The Second Law of Thermodynamics

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1 CHEM 331 Physical Chemistry Fall 2017 The Second Law of Thermodynamics We have now reached a major milestone for the course; and are ready to enunciate the Second Law of Thermodynamics. The 0 th and 1 st Laws are fine, as far as they go. However, it is the 2 nd Law which will give us the tool whereby we can predict the equilibrium position for a given system. In the run-up to our discussion of the 2 nd Law, we have spent considerable time discussing heat engines. This is because historically the 2 nd Law was stated in terms of the efficiencies of heat engines. Although we are not particularly enamored of thermodynamic history, this approach to stating the 2 nd Law is most useful because of its ease and its closeness to the actual observations which gave rise to the 2 nd Law. Once stated, however, we will find that a statement of the 2 nd Law in terms of heat engines is of limited use for chemists. So, we will move from a discussion of heat engines to a definition of a new state function, the Entropy (S). This is important because the 2 nd Law can be reworked into a statement about the entropy. And at this point we will have severed the link to heat engines and will have established the 2 nd Law in terms that will be useful to us... all real changes have a direction which we consider natural. The transformation in the opposite sense would be unnatural; it would be unreal. In nature, rivers run from mountains to the sea, never in the opposite way. A tree blossoms, bears fruit, and later sheds its leaves. The thought of dry leaves rising, attaching themselves to the tree, and later shrinking into buds is grotesque. An isolated metal rod initially hot at one end and cold at the other comes to a uniform temperature; such a metal rod initially at a uniform temperature never develops a hot and a cold end spontaneously. Yet the first law of thermodynamics tells us nothing of this preference of one direction over the opposite one. The first law requires only that the energy of the universe remain the same before and after the change takes place. In the changes described above, the energy of the universe is not one whit altered; the transformation may go in either direction and satisfy the first law. It would be helpful if a system possessed one or more properties that always change in one direction when the system undergoes a natural change, and change in the opposite direction if we imagine the system to undergo an "unnatural change." Fortunately, there exists such a property of a system, the entropy, as well as several others derived from it. To prepare a foundation for the mathematical definition of the entropy, we must first divert our attention briefly to the characteristics of cyclic transformations. Having done that, we will return to chemical systems and the chemical implications of the second law. And divert our attention we will. Gilbert W. Castellan Physical Chemistry, 3 rd Ed. We start by revisiting the Carnot Engine and its associated Carnot Cycle. In the present case, we will analyze the efficiency of the Carnot Engine. This analysis will help us in defining and understanding the entropy function.

N.L. Sadi Carnot (1796-1832) reported an analysis of the factors involved in the production of mechanical energy from heat in his Reflections sur la puissance motrice du feu (Reflections on the

2 N.L. Sadi Carnot ( ) reported an analysis of the factors involved in the production of mechanical energy from heat in his Reflections sur la puissance motrice du feu (Reflections on the motive power of fire, 1824). The eldest son of Lazare Nicolas Carnot ( ), the Organizer of Victory during the Revolution, was familiar with his father's theoretical analysis of water driven machines. Young Carnot believed heat a fluid analogous in some ways to water. Just as his father showed that water could be representative of liquids in general, Sadi showed that steam was representative of gases in general. He treated the behavior of the heat engine as an idealized cycle of heat transfers with accompanying expansions and compressions. Using the waterfall analogy, where height and quantity of water determine the power production, he argued that in the steam engine the temperature drop and quantity of caloric fluid fulfilled a similar role. He hell into the error of supposing that not heat is lost or converted, but made the correct observation that power produced is dependent on change in temperature. Before his death during the cholera epidemic of 1832 he realized that some heat is converted to mechanical energy and is therefore lost, thus recognizing the failure of the water analogy. He abandoned the caloric viewpoint at this time but his views were not published until 1878, by which time they had been rediscovered by other investigators. Carnot's ideas were extended by Bnoit Paul EmileClapeyron ( ) in Clapeyron, who was a professor at the Ecole des Ponts et Chaussees in paris, translated Carnot's ideas into analytic form. He revived or rediscovered Watt's indicator diagram and showed that a measure of the work done in the cycle was afforded by the area under the pressure-volume curve. Aaron J. Idhe The Development of Modern Chemistry To Carnot's Engine. It is driven by the expansion of a hot gas in a piston, where the gas remains hot because of the piston's contact with a high temperature reservoir. Heat floods in from the reservoir to keep the gas hot. This allows for a maximum of work to be obtained from the expansion. Then the gas is allowed to continue to expand adiabatically in order to cool the gas. This will allow the piston to be compressed with less work than was obtained from its expansion. Now the piston, with its cold gas, is compressed isothermally, rejecting heat into a cold reservoir. This is followed by an adiabatic compression to heat the gas back to its original temperature, thereby completing the cyclic operation of the Carnot Engine. The Engine is idealized in that all these processes occur reversibly. And, the working fluid in the piston is assumed to be an Ideal Gas. As Carnot showed, this last refinement is not needed, but will make our analysis of the efficiency of the engine easier. Nicolas Leonard Sadi Carnot

