Thermodynamics treats the principles of energy transformations, and the laws of thermodynamics establish the bounds within which these transformations are observed to occur. The first law states the principle of energy conservation, leading to energy balances in which work and heat are included as simple and equivalent additive terms. Yet work and heat are otherwise quite different. Work is directly useful in ways that heat is not, e.g., for accelerating an object or moving it against an opposing force, such as gravity. Evidently, work is a form of energy intrinsically more valuable than an equal quantity of heat. This difference is reflected in a second fundamental law which, together with the first law, lays the foundation for all of thermodynamics. The purpose of this chapter is to:
∙ Introduce the concept of entropy, an essential thermodynamic property
∙ Present the second law of thermodynamics, which formalizes the observed difference between processes that can occur spontaneously and those that cannot
∙ Apply the second law to some familiar processes
∙ Relate changes in entropy to T and P for substances in the ideal-gas state ∙ Present entropy balances for open systems
∙ Demonstrate the calculation of ideal work and lost work for flow processes ∙ Relate entropy to the microscopic world of molecules
5.1 AXIOMATIC STATEMENTS OF THE SECOND LAW
The two axioms presented in Chapter 2 in relation to the first law have counterparts with respect to the second law. They are:
Axiom 4: Entropy1 S is an intrinsic property of any system at internal equilibrium that is functionally related to the measurable state variables
1Pronounced en’-tro-py to distinguish it clearly from en-thal’-py.
that characterize the system. Differential changes in entropy are given by:
dS t = dQ rev /T (5.1) where St is the system (rather than the molar) entropy.
Axiom 5: (The Second Law of Thermodynamics) The entropy change of any system and its surroundings, considered together, and resulting from any real process, is positive, approaching zero when the process approaches reversibility. Mathematically,
Δ S total ≥0 (5.2)
The second law summarizes the universal observation that every process proceeds in such a direction that the total entropy change associated with it is positive, the limiting value of zero being attained only by a reversible process. No process is possible for which the total entropy decreases. The practical utility of the second law is illustrated by application to two very common processes. The first shows its consistency with our everyday experience that heat flows from hot to cold. The second shows how it establishes limits for the conversion of heat to work by any device.
Application of the Second Law to Simple Heat Transfer
First, consider direct heat transfer between two heat reservoirs, bodies imagined capable of absorbing or rejecting unlimited quantities of heat without temperature change.2 The equation for the entropy change of a heat reservoir follows from Eq. (5.1). Because T is constant, integration gives:
ΔS = __Q
T
A quantity of heat Q is transferred to or from a reservoir at temperature T. From the res- ervoir’s point of view the transfer is reversible, because its effect on the reservoir is the same regardless of source or sink of the heat.
Let the temperatures of the reservoirs be TH and TC with TH > TC. Heat quantity Q, transferred from one reservoir to the other, is the same for both reservoirs. However, QH and QC have opposite signs: positive for the heat added to one reservoir and negative for the heat extracted from the other. Therefore QH = −QC, and the entropy changes of the reservoirs at TH and at TC are:
Δ S Ht = Q___ H T H = − _____Q C
T H and Δ S Ct = Q___ C T C These two entropy changes are added to give:
Δ S total = Δ S Ht + Δ S Ct = _____ − Q C T H +
Q C
___T C = Q C ( _______T H − T C T H T C )
2The firebox of a furnace is in effect a hot reservoir, and the surrounding atmosphere, a cold reservoir.
5.1. Axiomatic Statements of the Second Law 179 Because the heat-transfer process is irreversible, Eq. (5.2) requires a positive value for ΔStotal, and therefore,
Q C (T H − T C ) > 0
With the temperature difference positive, QC must also be positive, which means that heat flows into the reservoir at TC, i.e., from the higher to the lower temperature. This result conforms to universal experience that heat flows from higher to lower temperature. A formal statement conveys this result:
No process is possible which consists solely of the transfer of heat from one temperature level to a higher one.
Note also that ΔStotal becomes smaller as the temperature difference decreases. When TH is only infinitesimally higher than TC, the heat transfer is reversible, and ΔStotal approaches zero.
Example 5.1
A 40 kg steel casting (CP= 0.5 kJ⋅kg−1⋅K−1) at a temperature of 450°C is quenched in 150 kg of oil (CP= 2.5 kJ⋅kg−1⋅K−1) at 25°C. If there are no heat losses, what is the change in entropy of (a) the casting, (b) the oil, and (c) both considered together?
Solution 5.1
While the overall process described here is irreversible, one can imagine the cool- ing of the casting from its initial to final temperature by reversible heat transfer and the corresponding heating of the oil by reversible heat transfer. In each case, the entropy change is given by Eq. (5.1). If the oil and the casting are both incom- pressible, then in each case dQrev = dUt = C pt dT. For an incompressible substance, all processes are constant volume processes, so Cp and Cv are equal and no revers- ible work is possible. Integrating Eq. (5.1) from an initial temperature T1 to a final temperature T2, for constant C pt , then gives the change in entropy as:
ΔSt= ∫ T1
T2
dQrev
_____ T = ∫T1
T2
Cp t dT
_____
T = Cp t ln ___T2 T1
The final temperature t of the oil and the steel casting is found by an energy balance.
Because the change in energy of the oil and steel together must be zero, (40)(0.5)(t − 450) + (150)(2.5)(t − 25) = 0
Solution yields t = 46.52°C.
(a) Change in entropy of the casting:
Δ S t = m
∫ C_____ P dT
T = m C P ln ___T 2 T 1
= (40)(0.5) ln 273.15 + 46.52_____________ 273.15 + 450 = −16.33 kJ⋅ K −1
(b) Change in entropy of the oil:
Δ S t = ( 150 ) ( 2.5 ) ln ____________273.15 + 46.52 273.15 + 25 = 26.13 kJ⋅ K −1 (c) Total entropy change:
Δ S total = −16.33 + 26.13 = 9.80 kJ⋅ K −1
Note that although the total entropy change is positive, the entropy of the casting has decreased.
Application of the Second Law to Heat Engines
Heat can be used far more productively than by simple transfer from one temperature to a lower one. Indeed, useful work is produced by countless engines that employ the flow of heat as their energy source. The most common examples are the internal-combustion engine and the steam power plant. Collectively, these are heat engines. They rely on a high-temperature source of heat, and discard heat to the environment.
The second law imposes restrictions on how much of their heat intake can be converted into work, and here we aim to establish this relationship. We imagine that the engine receives heat from a higher-temperature heat reservoir at TH and discards heat to a lower-temperature reservoir TC. The engine is taken as the system and the two heat reservoirs comprise the sur- roundings. The work and heat quantities in relation to both the engine and the heat reservoirs are shown in Fig. 5.1(a).
Figure 5.1: Schematic diagrams. (a) Carnot engine. (b) Carnot heat pump or refrigerator.
(a) (b)
Hot Reservoir Hot Reservoir
Cold Reservoir Cold Reservoir
QH
W W
QC QC
QH