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Home , Entropy (arrow of time), Heat, Laws of thermodynamics, Thermal expansion, Work (thermodynamics)

Jorgen Lovland Professor emeritus

Department of NTNU 2007

THE DEVELOPMENT OF

Table of contents

Volumetric and thermal properties…………………….. 2 Volumetric properties …………………………. 2 Thermal properties ……………………………. 3 The laws……………………………………….. 3 The second law………………………………… 4 The first law …………………………………… 4 Energy and …………………………….. 5 Equilibrium criteria…………………………………….. 6 Modern thermodynamics………………………………. 7 Experimental data……………………………… 7 Mixture models………………………………… 8 Activity coefficients……………………. 8 Equations of state………………………. 8 Process simulators……………………………… 9 Global stability…………………………………. 9 Molecular calculations…………………………. 10 Status today…………………………………….. 10 References……………………………………………… 11 2

The Development of Thermodynamics

Engineers use thermodynamics for process calculations, in particular for initial calculations where material and energy streams are determined for (most often) a number of process configurations. Physical streams in a process will always include mixtures, even if a few calculations could involve only pure components. Thermodynamics are used to find • Simple physical properties, like when and is known, or vapor pressure at a given temperature. • Energy functions to determine energy needs: heating, cooling, of , compression, expansion. • equilibria, involved in the majority of separations. • Reaction equilibria: maximum or actual conversions in reactors

Thermodynamicists to make such calculations possible for steadily more complex systems, by measurements and by including more and more fundamental phenomena in the calculations models that are used routinely by others.

Chronologically, the development of thermodynamics followed somewhat the points above, but the history is better summarized by these points: • Volumetric and thermal properties, at first only for pure components (mid-1600’s- today) • The energy laws (1824-1855) • Equilibrium criteria (1876) • Modern thermodynamics (~1900 – today)

Volumetric and thermal properties

These properties can be measured, which sets them apart from other properties that are derived from fundamental principles and that cannot be measured, only calculated. In addition, mass (weight) and composition can be measured.

Volumetric properties

By this is meant connected values of volume, pressure and temperature. The measurements go back to the invention of the and the manometer in the mid-1600’s. The first thermometer with a sealed stem was made by Grand Duke Ferdinand II of Tuscany in 1641. Much improved versions were made by G. D. in 1714, using first alcohol and then as the .

The mercury manometer was invented by Torricelli in 1643, while Otto von Guernicke invented a piston vacuum pump around 1650. Robert Boyle made an improved pump, and did experiments on gases establishing Boyle’s law: Pressure volume is constant at any given temperature. Later, around 1786, J. A. C. Charles studied of several gases, and established Charles’ law: All gases expand identically with increasing temperature. Only in 1835 did Clapeyron combine Boyle’s and Charles’ law to the law. 3

In 1809, A. Avogadro proposed that equal of gases at the same pressure and temperature contain the same number of atoms or . This correct idea (at low pressure) was not generally accepted until about fifty years later, but did a lot to establish a correct view of atoms and molecules.

The critical point of propane was discovered in 1822. T.Andrews in the 1860’s made more systematic measurements of the pressure-temperature-volume connections of dioxide, including an exploration of the critical point. The stage was then set for the so-called (a relationship between pressure, volume and temperature) by J.D.van der Waals in 1873. His equation of state covers both the gas and the liquid state at all . His equation is not too accurate it turned out, but it nevertheless increased our understanding of liquid and gas behavior very much, and became a starting point for a rush of similar equations in the second half of the 1900’s. The equation was: 2RTaPvbv=−− (1) where a and b are parameters found from the critical point behavior. Thus, the equation was “anchored” in the critical point, establishing the so-called corresponding states principle. Corresponding states mean states that have temperature and pressure in equal ratio of the critical values, and the principle have been used in most later equations of state. Eq. (1) is cubic (third order) in v, and such equations are referred to as cubic equations of state.

Thermal properties

The initial advance in determining thermal properties is above all associated with Joseph Black He made the conceptual differentiation between heat and temperature, and in 1763 established specific heat and (the heat of and vaporization being “latent” in the liquid and vapor).

That the specific heat at constant pressure is larger than the one at constant volume seems to have been pointed out first by R.J. Haüy in 1806.

The energy laws

Thermodynamics as is based on the first, second and third . Of these, the second law came first in 1824, the first one came around 1850 and the third in 1906. 4

The second law

Based upon Black’s work as well as work by others, the of heat was developed in the late 1700’s. This theory held that heat is a fluid that can not be created or destroyed, a so-called conserved property as for instance mass is. Several phenomena could be well explained by this theory, and it was very logical from the knowledge available. However, already in 1798 B.Thompson (Count Rumford) published results from his cannon boring work, and pointed out that it was against all experience that a fluid should be produced in seemingly endless quantities from the metal being bored, and without changing the properties of the metal.