3 Clapeyron's indicator diagram for the Engine is as depicted below: In this case it is known as the Carnot Cycle. Since the Engine operates cyclically: U cycle = 0

4 Also, since all the processes are reversible: W expan = Maximum Work Output W comp = Minimum Work Input Now to the work provided by the Engine in a given cycle: W isotherm, expan = - RT h ln (V 2 /V 1 ) W adiabat, expan = W isotherm comp = - RT c ln (V 4 /V 3 ) W adiabat comp = The adiabatic work terms cancel, leaving us with: W cycle = - RT h ln (V 2 /V 1 ) - RT c ln (V 4 /V 3 ) However, the volumes V 2 and V 3 and V 4 and V 1 are connected by adiabats and are thus related to each other via: T h V 2-1 = T c V 3-1 & T h V 1-1 = T c V 4-1 This gives us: W cycle = - R (T h - T c ) ln (V 2 /V 1 ) Now, the heat drawn in during the isothermal expansion of the hot gas can be determined via: Q h = U h - W h = 0 - (- RT h ln (V 2 /V 1 )) = RT h ln (V 2 /V 1 ) And now, we have what is needed to calculate the efficiency of the engine. The efficiency of any heat engine is defined as: = For the Carnot Engine its efficiency is given by: rev = - R (T h - T c ) ln (V 2 /V 1 ) / RT h ln (V 2 /V 1 ) =

Note that this is dependent only on the temperatures of the reservoirs. (I have added "rev" to emphasize that the Carnot Engine is operated reversibly.

5 Note that this is dependent only on the temperatures of the reservoirs. (I have added "rev" to emphasize that the Carnot Engine is operated reversibly. Okay, with the preliminaries out of the way, let's turn to the two different observations that constitute the different but equivalent statements of the Second Law of Thermodynamics. Both observations are in terms of the efficiencies of heat engines, where in one case the engine is reversed to operate as a refrigerator. The engine operates by drawing heat energy (Q h ) out of a hot reservoir at temperature T h, doing some net work (W) and rejecting heat (Q c ) into a cold reservoir at temperature T c. The refrigerator does the opposite. It has some work (W) done on it, drawing heat (Q c ) out of a cold reservoir at T c and rejecting heat (Q h ) into a hot reservoir at T h. Statements of the 2 nd Law Our first statement of the 2 nd Law derives from the observation that although more efficient steam engines were being constructed, no steam engine was ever found to be 100% efficient. It is impossible for an engine working in a cycle to produce no other effect than that of extracting heat from a reservoir and performing an equivalent amount of work. Stated by Lord Kelvin circa 1850, it says we cannot build what I will call an impossible engine of the type pictured below.

Lord Kelvin Our second statement of the 2 nd Law was presented to the

other effect than the transfer of heat from a colder body to a hotter

6 Lord Kelvin Our second statement of the 2 nd Law was presented to the Berlin Academy in 1850 by the mathematical physicist Rudolf Julius Emmanuel Clausius. It is impossible for a refrigerator working in a cycle to produce no other effect than the transfer of heat from a colder body to a hotter body. Again, this statement means that we cannot build an "impossible" refrigerator. Rudolf Clausius In these statements of the Second Law, the phrase working in a cycle must be emphasized. The cycle allows us to specify that the working substance returns to exactly it initial state, so that the

process can be carried out repeatedly. It is easy enough to convert heat into work isothermally if a cyclic process is not required: for example, simply expand a gas in contact with a heat reservoir.

Although the 2 nd Law was first stated in two different contexts, there is only one 2 nd Law and it can be shown that the Kelvin and Clausius statements are equivalent. This is what we shall do now.

