Friday, 18 November 2016

Entropy (3) Definition of entropy

Cambridge A level H432 from 2015

Entropy
.    (a)  explanation that entropy is a measure of the dispersal of energy in a system which is greater, the more disordered a system.


What is entropy?

As you can see from this learning objective the definition of entropy is about both the disorder of the particles of a system and the distribution of the energy quanta in that system.

Let’s use an argument from probability to illustrate the way in which energy quanta are distributed among the particles in any given substance.

The first assumption that we can make is this: if energy quanta are available to share between molecules they will be shared in every way possible between all the available molecules. 

There is assumed to be no restriction on the way in which the energy quanta are shared.

So if there are three molecules with three energy quanta to share between them then they will share their quanta a total of 10 ways:

Number of quanta
Molecule 1
Molecule 2
Molecule 3
3
0
0
2
1
0
1
1
1
2
0
1
1
0
2
0
3
0
0
0
3
0
1
2
1
2
0
0
2
1

Here is the formula for calculating the number of ways of arranging quanta on a given number of molecules.


Where q is the number of energy quanta and m is the number of molecules over which the quanta are shared.

The example above is very simple and hardly representative of true states of molecules and quanta because the real numbers are huge! 

A mole of molecules contains 6.02 ×1023 molecules and the quanta they may possess may well be of a similar number.

To give you an idea of the size of these numbers consider this situation: (you can try and calculate these values to check for yourself if you want to!!)

Number of molecules (m)
Number of quanta (q)
W the number of ways the quanta are shared.
100
10
1013
100
100
8×1059
200
110
1086

The point of this exercise in statistical probabilities is this:  the chances of all the quanta being arranged in a particularly organised way of say one quanta per atom is highly unlikely.

As we can see from the table above the chances of 100 quanta being arranged over 100 atoms with one quantum per atom is 1 in 8×1059

That situation is very unlikely to occur.

From this statistical model we conclude that changes that happen by chance tend to go in the direction that will increase the number of ways of distributing the molecules and energy quanta. 

The entropy of a system is related to W the number of ways in which the energy of a system is dispersed over its molecules. 

S  =   k  ln  W

Where S is the entropy of a system, ln W is natural logarithm of the number of ways of arranging the energy quanta in the system and k is a constant called Boltzmann’s constant.
So from the Boltzmann formula we can see that if W increases so will S the entropy of a system. 

This means that chemical change happens in the direction of an increase in W and S an increase in the total entropy of the system

For a reaction to be feasible the total change of entropy of both the system and surrounding has to be positive.  

ΔStotal  =  ΔSsystem +  ΔSsurroundings   has to be positive

Now it ought from this to be clear that entropy increases with temperature since any given substance has a greater number of quanta at a higher temperature.

As a substance changes from solid to liquid to gaseous state its molar entropy value will increase.  




Note that there is jump in entropy values when the substance changes state and melts or boils. 

All this suggest that there is a temperature at which the entropies of all substances is 0 and that that temperature is 0 Kelvins or absolute zero or —273K.

At this temperature all molecular motion has ceased completely. 

All substances have no energy quanta to disperse over their molecules. 

And as you can probably guess this situation is never realised in practice anywhere in the Universe. 


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