Entropy

Apparently next week actually means next month. Sorry for the delay, I have been busy!
Following the last post regarding entropy and Gibb's calculations I decided to change the aim of this to entropy. This was originally written as an essay for physics so please excuse any dodgy wording or citation marks. Furthermore, it is a 1st draft - so it may have mistakes and other issues. One good side effect of this is it has a proper list of sources at the end!

Entropy is a measure of the multiplicity of a thermodynamic system, or how disordered a system is - the higher the entropy, the more ways in which it can be arranged, and generally this means that it will be more disordered. Imagine a brick wall - it is ordered, so has a low entropy, but if knocked down it will become more disordered - it will have a higher multiplicity as the fragments of wall can be ordered in more ways than the original wall, and hence a high entropy. It is often easier to knock down a wall than build a wall, meaning that generally entropy increases. It can also be seen as the amount of energy unavailable to do work. If energy is stored in a low entropy glucose molecule it can be used to work. If the energy is spread throughout some paper as heat then it is much harder to be used to do work, and so the paper has higher entropy.
According to the second law of thermodynamics the entropy of the universe will always increase, and over time the multiplicity of the universe will increase. This means that entropy is one of the only physical quantities which works in a direction in time. If time travel ever occurred entropy would be running in reverse. For example, while a thermometer can go up and down, it would be very unlikely to see a smashed window suddenly reassemble, yet as entropy would be in reverse (ie decreasing) it would reassemble easily.
Due to the second law of thermodynamics predicting the increase of entropy it has an important role to play in the predicting how chemical reactions will occur. The first section of this report will concern itself with how entropy changes work, moving on to Gibb’s Free Energy, and the last section will be about the heat death of the universe and black holes.

Physical Uses of Entropy

The idea that entropy always increases may appear rather odd - if entropy always increases, how come water can be frozen? How come brick walls can be built? The reason these things can occur is due to the fact that it is the universe’s entropy, \Delta S_{univ}, which increases, not necessarily the entropy of the system, \Delta S_{sys}. As long as the sum of the change in entropy of a system and the change of entropy in its surroundings is positive, ie the entropy over all increases, it can take place. Therefore we can say that

\Delta S_{univ} = \Delta S_{surr} + \Delta S_{sys} > 0

And in order for a reaction, or physical action to take place, \Delta S_{univ} must be positive. For example, when freezing ammonia the reaction is exothermic, so heat is lost to the surroundings. This means that the surroundings get more energy and so become less ordered, therefore while the entropy of the frozen ammonia, \Delta S_{sys}, has decreased, in doing so the entropy in the surroundings has increased, meaning the overall change in the universe is positive. On the other hand, when ammonia is melting the \Delta S_{surr} is negative - it is endothermic so takes in heat. Luckily for the Laws of Thermodynamics, solid ammonia melting produces liquid ammonia, and the molecules in liquid ammonia can be arranged in more ways than in solid (they are free to move, so have higher entropy) - so \Delta S_{univ} is still positive.
This relationship is very useful for working out the temperatures at which reactions like this will occur. Another example may be the brick wall. In order for a brick wall to be built by hand, carbohydrates are broken down by the body which increases the multiplicity of molecules in the body - the multiplicity of the surroundings increases, so the total entropy change of the universe is positive.
In relation to solid ammonia and liquid ammonia, as above, entropy allows for the calculation of the freezing point of ammonia. The entropy change of the surroundings of a material can be calculated as follows;

\Delta S_{surr} = \frac{\Delta q_{surr}}{T_{surr}}

Where q_{surr} is the heat absorbed by the surroundings - the enthalpy change - (measured in Joules), and T_{surr} is the absolute temperature of the surroundings (in Kelvin). Typically, q_{surr} is given per mole which gives the entropy per mole. The reason T_{surr} is the divisor is because the hotter something is, the lower the change in entropy for a given additional amount of energy - imagine a brick wall being pushed over, and a pile of bricks being pushed over - the brick wall has a larger entropy change as it is going from very orderly to not.
Given that \Delta S_{univ} must be positive, this allows the calculation of the freezing/melting point of ammonia. From looking in the US NIST Chemical WebBook we can find that the entropy change of fusion (this is the entropy change when melting - fusion is another name for melting) for ammonia is 28.93JK^{-1}mol^{-1}. As we are looking at when it is freezing we can change the sign - the entropy change for melting is opposite the entropy change of freezing. We then need the standard enthalpy change of fusion , which the same data source tells us is 5653Jmol^{-1}. Putting this into the above formulas gets us the following;

\Delta S_{univ} = \Delta S_{sys} + \frac{\Delta q_{surr}}{T_{surr}} = -28.93 + \frac{5653}{T_{surr}}

And by plotting this we can see where the entropy of the universe is positive, and therefore the fusion point of ammonia.

Screenshot from 2015-10-08 18:32:37

As we can see in Figure 1, the entropy change of the universe is positive whenever T_{surr} < 195.4K. From looking in the aforementioned data book we find that the temperature of fusion, T_{fus} is 195.4K - so we have the correct point. This same relationship works for any other thermodynamic system.