For a number of reasons his objections were not accepted. One of the reasons was the work of Fourier on heat conduction – where heat is conserved - in which he invented new mathematical methods (Fourier series) to solve a variety of heat conduction cases.

A number of French scientists in the early 1800’s did work on the efficiency of the steam . One of them, the engineer Sadi Carnot published “Reflections on the motive of heat” in 1824, one of the landmark papers of thermodynamics. He developed the concept of a cycle and of reversible processes, and showed that the maximum efficiency (Work W divided by heat Q entering) would only take place for a reversible process and be given by only the upper and lowerhighlow temperaturehighTTWQT (T)−= according to the formula (2)

This formula is one manifestation of what is called the second law of thermodynamics. Interestingly, he got this correct result assuming the caloric theory to be true. He used an analogy to power where water is conserved but falls in height, so heat did not change but produced work by falling from higher to lower temperature in this view.

Based upon Carnot cycles E. Clapeyron derived how vapor pressure changes with temperature in 1834. In 1848 W.Thomson (later Lord ) proposed an absolute temperature scale based upon the equation above. It is seen that if Tlow in Eq. (1) becomes zero the efficiency becomes 1, and this zero cannot be an arbitrary point as for instance 0 oC, but must be a unique lowest point. His suggested zero equal to -273 oC is only 0.15 oC above the present value.

The first law

After Count Rumford, Robert Mayer was the next to state that the caloric theory had to be wrong, and that heat could be converted to work. Again, these objections were not believed, in part because, it seems, his arguments were incomplete.

It was left to James P. to perform the experimental work that finally convinced scientists that heat is not conserved, but that energy is. Already in 1840 he had shown that heat evolved in a wire conducting was proportional to the resistance and the electric flow squared, which is now called Joule’s law. He went on to show that this heat could not come from outside the wire, and that heat thus had to be produced in the wire itself. In the years 1841-1850 he performed a number of different and very accurate experiments converting various forms of work to heat. (Some of his equipment can be seen in Science Museum in 5

London.) As a result of his work he claimed that heat and work is one and the same thing. Using his units for work (lbft – pound-foot) and heat (Btu – ) his experiments gave the values for the conversion factor between heart and work, the so-called mechanical equivalent of heat, listed in the table. (Using our units the conversion factor is 1, since we all in Joule)

Table 1. Joule’s values of the mechanical equivalent of heat

Year Value, lbft/Btu Experimental method 1843 838 Falling weight driving el. generator 1843 770 : Forcing water through holes 1845 823 Gas compression 1845 819 Weights driving wheel in water 1847 772 Weights driving paddle wheel in water

Today 778.14

We see that his last value (which he actually gave with three decimals) is about 1 % off the correct one. Even if some doubt remained about the exact value, naturally enough, his main point was to show unequivocally that work was converted to heat. One hurdle in accepting Joule’s results was how to reconcile them with Carnot’s work which was now generally accepted. In 1850 R. Clausius explained that it was a of two separate laws. The is what was soon to be called the first law of thermodynamics.

Energy and entropy

The first law gave rise to a new and more correct understanding of energy. Energy is now a more elusive property, and not just the ability to do work as it is still often defined. Energy has no definite value, and only energy differences can be measured by the heat and work removed or added. Heat and work are not stored as such anywhere, but are the two forms of energy transfer. Unfortunately, our everyday language has not kept up with this scientific advance, and we still talk about for instance heat content.

The energy difference in a system (= the part of the world we are studying) as a result of heat and/or work transfer is termed the . Since in all flow processes and in non-flow ones at constant pressure the atmosphere is part of the work by pushing on or being pushed away, a combination of internal energy and atmospheric work is more convenient to use. This combination was given the name by H. Kammerlingh Onnes in or before 1909. For example, what is often called heat of reaction is actually the enthalpy of reaction.

The second law gave rise to a brand new property: entropy. The property arises from the second law after a small bit of rewriting. First we include an inequality sign in Eq.(2) to make it correct also for real processes with lower efficiency that the reversible process. Then we use the first law to substitute Qin – Qout for W. We then get directly inouthighlowQQTT≤ (3) 6

Thus, for all real processes a quantity Q/T will always increase. This ratio was what R. Clausius named entropy in 1865, when he also formulated the two laws very concisely as “The energy of the world is constant, the entropy strives to a maximum”. It should be added that the entropy maximization holds for isolated systems, so in most cases the system and its surroundings must be considered as a hypothetical .

