GENERAL I ARTICLE

The Optimal Use of Entropy and Enthalpy

M S Ananth and R Ravi

Chemical engineers juggle entropy and enthalpy changes to produce chemicals with the minimal expenditure of work. In a free market, the prices of common chemicals correlate very well with the thermodynamic work required to produce them!

M S Ananth is a Professor Introduction of Chemical Engineering at the Indian Institute of The majority of the elements on earth are present in the form of Technology, Madras. His compounds, mainly oxides. Oil, natural gas, coal, biomass, rock, research interests are in salt, sulphur, air and water are the primary raw materials that th~ thermodynamics and chemical industry depends on. From distillation of crude oil mathematical modelling. He is deeply concerned chemical engineers produce the major fuels such as LPG, gaso­ about the quality and line, diesel and kerosene. There are about 20 base chemicals reach of engineering including ethene, propene, butene, benzene, synthesis gas, am­ education in India. monia, methanol, sulphuric acid and chlorine and 300 interme­ diates like acetic acid, formaldehyde, urea, acrylonitrile, acetal­ dehyde and terephthalic acid. These base and intermediate chemicals are often referred to as bulk chemicals. The very large number of speciality chemicals are manufactured using these bulk chemicals as the raw materials [1]. R Ravi is an Associate Professor in the Depart­ Figure 1 indicates schematically the typical components of the ment of Chemical chemical industry: reactors, separators, heat exchangers and Engineering at the Indian Institute of Technology, utilities. Industrial separation processes represent a major por­ Madras. His research tion of the manufacturing cost for most chemicals. Of these interests are in thermody­ (Figure 2) about 55% can be described as equilibrium separation namics and statistical processes and are, as the name implies, amenable to quantitative mechanics. thermodynamic analysis. It is estimated that, on the average, 70% of the capital investment in the petroleum and petro­ chemical industry is on separation equipment alone. The energy consu.med by such prqcesses in the USA in 1976 was of the order of 1016 Joules!

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Rate governed separation process

separation process process

Figure 1. (left) Components In any chemical industry the raw materials enter at room tem­ ofa typical chemical indus­ try. perature and the products leave at room temperature. We can Figure 2. (right) Broad clas­ therefore look upon each chemical industry as a black box sification ofindustrial sepa­ representing an isothermal process for converting a set of raw ration process. materials into a set of finished products. Thermodynamics en­ ables us to calculate the minimum work that has to be put into the industry in order to achieve the change of state from raw materials to finished products.

In the following section we discuss some thermodynamic pre­ liminaries. Then we derive expressions for extrema in work which exhibit explicitly the central role of the Gibbs free energy. In order to apply these results to engineering systems we then describe the model for the Gibbs free energy of an ideal mixture. The calculation of the work of separation and the relative importance of enthalpic and entropic contributions are The major then illustrated using examples. The major component of the component of the cost of many chemicals arises from the separation steps involved cost of many in their manufacture. For a few such chemicals it is shown that chemicals arises the thermodynamic calculation of the work required to extract from the them from their naturally occurring state correlates well with separation steps their prices in a free market. We also describe briefly the concept involved in their of 'zero work' cycles. We then derive criteria of equilibrium in manufacture. terms of the Gibbs free energy and use examples to demonstrate

-68------~------RESONANCE I September 2001 GENERAL I ARTICLE the role of energy and entropy in the attainment of equilibrium in some reacting systems. We touch upon the interpretation of energy and entropy in statistical thermodynamics in order to help visualise mixing processes. Finally we trace the history of the development of the concept of entropy for the sake of completeness.

Thermodynamic Preliminaries

We shall recall the definitions of a few basic terms. A system is simply a region of interest. It is separated from the surroundings by boundaries. Systems are conveniently classified into isolated, closed or open systems depending on their interaction with the surroundings. An isolated system has no interaction whatever with its surroundings; a 'closed' system exchanges energy but no mass with its surroundings; an open system exchanges both energy and mass with its surroundings.