7 process can be carried out repeatedly. It is easy enough to convert heat into work isothermally if a cyclic process is not required: for example, simply expand a gas in contact with a heat reservoir. Walter J. Moore Physical Chemistry, 4 th Ed. Although the 2 nd Law was first stated in two different contexts, there is only one 2 nd Law and it can be shown that the Kelvin and Clausius statements are equivalent. This is what we shall do now. We start by assuming the Clausius statement is incorrect and that we can indeed build an impossible refrigerator, operating between the same temperature baths T h and T c. We will allow a standard Kelvin engine to operate between the two temperature baths. I have attached numbers to W, Q h and Q c for each machine. No particular significance should be attached to these values; they are simply illustrative. The point is that if Q c absorbed by the impossible Clausius refrigerator is equivalent to Q c rejected by the Kelvin engine, 60 units in this illustration, then the net effect of operating the two machines is equivalent to the operation of an impossible Kelvin engine. A contradiction has developed as a result of our assumption that the Clausius statement is incorrect. This contradiction is that we can violate Kelvin's prescription for engine construction. In other words, if the Clausius statement of the 2 nd Law is incorrect, then the Kelvin statement is also incorrect. Now for a logical reversal of things. Assume the Kelvin statement is incorrect. We can operate our Kelvin engine with 100% efficiency.

8 Then, if we use the work output of this engine to drive a Clausius refrigerator operating between the same temperature baths, the net effect is to have built an impossible Clausius refrigerator. (Numbers attached in the illustration above are again for illustrative purposes.) Again, our assumption has lead to a contradiction. In other words, if the Kelvin statement is incorrect, then so to is the Clausius statement. Logically, this set of contradictory results translates into an equivalence of the Kelvin and Clausius statements of the 2 nd Law. Can Q h and Q c Have the Same Sign? For our cyclically operated heat engine, we have: U cycle = 0 This is nothing more than a recognition that U is a state function. This means that: W cycle = - Q cycle = - Q h - Q c Thus, we can calculate the efficiency of the engine by: = Now, is it possible for Q h and Q c to have the same sign. If so, then > 100%. Well, let's assume they can have the same sign. I'll tip my hand by saying that we have constructed an "impossible" engine; for instance, drawing heat from both the hot and cold temperature reservoirs and doing work. Again, numerical values are illustrative only. However, suppose our impossible engine draws 40 units of heat out of the cold bath. Let's arrange to have 40 units of heat transferred from the hot bath to the cold bath. This is easy enough to do; heat flow directly from hot objects to cold objects occurs all the time. But, now we have a contradiction. The net effect of this rejection of

heat from the hot to the cold bath and the operation of the engine is equivalent to having constructed an impossible Kelvin engine; a violation of the 2 nd Law.

9 heat from the hot to the cold bath and the operation of the engine is equivalent to having constructed an impossible Kelvin engine; a violation of the 2 nd Law. Hence, our assumption must be wrong and Q h and Q c must have opposite signs. Whew, the world is safe again. Can an Irreversible Engine Outperform a Reversible Engine? Now we turn to a seemingly odd question, can an irreversible heat engine outperform a reversible one? More accurately, can an engine operating irreversibly between two temperature reservoirs be more efficient that a reversible engine operating between the same two reservoirs? Let's set up two engines, one operating reversibly between reservoirs set at temperatures T h and T c, the other will operate irreversibly. The numbers of units of energy in the illustration are illustrative but also informative in this case. Note the reversible engine is operating with an efficiency of: rev = = 50% and the irreversible engine with: irrev = = 62.5% This means we are assuming: irrev > rev Now, reverse the reversible engine and operate it as a refrigerator and drive it with the irreversible engine.

The net effect of having coupled these two engines is to have produced an impossible refrigerator; a contradiction becasue this would violate the 2 nd Law.