Although well defined, it was difficult to associate anything physical with the quantity Q/T. But in 1877 L. Boltzmann gave a statistical definition of entropy

S = k lnW (4) where k is a constant (later called Boltzmann’s constant) and W is the number of “microstates” a system has for a given “macrostate”. Macrostate is what we determine as volume, pressure, temperature and mass, microstates are related to how the atoms and molecules are “spread out” on energy levels, mixing, and space. In a more colloquial way of speaking, entropy is a measure of internal disorder. Since heat gives increased molecular , it is reasonable to us today that a measure of disorder is related to heat, as in Eq.(3). However, we can have increased disorder in other ways too, for instance by mixing without heat effects. On the other hand, because of the energy levels, we can have an increase in entropy when the system seemingly changes to more order; an example is precipitation of sulfate from a supersaturated solution which both gives more orderly and cools the system (Lambert 2002).

Boltzmann’s definition utilizes molecules directly. This is unproblematic today, but when Boltzmann published his work, it was still a topic of much debate if molecules really existed. Most were keen to emphasize that thermodynamics was a macroscopic science which did not depend on whatever was the correct fine structure of matter.

A little later, in 1905, W. Nernst formulated the third law: At temperature the entropy is zero. Actually, some entropy remains, but this entropy is more or less constant on heating, so it is now usual to define “practical entropy” which is zero at zero temperature. This law enables us to calculate absolute values of entropy, and to get consistent values for various energy functions needed for equilibrium calculations.

Equilibrium criteria

Much experimental work went on in the 1800’s on various chemical systems and on phase equilibria. A central question was how to determine the equilibrium values of composition at a given temperature and pressure, the two variables easiest to control and to measure. Since most reaction give off or absorb heat, it was natural to seek a connection between the heat of reaction and equilibrium. It was also natural to use concentrations directly in any correlation, as was used in the mass action law in 1864.

The problem was solved for all types of equilibria by Josiah W. Gibbs in 1878, in his now famous paper “On the equilibrium of heterogeneous substances”. He showed that it follows from the second law that equilibrium at fixed temperature and pressure occurs at a minimum of a new energy combination, which is now called Gibbs’ energy. He also introduced 7 which relates the total Gibbs energy to the components present, and which is equal for each component in each phase in phase equilibria. He pointed out that the condition for local stability is that the Gibbs energy function curves upwards in the equilibrium point. Stability means whether a phase is stable or will split into more phases.

Note that thermodynamics tells us about equilibrium states, but it does not and cannot say anything about how fast anything happens. is not a variable in thermodynamics. However, it is fully possible to calculate changes in a system, e.g. in a gas being compressed. But here too, thermodynamics gives us the states, not how fast they are reached.

Modern thermodynamics

Based on the three laws, thermodynamics gives a number of exact relationships between a number of properties, for instance how to go from volumetric behavior (an equation of state) to an energy function. These relationships were known by around 1900 with one exception. The laws and their consequences were rightly hailed as one of the great intellectual achievements of the 1800’s. However, they are a theoretical construct, and the new properties cannot be measured, just calculated. To get numerical answers, the theoretical framework has to be connected to the behavior of matter through properties that can be measured. Thus, from now on the effort was on describing matter, in particular mixture behavior which cannot be predicted from the constituent components. However, before the advent of computers only limited descriptions were possible.

Experimental data

For pure components, the following properties are typical of those that are measured: - vapor pressure - specific - heat (enthalpy) of melting and vaporization - freezing point and critical point - /volume at given pressure and temperature

For mixtures, some of the same properties may be measured, and in addition or instead - composition at equilibrium at given pressure and temperature; most commonly - electric conductivity (for electrolytes) - osmotic pressure, elevation, freezing point and vapor pressure reduction - boiling point, dew point

The experimental data in turn have to be correlated somehow. Earlier graphical methods were of necessity much used, and visual presentations will always have a role. are still in use for visualizing and calculating simple processes (e.g. a refrigeration cycle) using only one component.

The use of computers starting around 1960 meant a gradual but dramatic change in the way thermodynamics was practiced. It became increasingly possible to correlate data in proper models (i.e. determining parameters in the models) and then to use these models in 8 combination with the rigorous thermodynamic relations to get better answers. Almost all thermodynamics now rely on computers.

Mixture models

Most thermodynamic research deals with mixture properties. Two main methods have been developed to deal with fluid mixtures: - activity coefficients - equations of state

Activity coefficients are in principle a correction factor applied to the liquid phase fugacity of each component, where the fugacity is a logarithmic transformation of chemical potential first introduced by G. N. Lewis in 1908. Activity coefficients were introduced by J.J. van Laar in 1910. This is a way to describe at low enough pressure for the pressure not to influence the liquid. With a vapor as the other phase this would be described either as ideal gas or by the simplest so-called virial equation of state.