A certain minimum number of variables must be specified in order to describe a system completely. These variables, which describe the present state of the system without any reference to its history, are called state variables (or functions of state or simply properties of the system). It follows that the change in the value of any state variable between two given states of the system is independent of the path. The most common examples are the measurable properties like pressure, volume, tempera­ ture, and composition.

Functions of state playa very important role in thermodynam­ The discovery of ics. For all practical purposes the discovery of the laws of the laws of thermodynamics can be looked upon as the discovery of two thermodynamics important functions of state, the internal energy U and the can be looked entropy S. upon as the discovery of two Indeed Gibbs [2] introduced the 'fundamental thermodynamic important functions equation' and its differential form of state, the U=U(S, V) (1) internal energy U dU=TdS-PdV (2) and the entropy S.

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Thermodynamics from which 'may be derived all the thermodynamic properties of provides criteria for the fluid (so for as reversible processes are concerned),. equilibrium, In chemical engineering applications the most important func­ typically telling tions are the enthalpy H defined as U + PVand the Gibbs free chemical energy G defined as H - TS. engineers when a reaction or a The concepts of heat (Q) and work (W') are central to thermody­ separation can namic analysis and are as difficult to define precisely as they are progress no familiar. Unlike the functions of state discussed above they are further. 'energies in transit' and are path dependent quantities.

There are primarily two concerns, central to chemical engineers, that thermodynamics addresses. Firstly, thermodynamics en­ ables us to identify reversible paths, along which Q and W take on maximum or minimum values. Secondly thermodynamics provides criteria for equilibrium, typically telling chemical en­ gineers when a reaction or a separation can progress no further.

Extrema in Work

Consider a closed system. The two laws of thermodynamics can be expressed mathematically as follows.

dU= 8Q-bW 8Q ~ TdS.

Combining the two laws, rearranging and integrating between any two states of the system we deduce that the maximum work that can be done by a closed system in the isentropic case is given by

w' max = -IJ.U.

Conversely the minimum work that must be put into a system for an isentropic change of state is clearly given by

(-W)min = ~U.

While the work done by a closed system during an adiabatic process is always given by -~U, the maximum work is obtained

------~------70 RESONANCE I September 2001 GENERAL I ARTICLE from the reversible case i.e. the isentropic process. Another case The total mass of practical interest is that of a process in which the system of inflow is effectively interest exchanges just enough heat with the surroundings so as equal to the total to remain at constant temperature. In this case the correspond­ mass outflow. ing results are:

Wmax=-M (-W)min = M, where A = U - TS is the Helmholtz free energy, the adjective 'free' emphasising the fact that, in an isothermal process, a part of the change -t1U in the internal energy namely + Tt1S is 'unavailable', while the remainder, -M, is 'free' for conversion to useful work.

The discussion following the equations for closed systems can be easily extended, mutadis mutandis, to open systems for two special cases: systems operating at steady state or with mass hold-up that is negligibly small compared to the mass flowing through the system. In either case the total mass inflow is effectively equal to the total mass outflow. In the isentropic and isothermal cases the results are simply

W s max / m =-M W smaxm=-g/ . t1 (3) where Ws is mechanical 'shaft' work defined as work other than that due to changes in P or V. Small case letters refer to pro­ perties per unit mass and t1 denotes the difference in values between outlet and inlet. The left hand side of these equations represents the rate of work done divided by the mass flow rate or simply the work done per unit mass flowing through the system.

