Entropy 2

A bit more on Entropy.

We observed in the first entropy post that, for a spontaneous (i.e. natural, non-coerced) thermodynamic process, the net change in entropy is always positive.

In the example we cited, this was because the entropy decrease caused in a warm body by the withdrawal of heat was less than the entropy increase caused in a cool body absorbing the same quantity of heat.

Unfortunately, I don’t think these words always relay an intuitive sense of what entropy is for most beginning engineers.  Clearly temperature somehow relates to entropy, but in what way? It’s kind of a head scratcher.

While I have seen many attempts at plain English analogies for entropy, I have not personally found one that really worked for me, but I am offering my own analogy here.  I imagine this analogy has been used elsewhere, though I have not run across it.  Also, I am working without a net, so there may be some conceptual errors that I am not seeing.  Still, even if imperfect, it promotes thinking about entropy.

Let’s ignore work energy for now, and recall the expression for internal energy and heat flow from our last post:

dU = dQ = TdS

And if heat is transferred exclusively between two bodies of different temperatures the law of conservation of energy dictates that:

T(1) * dS(1) = T(2) * dS(2)

If T(1) is greater than T(2) we see dS(1) must be less than dS(2) for the equality to hold.  Thus the change in entropy must be greater than zero as the system tends towards thermodynamic equilibrium.

***

Here is our analogy.

Imagine a country of altruistic and rational people

Imagine that the country has a fixed amount of money – money is never created or destroyed.

Imagine that people in this country initially tend to have either lots of money, or not much money.

The people who have lots of money are “Wealthy”, those without much money are “Less Wealthy”.

***

As I said, our imaginary citizens are altruistic and rational.

As citizens interact and become friends, the Wealthy citizens become inclined to give some of their money to their less well off friends.

The wealthiest (those with the “densest” bank accounts) can afford to be the most generous.

Now, to the Wealthy citizens, giving away some money will diminish their wealth slightly, but it improves the Wealth of their less well off companions much more noticeably.  Thus, the overall Wealth of society increases when Wealthy citizens share their wealth with the Less Wealthy.  Perhaps it prevents children from going hungry in Less Wealthy families, for instance.

In addition, our citizens are rational, so Less Wealthy citizens do not give money to their Wealthier friends.  It would make no financial sense, and it diminishes the overall Wealth of society at large since it beggars the Less Wealthy at the expense of the already Wealthy.  So overall, the Wealth of society tends to increase in all “natural” transactions.

Observe that as these altruistic citizens continue to exchange money, they will eventually reach a point where everyone will have the same amount of wealth.  Transactions between the citizens will no longer occur, and the wealth of the average citizen will have been maximized.

Note that when a Wealthy citizen shares their wealth with a Less Wealthy person, the amount of money in the society remains unchanged even though societal wealth can increase.

***

As is hopefully obvious from the above the analogy components are:

  • Money = Energy
  • Bank Account = Temperature
  • Giving Money = Heat Flow
  • Wealth = Entropy

If this analogy is valid, what it suggests is that entropy somehow relates to the “wealth” of a substance, but it is different from the specific amount of the energy in a substance (the number of dollars in the bank account.)

Can we think of entropy as the “wealth” of a substance, denoting a capability to give, whereas energy is like the exact amount of money?  I would say that, loosely speaking, yes we can.

We can notice something else.  Over time, the money will be evenly distributed, the sharing will cease and the wealth of the citizens will be maximized overall.  This is equivalent to our system reaching thermal equilibrium.  At this point, heat flow stops and entropy is at a maximum for the system.

Note too that as citizens give away money and their Bank Accounts become more similar, the effects on society’s wealth become smaller.  For instance, when a citizen with $1,000,001 dollars gives $1 to a citizen with $999,999 dollars, the change in wealth to either one and to society is vanishingly small.  So too, when the temperature differences in a thermodynamic system are very small, so is the creation of entropy.  If this difference could be made to be zero, the entropy change would be zero and the process would be reversible (see last post.)

***

As a physical example, if you combine a pot of cold water and a pot of hot water, the net entropy will increase as the temperatures tend towards each other.  But they will be at a maximum only when the the combined water reaches a fixed, stable equilibrium temperature and no further heat flow occurs.

***

A few closing comments.

An excellent, though somewhat more complicated (at least to me), analogy is provided by Van Ness in his great book Understanding Thermodynamics.  Worth a look see.

It is worth mentioning the units we are working with.

The internal energy of a substance is reported as energy per unit mass.  Examples include kiloJoules/kilogram and Btu/Pound-mass.

As you might expect (since Q = T * dS), entropy is reported as energy per unit mass – degree.  Examples include kiloJoules/kilogram-Kelvin  and Btu/Pound-mass-Rankine.

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One comment on “Entropy 2

  1. lark says:

    Better. Rather strange, but much better.

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