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  • Define entropy and calculate the increase of entropy in a system with reversible and irreversible processes.
  • Explain the expected fate of the universe in entropic terms.
  • Calculate the increasing disorder of a system.
Photograph shows a glass of a beverage with ice cubes and a straw, placed on a paper napkin on the table. There is a piece of sliced lemon at the edge of the glass. There is condensate around the outside surface of the glass, giving the appearance that the ice is melting.
The ice in this drink is slowly melting. Eventually the liquid will reach thermal equilibrium, as predicted by the second law of thermodynamics. (credit: Jon Sullivan, PDPhoto.org)

There is yet another way of expressing the second law of thermodynamics. This version relates to a concept called entropy    . By examining it, we shall see that the directions associated with the second law—heat transfer from hot to cold, for example—are related to the tendency in nature for systems to become disordered and for less energy to be available for use as work. The entropy of a system can in fact be shown to be a measure of its disorder and of the unavailability of energy to do work.

Making connections: entropy, energy, and work

Recall that the simple definition of energy is the ability to do work. Entropy is a measure of how much energy is not available to do work. Although all forms of energy are interconvertible, and all can be used to do work, it is not always possible, even in principle, to convert the entire available energy into work. That unavailable energy is of interest in thermodynamics, because the field of thermodynamics arose from efforts to convert heat to work.

We can see how entropy is defined by recalling our discussion of the Carnot engine. We noted that for a Carnot cycle, and hence for any reversible processes, Q c / Q h = T c / T h size 12{Q rSub { size 8{c} } /Q rSub { size 8{h} } =T rSub { size 8{c} } /T rSub { size 8{h} } } {} . Rearranging terms yields

Q c T c = Q h T h size 12{ { {Q rSub { size 8{c} } } over {T rSub { size 8{c} } } } = { {Q rSub { size 8{h} } } over {T rSub { size 8{h} } } } } {}

for any reversible process. Q c size 12{Q rSub { size 8{c} } } {} and Q h size 12{Q rSub { size 8{h} } } {} are absolute values of the heat transfer at temperatures T c size 12{T rSub { size 8{c} } } {} and T h size 12{T rSub { size 8{h} } } {} , respectively. This ratio of Q / T size 12{Q/T} {} is defined to be the change in entropy     Δ S size 12{ΔS} {} for a reversible process,

Δ S = Q T rev , size 12{DS= left ( { {Q} over {T} } right ) rSub { size 8{"rev"} } } {}

where Q size 12{Q} {} is the heat transfer, which is positive for heat transfer into and negative for heat transfer out of, and T size 12{T} {} is the absolute temperature at which the reversible process takes place. The SI unit for entropy is joules per kelvin (J/K). If temperature changes during the process, then it is usually a good approximation (for small changes in temperature) to take T size 12{T} {} to be the average temperature, avoiding the need to use integral calculus to find Δ S size 12{ΔS} {} .

The definition of Δ S size 12{ΔS} {} is strictly valid only for reversible processes, such as used in a Carnot engine. However, we can find Δ S size 12{ΔS} {} precisely even for real, irreversible processes. The reason is that the entropy S size 12{S} {} of a system, like internal energy U size 12{U} {} , depends only on the state of the system and not how it reached that condition. Entropy is a property of state. Thus the change in entropy Δ S size 12{ΔS} {} of a system between state 1 and state 2 is the same no matter how the change occurs. We just need to find or imagine a reversible process that takes us from state 1 to state 2 and calculate Δ S size 12{ΔS} {} for that process. That will be the change in entropy for any process going from state 1 to state 2. (See [link] .)

The diagram shows a schematic representation of a system that goes from state one with entropy S sub one to state two with entropy S sub two. The two states are shown as two circles drawn a distance apart. Two arrows represent two different processes to take the system from state one to state two. A straight arrow pointing from state one to state two shows a reversible process. The change in entropy for this process is given by delta S equals Q divided by T. The second process is marked as a curving arrow from state one to state two, showing an irreversible process. The change in entropy for this process is given by delta S sub irreversible equals delta S sub reversible equals S sub two minus S sub one.
When a system goes from state 1 to state 2, its entropy changes by the same amount Δ S size 12{ΔS} {} , whether a hypothetical reversible path is followed or a real irreversible path is taken.
Practice Key Terms 3

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Source:  OpenStax, College physics ii. OpenStax CNX. Nov 29, 2012 Download for free at http://legacy.cnx.org/content/col11458/1.2
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