Gibbs Free Energy of a Mixture

In order to apply (3) it is necessary to relate changes in the Gibbs free energy to changes in measurable quantities. This involves the repeated use of differential calculus and the two laws. These relations are derived in standard textbooks [3]. We will merely

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Thermodynamics state a few results that help explain some typical chemical does not specify engineering applications. The Gibbs free energy of a mixture is the composition given by dependence of the chemical potential

completely [3]. It where ni is the number of moles of i and r is the number of leaves one components. The chemical potential, f-Li' of the ith component unspecified degree is defined as: of freedom that is exploited to build l1i = (~Jan' 'admissible t T,P,{n j'F i} models'. In a pure substance f-L is the same as g, the specific Gibbs free energy. Thermodynamics does not specify the composition de­ pendence of the chemical potential completely [3]. It leaves one unspecified degree of freedom that is exploited to build 'ad­ missible models', the simplest of which is the 'ideal mixture' model:

f-Li == f-Lyure (T, P) + Rt In (xi) ,

where Xi is the mole fraction of i in the mixture. It follows from these equations that the enthalpy, entropy and Gibbs free en­ ergy changes that occur due to ideal mixing at constant Tand P are given by

M,idmix =0

flgidmix - (4) RT

In real mixtures the changes due to mixing are conveniently written in terms of excess functions, which represent the depar­ ture of real from ideal mixing. Many models for the composition dependence of the excess Gibbs free energy of liquid mixtures have been developed.

Work of Separation

The minimum work required to separate an ideal mixture into

72------~------RESONANCE I September 2001 GENERAL I ARTICLE its components at given T and P is simply minus of the differ­ In a free market ence between the Gibbs free energy per mole after mixing and we expect the before mixing. For an ideal mixture it follows from (3) and (4) prices of that chemicals to depend (5) monotonically on the work required The factor {-RTln(x/)} represents the work required to produce to produce them! 1 mole of pure i from the mixture state of mole fraction Xi in which it occurs naturally. In the case of real mixtures it is necessary to account for the departure from ideal behaviour due to the finite size of the molecules and the intermolecular inter­ actions. Nevertheless the significance of this result is demon­ strated in Figures 3 and 4. In a free market we expect the prices of chemicals to depend monotonically on the work required to produce them! The latter is given by {-RT In(x/)} per mole. Figures 3 and 4 show a remarkable confirmation of this expecta­ tion in two different contexts: one for India and another for the USA [4]. These are log-log plots. The correlation is indeed Figure 3. Indian market remarkable. Petrol price in India includes a large component of price vs concentration in excise duty to discourage its use, while the prices of sugar and naturally occurring raw LPG are kept low thr~ugh subsidies. For urea the pre-subsidy materials.

2.8~ ______~

~trol • Ie G> Soda '0 E 2.3 t-!- :E Pig· I/) Q) Iron Q) Co ::l 0::: 1.8 en 0 Sugar ..J •

1.3 0.0 0.5 1.0 1.5 2.0 - Log ( Weight Fraction)

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Figure 4. US market price' vs concentration in natu­ rally occurring raw materi­ Urokinase. also Reproduced with per­ 8 mission from [6}. 7

6

:c 5 ~ Insulin. oi .~ 4 0...

0) ~ 3

2

o

-1 -2 -3 -4 -5 -6 -7 -8 -9 Log (Weight Fraction in Substrate)

price is available and has been used. The correlation in Figure 4 between market price and thermodynamic work in the US is even more impressive. These figures explain why non-idealities of mixing are well worth investigating in order to quantitatively predict the market price of chemicals in the free market.

A few examples are i~cluded here to illustrate the relative importance of the entropic and enthalpic. effects in the separa­ tion of real mixtures. Consider the separation of a hydrocarbon mixture [5] with components sufficiently similar for the mix­ The correlation in ture to be near ideal. The separation of the feed mixture is to be Figure 4 between effected by distillation at 1 atm pressure and appropriate cooling market price and into three product streams. All the streams are liquids at 421 K. thermodynamic The flow rates and compositions of the streams are specified in work in the US is the adjoining Table 1, taking the feed as 1 mole. The minimum even more work required per mole of the feed F can be conveniently impressive. calculated as the algebraic sum of four terms involving free

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Stream Feed (F) Overhead(P1) Middle(P2) Bottom (P3) Component

Ethyl Benzene 0.15 0.9816 0.0096 0 p-xylene 0.19 0.0065 0.3018 0.0041 m-xylene 0.43 0.0117 0.6830 0.0112 o-xylene 0.23 0 0.0055 0.9847 Moles 1 0.147 0.623 0.230

energies of mixing: Table 1. Composition of feed and product streams -~(F) + 0.147 ~(Pl)+O.623 ~(P2)+O.230 6g(P3). in distillation.