10 The net effect of having coupled these two engines is to have produced an impossible refrigerator; a contradiction becasue this would violate the 2 nd Law. This means our assumption is incorrect and we instead have: rev > irrev (I am being a bit cavalier here. Could it be that irrev = rev? No; and I'll leave the proof to you.) This means that Carnot's engine provides for an upper limit on the efficiency for any engine operating between two temperature reservoirs at T h and T c since it is operating reversibly. And, note there is no restriction here on the nature of the working fluid in the Carnot engine, so our choice of an Ideal gas as a working fluid will not affect our results. So, rev = 1 - = 1 + Note that Q h and Q c must be extracted and rejected respectively from/to their appropriate baths by an engine operating reversibly. The Entropy Defined Now we move towards a definition of a new state function; the Entropy (S), a term introduced by Clausius, meaning randomness. From the above:

11 1 - = 1 + Rearranging, we obtain: Thus, over the cycle: Or, on a differential basis: Meaning, we can define our new state function as: ds = and be assured that it is in fact a state function since: Now, have we leapt too far too fast. Is the generalization: in fact valid. Now that we have established the generality of the conclusions of the Carnot cycle, it is very useful to put the cycle itself on a more abstract and general basis. The [figure below] depicts a reversible cycle between two states A and B. The paths A B and B A are abitrary but can be approximated by a set of Carnot cycles. The interior portions cancel, and so the net results is to give the stepped path traced in heavy lines. Since: applies to each cycle, it must be true for the set of cycles that: This sum can be divided into two parts, that for the terms that are made up by the contour from A to B and that for the terms made up by the contour of the path from B to A:

12 Carnot cycles can be made as small as desired, so that in the limit we may reproduce the arbitrary path as closely as we please. For an infinite number of small steps, [the above result] takes an integral form: or: So, we are safe in defining the entropy as: ds = Arthur W. Adamson A Textbook of Physical Chemistry, 2 nd Ed. The Clausius Inequality Now, we can certainly have a lot of fun with our new state function. But the real fun comes from tying this new function to the statements of the 2 nd Law. This connection was first published by Clausius in his memoir " On the Application of the Theorem of the Equivalence of Transformations to Interior Work". So, let's proceed by considering two engines, one reversible and operating between baths at T h and T c. The other is operating irreversibly between baths at T h ' and T c '.

13 We know that for the reversible engine: W cycle = - & For the irreversible engine we have only: Now assume: W cycle = - is true for the irreversible engine. We want now to reverse the reversible engine and couple it to the irreversible engine. For the composite engine:

14 W = 0 = = Q h + Q c + Q h ' + Q c ' positive with the indicated terms being positive. Also for the composite engine, by our assumption: This means: must be larger than negative terms In order that this new requirement stated directly above, and the requirement of the work equation, be met, we have: as well as: T c = small # & T h ' = large # T h = small # & T c ' = large # This leads to the contradiction that our composite engine will be drawing heat out of a cold reservoir and rejecting it into a hot reservoir; a violation of the 2 nd Law. Hence our assumption must be incorrect and we instead have: Now consider a system that cycles between two states, State 1 and State 2. The path from State 1 to State 2 is irreversible and that from State 2 to State 1 is reversible.

15 By the above result: Applying the definition of the entropy: Or, - ds < 0 This leaves us with the inequality of Clausius: ds > The importance of this result cannot be overstated. This is the Clausius inequality, which is a fundamental requirement for a real transformation. The inequality enables us to decide whether or not some proposed transformation will occur in nature. We will not ordinarily use [the inequality] just as it stands but will manipulate it to express the inequality in terms of properties of the state of a system, rather than in terms of a path property such as Q irrev. Gilbert W. Castellan Physical Chemistry, 3 rd Ed. We have now provided ourselves with a new state function, the entropy, and have shown how the 2 nd Law can be reformulated in terms of this state function; the Clausius Inequality. This allows us to divorce ourselves from 2 nd Law statements in terms of heat engines and refrigerators. Heat engines have served us well, but now they must be gone. We are interested in chemistry and what the 2 nd Law tells about our chemical systems. This is accomplished via the above arguments about the entropy. For one last example, consider an isolated system such as the Universe. Since the system is completely isolated: and therefore: Q irrev = 0 ds > 0 for some transformation which occurs within the Universe.

16 This means that at equilibrium, the entropy of the isolated system is at a maximum. In other words, the entropy of the Universe must increase for all naturally occurring processes. Well, what if our system is not isolated? How can we leverage the Clausius Inequality to tell us something about the equilibrium position of the system? This is where Gibbs enters the picture. More on this in a bit.

Lecture 2 Entropy and Second Law

Lecture 2 Entropy and Second Law Etymology: Entropy, entropie in German. En from energy and trope turning toward Turning to energy Motivation for a Second Law!! First law allows us to calculate the energy