Gibbs Free Energy

Gibbs Free Energy is a way of doing the same calculations as we just did to work out the fusion point of ammonia, only it considers purely the system and not the surroundings which makes it easier to deal with. The Gibbs Free Energy is in essence the amount of energy needed on top of the energy contained within the system for a reaction to go - so if it negative then the system has sufficient energy, and some free energy, however only works at constant pressure. The equation for Gibbs Free Energy is given as follows;

\Delta G = \Delta q_{sys} - T_{sys}\cdot \Delta S_{sys}

This is derived from the same equations we used previously by assuming that T_{surr} = T_{sys} - as both are in the same environment - and the knowledge that q_{surr} = -q_{sys} (because energy leaving the system goes to the surroundings). This allows the following to occur;

\Delta S_{univ} = \Delta S_{sys} + \frac{q_{surr}}{T_{surr}} = \Delta S_{sys} - \frac{q_{sys}}{T_{sys}}

And by multiplying this through by -Tsys we gain the following;

-\Delta S_{univ}\cdot T_{sys} = q_{sys} - \Delta S_{sys}\cdot\Delta T_{sys} = \Delta H_{sys} - \Delta S_{sys}\cdot \Delta T_{sys} = \Delta G_{sys}

Which gives us the Gibbs free energy. If this is lower than 0 then it follows that the reaction can is thermodynamically favourable and will occur spontaneously as there must be a decrease in the overall energy in the system for the reaction to be spontaneous at a specific temperature. For example, we can show the fusion point of Ammonia again;

\Delta G = -5653 - (195.4\cdot -28.93) = 0Jmol^{-1}

As \Delta G is 0Jmol^{-1} we know that this is the fusion point - if T_{sys} was any lower then \Delta G would be negative and it would freeze (given that these are the enthalpy change of freezing and the entropy change of freezing).

Heat Death of the Universe

Given that \Delta S_{univ} is always positive it follows that the amount of entropy, or disorder, in the universe is always increasing. This leads to a theory regarding how the universe will end because the increasing entropy implies that as time goes by the universe will be reaching its thermodynamic equilibrium and will reach its maximum entropy - at the moment it is not in equilibrium as it requires more entropy. At this point there will be no more free energy, as needed for spontaneous reactions and as such no more reactions will be able to progress, and as such the universe will die.

For an example of this, imagine you have a room into which you spray an aerosol. The fumes from the aerosol will diffuse into the room and you will eventually reach a point at which the aerosol has fully diffused and so there is no longer an area with a large quantity of aerosol and another without. In the same way, eventually it is theorised that the universe will eventually balance out and reach an equilibrium point at which point there it cannot do anything more.

Conclusion

Entropy is one of the major ideas from Thermodynamics - it is in essence, the 2nd Law of Thermodynamics, and therefore is very important for both physics and chemistry. Entropy plays a part in cosmology as shown in the section on black holes, and it could be a cause for the end of the universe - as the universe reaches thermodynamic equilibrium and no more free energy remains. In terms of chemistry entropy is very useful for calculating whether or not a reaction will occur.

References


[1]   
NIST National Institute of Standards and Technology
http://webbook.nist.gov/cgi/cbook.cgi?ID=C7664417&Units=SI&Mask=4#Thermo-_Phase


[2]   
ChemWiki Standard Enthalpy Changes of Formation http://chemwiki.ucdavis.edu/Reference/Reference_Tables/Thermodynamics_Tables/T1\%3A_Standard_Thermodynamic_Quantities_for_Chemical_Substances_at_25\%C2\%B0C


[3]   
ChemWiki Calculating Entropy Changes http://chemwiki.ucdavis.edu/Physical_Chemistry/Thermodynamics/State_Functions/Entropy/Calculating_Entropy_Changes


[4]   
4College Entropy and the Second Law of Thermodynamics
http://www.4college.co.uk/a/O/entsurr.php


[5]   
EntropyLaw Entropy and the Second Law of Thermodynamics
http://www.entropylaw.com/entropy2ndlaw.html


[6]   
LiveScience What is the Second Law of Thermodynamics?
http://www.livescience.com/50941-_second-_law-_thermodynamics.html Jim Lucas May 22,
2015


[7]   
HyperPhysics Second
Law: Entropy http://hyperphysics.phy-_astr.gsu.edu/hbase/thermo/seclaw.html


[8]   
HyperPhysics Entropy as Time’s Arrow
http://hyperphysics.phy-_astr.gsu.edu/hbase/therm/entrop.html


[9]   
Texas A&M University Thermodynamics: Gibbs Free Energy
https://www.chem.tamu.edu/class/majors/tutorialnotefiles/gibbs.htm


[10]   
Bodner Research Web Gibbs Free Energy
http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch21/gibbs.php


[11]   
UCDavis ChemWiki Gibbs Free Energy http://chemwiki.ucdavis.edu/Physical_Chemistry/Thermodynamics/State_Functions/Free_Energy/Gibbs_Free_Energy


[12]   
HyperPhysics Gibbs Free Energy
http://hyperphysics.phy-_astr.gsu.edu/hbase/thermo/helmholtz.html


[13]   
How Chemical Reactions Happen James Keeler, Peter Wothers March 27, 2003


[14]   
HyperPhysics Gibbs Free Energy and the Spontaneity of Chemical Reactions
http://hyperphysics.phy-_astr.gsu.edu/hbase/chemical/gibbspon.html


[15]   
UCDavis ChemWiki Second Law of Thermodynamics http://chemwiki.ucdavis.edu/Physical_Chemistry/Thermodynamics/Laws_of_Thermodynamics/Second_Law_of_Thermodynamics


[16]   
A Larger Estimate of the Entropy of the Universe Chas A. Egan, Charles H. Lineweaver

http://arxiv.org/abs/0909.3983


[17]   
PhysLink What exactly is the Heat Death of the Universe?
http://www.physlink.com/education/AskExperts/ae181.cfm


[18]   
AskAMathematician After the Heat Death of the Universe will anything ever happen again?
http://www.askamathematician.com/2015/03/q-_after-_the-_heat-_death-_of-_the-_universe-_will-_anything-_ever-_happen-_again/


[19]   

On the Thermodynamic Equilibrium in the Universe F. Zwicky
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1085617/