The minimum work in the ideal case is -2.91 kJ while the actual work is -1.31 kJ emphasizing the need for taking into account the non-idealities of mixing. The enthalpic contribution in this case is negligible and the actual work is almost entirely entropic.

An example of a liquid-phase separation in which enthalpic effects are likely to be dominant is that of a 35 mole% mixq.lfe of acetone in water into products containing 99 tnole%acetone and 98 mole% water, respectively [5]. This requires a minimum work input of 0.41 kJ/mol of feed. The calculation can be broken up into two components:

The data for the acetone-water system shows that the enthalpic contribution is 0.31 kJ/mol while the entropic contribution is -0.1 kJ/mol. Hydrogen is the cleanest fuel Zero Work Cycles available from an environmental Hydrogen is the cleanest fuel available from an environmental point of view. The point of view. The most abundant source ofhydrogen is clearly most abundant water. Consider a multi-step steady state chemical process to source of carry out the decomposition of water with a heat source available hydrogen is clearly at TH and a heat sink available at To' The change in the Gibbs water.

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free energy can be written for step i operating at Tj as :

(6)

where ,':lSi is the entropy change associated with step i and for convenience is treated as constant over the temperature range in question. Shaft work is derived from thermal energy and this conversion is subject to Carnot type limitations. Thus, assum­ ing TH = 1000 K (corresponding to a high temperature gas cooled nuclear reactor) and To = 300 K, conversion of thermal energy to work involves a loss of 300/1000 or 30% of the energy. So it is interesting to investigate the possibility of choosing the temperatures Ti such that the net work is zero. Since G and S are functions of state it is clear that

where I!1Go and I!1So are the standard free energy and entropy changes for water decomposition at temperature To, taken usu­

, ally as 298 K. Recalling that Wsj is equal to I!1G j the zero work cycle stipulates that

Substituting these three results into (6) we get the criterion for

choosing the set of temperatures Tj •

For .water splitting at 25 0 C and 1 atm, I!1Go and I!1So are 229 kJ/ mQl and 0.0443 kJ/mole K. A single step process requires a T} of 5170 K! If 1000 K is taken as a practical upper limit for any step, a two step process will involve entropy change greater than 0.234 kJ/mole K which is too farge an entropy change for most realistic reactions. As of now a few four step cycles have been devised and studied [6].

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Criteria of Equilibrium All processes that The second question of interest to engineers is the identification lead to a decrease in, Gibbs free of the equilibrium state of a system that is suitably constrained. In the absence of shaft work, we have energy occur spontaneously. dG ~-SdT + VdP.

If T and P are held constant, then according to the above inequality the Gibbs free energy of the system cannot increase. Such a system dearly reaches equilibrium only when its Gibbs free energy is a minimum.

G ==H -TS == minimum. (7)

This criterion for equilibrium assumes special significance be­ cause the constraints involved are in terms of measurable quan­ tities that are often held constant in the industry.

All processes that lead to a decrease in G occur spontaneously., This can occur in two ways - through a reduction in enthalpy or an increase in entropy or both. In pure component systems these results are of no consequence since such systems have only two degrees of freedom. However it is multicomponent systems that are central to the chemical engineers' interest.

Reaction Equilibria

We now illustrate the interplay of enthalpic and entropic effects in two cases, one endothermic and the second exothermic. The dehydrocydisation of n-octane to p-xylene at I atm is an endot­ hermic reaction [7].

