The entropy of a perfect crystal is zero at absolute zero. This is the Third Law of Thermodynamics, and is a starting point for defining absolute values of entropy.

A second approach is to define it through a statistical approach. Here, entropy is a function of the number of ways of arranging particles in the system. If there is only one state and one particle, there is only one arrangement.

If there are two states and two particles, there are two arrangements. These arrangements are known as microstates, and entropy is a function of the number of them.

Entropy is defined as the natural log of the number of microstates, multiplied by the Boltzmann constant.

[S=k_BlnOmega]

You should remember that the Boltzmann constant is a ‘per molecule’ version of the gas constant, which is ‘per mole’.

There is a lot more detail on this subject, but this page is here to remind you that entropy is more than just disorder.

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Introduction to Thermodynamics

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