We take 1 mole of n-octane as the basis and let ~ represent the extent of reaction, i.e., moles of Cs HIS that have reacted at equilibrium. The free energy change of such a reaction mixture can be expressed as follows:

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where !J.G mix is the change in G involved in mixing (l-~) moles of n-octane with q moles of p-xylene and 4q moles of Hz. MJO and !J.So are the 'standard' enthalpy and entropy change associ­ ated with the reaction. At equilibrium '!J.G' is a minimum and hence d(!J.G) =0. d~

For a given temperature !J.Go == MJO- T!J.S° may be computed and hence the equilibrium extent of reaction calculated. Figure 5 is a plot of the extent vs temperature. In an endothermic reaction, the extent increases with increase in temperature.

For the heterogeneous exothermic reaction

2CO(g) ~ CO2 (g) +C(s) both the standard enthalpy and the standard entropy of the reaction are negative. The reaction becomes spontaneous below Figure 5. (left) Extent of re­ about 975 K when the entropic effect exactly balances the action vs temperature. exothermicity of the reaction (Figure 6). Further the extent Figure 6.(right) Standard decreases with increase in temperature (Figure 7). The higher energy and entropy chan­ the temperature the faster the reaction and consequently an ges as a function of tem­ optimal temperature has to be chosen as a compromise between perature. the thermodynamic conversion and the kinetic rate.

1 30 0.9 ' .' -' ,. ' . 0.8 ,-' -20 .'.# • ./ 0.7 . ' , , -. -- MJO 0.6 -70 O (5 -- T!J.S ~ 0.5 .€ ...... !J.GO 0.4 ...ll<: -120 0.3 0.2 -170 0.1 0 -220 450 500 550 600 700 800 900 1000 1100 1200 TK TK

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1 , Figure 7. Extent of reaction 0.9 . ~ as a function of temperature. 0.8 ~ 0.7 j ~ ~:: li 0.4 , I 0.3 -j 0,2 1 0.11 o ~ 700 800 900 1000 1100 1200 TK

Mixing and M.olecular Description of the Roles of U andS

The results of classical thermodynamics are independent of the molecular structure of matter. Nevertheless some elementary molecular concepts help in the understanding of the concepts of energy and entropy. The energy of a single molecule, that is of concern to chemical engineers, is of three ki~ds - translational, rotational and vibrational. Energy of all kinds is quantised. The size of the quanta varies with the kind of energy and is generally lowest for translational energy and highest for vibrational en­ ergy. In a system of molecules there is also the potential energy due to interaction between molecules or due to an external field. The sum of these energies appropriately weighted and averaged according to the laws of statistical mechanics yields the internal The energy of a energy U. The enthalpy H is, as noted earlier, simply U + PV. It single molecule, is a measure of the energy 'content' of a system. that is of concern The concept of entropy is more subtle. In statistical mechanics to chemical' the entropy of a closed system is given by the following formula engineers, is of [8] three kinds - translational, S=-RL. P .ln (P.), J I J rotational and vibrational. where Pj is the probability that the system is in state j. Since

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Suggested Reading probabilities are always less than unity S is always non-negative. It is a postulate of statistical mechanics that all the states of an [1] J A Moulijn, M Makkee, A van Diepen, Chemical Pro­ isolated system are equally probable. In this case, the probabili­ cess Technology, John Wiley ties reduce to n-l for allj and the expression for S reduces to the & Sons, New York, 2001. well-known Boltzmann-Planck formula [9] [2] J W Gibbs, The Scientific papers of J Willard Gibbs, S=R In cn), Thermodynamics, Dover Publications, New York, Vol. 1,1961. where n is the number of microscopic states available to the [3] K Denbigh, The Principles of system. A system that can exist in only one state (n = 1) has no Chemical Equilibrium, Cam­ entropy. It is often described as a completely 'ordered' state. An bridge University Press, Cambridge, 1968. increase in n (the number of ways that energy can be distributed [4] G E Keller II, AIChE Mono­ among the many quantized levels) means an increase in disorder graph Series, Vol. 83, p. 17, and in the entropy. 1987. [5] J D Seader and E J Henley, The mixing of gases is essentially entropy driven. When gas Separation Process Prin­ ciples, John Wiley and Sons, molecules are mixed, they have more volume to move in, more New York, 1988. closely-spaced translational! rotational energy levels in which to [6] K F Knoche and J E Funk, spread out their original energy. Their entropy increases and the Entropy Production, Effi­ ciency, and Economics in the process occurs spontaneously. Thermochemical Genera­ tion of Synthetic Fuels, Inti. When crystals dissolve in water there are two opposing tenden­ J. Hydrogen Energy, Vol. 2, cies: the entropy of the salt molecules increases while that of the p. 387,1977. solvent molecules decreases. In the crystalline state ions vibrate [7] A S Kazanskaya and V A Skoblo, Calculations of about their mean positions in the lattice. In solution, the ions Chemical Equilibria, Ex­ have access to the more closely spaced translational and rota­ amples and Problems, Mir tional states leading to an increase in entropy. So the process Publishers, Moscow, 1978. [8] E T Jaynes, Papers on Prob­ should occur spontaneously. On the other hand the water mol­ ability, Statistics and Statis­ ecules have access to the closely spaced translational and rota­ tical Physics, Edited by R D tional states constrained only by the hydrogen bonds, that form Rosenkrantz, D Riedel Pub­ lishing Company, Dorc­ and break continually. In the solution, the polar water molecules hecht, Holland,1983. gather relatively tightly around each ion because of electrostatic [9] R K Pathria, Statistical Me­ forces restricting their movement and decreasing the number of chanics, International Series in Natural Philosophy, Vol. readily available energetic states. So there is a decrease in en­ 45, Pergamon Press, Oxford, tropy. Overall, there is usually a net entropy increase making the 1972. dissolution of salts a spontaneous process. [10] C Truesdell, The Tragicomi­ cal History ofThermodynam­ ics, 1822-1854, Springer­ However there are energy considerations as well. The tempera­ Verlag, New York, 1980. ture of the solution usually decreases when most common salts

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Box 1.

Boltzmann was the first to recognize that the entropy of a macroscopic state is related to the number of 'micro' states consistent with that macrostate. The quantum interpretation of Boltzmann's idea was due to Planck [9]. The concept of entropy was further generalized by Jaynes [8], who proposed that the probabilities of the various microstates are those that would maximize the entropy. Using this central idea, Jaynes derived all the previously known results of equilibrium statistical mechanics. What is even more remarkable is that this idea has been shown to be sufficient for nonequilibrium processes as well. dissolve. For example, energy is required to break the strong bonds among the ions in the crystal while energy is released when the ions form weak bonds with the polar water molecules. In the case of sodium chloride, the net energy required is very small. In the case of ammonium nitrate the energy released is much less than that needed to break the bonds in the crystal. The dissolu~ion occurs with such rapid cooling of the solution that ammonium nitrate is used as an instant athletic 'ice-pack' for sprains and bruises. In the case of calcium chloride the reverse is true and solution gets heated.

History of Entropy

Classical thermodynamics, for the most part, was concerned with bodies capable of undergoing only reversible processes. While the formal relations containing the entropy were present in Rankine's early papers, he identified entropy in 1854 and called it 'a thermodynamic function'. This historical fact runs counter to the popular opinion that Carnot discovered entropy and that the second law therefore preceded the first law. Clausius was to rediscover the function eleven years later and coined the modern term 'entropy' [10].

Gibbs in his papers from 1873-1876 [2] gave entropy a central

place in his theory of thermodynamic equilibrium and stability Address for Correspondence and recognized that the relations, proved to hold for bodies M 5 Ananth and R Ravi undergoing reversible processes, should hold for all bodies in Department of Chemical equilibrium Gibbs formulated his famous minimum energy/ Engineering Indian Institute of Technology maximum entropy postulate and regarded entropy as the only Chennai 600 036, India. parameter required to take into account thermal effects.

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