Others proposed alternatives to van Laar’s equation for the activity coefficient. In 1923 P.Debye and E. Hückel determined the limiting law (i.e. at very low dilution) for ions in solutions. After computers had arrived, G.Wilson in 1964 proposed a local composition model for the activity coefficients of non-electrolyte solutions. This was developed into the “non- random two-liquid theory” (NRTL) by J.M.Prausnitz in 1967 and into the “universal quasi- chemical theory” (UNIQUAC) by Prausnitz in 1975. The basic idea behind these three models is that the liquid is not perfectly mixed, but is a mixture of two different liquids within the overall composition.

Based upon UNIQUAC, Prausnitz, A. Fredenslund and some others developed the UNIFAC (Uniquac functional activity groups) group contribution method in 1975. A group contribution method refers a molecular property to the functional groups making up the . Such groups could be ketones, alcohol, aldehyde etc. The underlying idea is that such groups influence their molecules in certain ways, which is the reason we talk of groups of components as for instance alcohols. The great advantage of a group contribution method is that it can predict properties for components where there are no or very few data, which typically is the case when new products are made.

The advantage of activity coefficients is that they can be used to describe mixtures with very strong interaction between the components. But the usual models are limited to moderate pressure and temperature, say below 5-10 bar and between 0 oC to 120-150 oC.

Equations of state are applied to all phases present, giving simpler derivation and programming. With equations of state, all changes in properties with temperature is performed in a hypothetical ideal state at 1 bar, and thus only the ideal state is needed. This can be determined from spectroscopic measurements or from estimates. The equation of state – which anyway needs to have the as the limit at very low pressure – is used to calculate changes with pressure from the ideal gas state to the real state. From the exact thermodynamic relations an equation of state gives values for enthalpy, entropy and phase equilibria, ensuring consistent values. 9

Although an improved version of the van der Waal cubic equation of state was published by Redlich and Kwong in 1949, calculations based on equations of state even more than those besed on activity coefficients rely on the use of computers. It was only with the publication of the Soave-Redlich-Kwong equation in 1972 and the Peng-Robinson equation in 1976 that such calculations really took off.

Cubic equations of state for pure components are made to fit the vapor pressure and the critical point. The parameters are “mixed” to provide one combined equation for the mixture, one in each phase. The so-called mixing rule was initially very simple, making equations of state calculations suitable for a limited range of components, typically hydrocarbons and light gases. Much development went on in and after the 1970’s to extend the mixing rules so more components could be included. It is now possible to include the UNIQUAC/UNIFAC parameters in the mixing rule, making calculations using activity coefficients superfluous except for electrolytes.

Cubic equations of state are fit to vapor pressure to give best results for vapor-liquid equilibrium calculations, which are very important in process calculations. However, since the cubic equations are fairly simple, they cannot represent all properties equally well. Typically, volumes and especially liquid volumes are less accurate. There exist a large number of other equations of state, used for instance for buying and selling , for calculations, for refrigerants or for steam. There has also been a development of equations of state for general process use that are more complex: Perturbed hard chain (1975) and Statistical associating fluid theory (1990). They give generally better all-round results.

Process simulators

Mainly using equation of state thermodynamics, process calculations have been automated in special programs like Hysys, Aspen, ProVision and others. Such program go back to around 1970, and have been growing in abilities ever since. They are now very user-friendly, including a selection of thermodynamic models, a long list of components, fast and robust algorithms, graphical interface, many optional calculations for the streams (e.g. hydrate formation, bubble point). These programs are very impressive, but must necessarily lag behind the latest research results.

Global stability

A globally stable system will not be able to change to more phases, while a locally stable system can sometimes split. In 1982, L.E.Baker and coworkers found a criterion for determining global stability. The same year M Michelsen presented the criterion in a much simpler way, and at the same time gave a smart algorithm to check if global stability had been reached. The criterion is called the tangent plane criterion. The idea is that if the tangent plane in a point is below the curve for Gibbs energy G the system is globally stable. If the tangent plane is also a tangent at another point, the tangent points represent equilibrium phases in global stability. However, if the tangent plane intersects the G-curve, the system can lower its Gibbs energy by splitting into more phases at the fixed overall composition. The new phases will have a common tangent plane. After a split, the new phases must each be checked for global stability in the same way to ensure they are stable. Such a test is now performed automatically in the process simulators. 10

The test follows from the criterion of minimum Gibbs energy at given temperature and pressure, and is a new rigorous relation following from the basic laws, the one that was not known in the early 1900’s.

Molecular calculations

Thanks to the increase in computer power, it is now possible to perform calculations on a large but limited number of actual molecules, perhaps around 500-1000 molecules. The molecules are computationally set in motion and collide and interact according to rules that are specified. Such calculations are useful as an exploratory step, for instance in looking for a solvent. It can also be used when experiments are impossible. For example, n-alkanes above C18 decompose at below the critical, but calculated values have been given for critical temperature and pressure.

Status today

Despite the impressive work over the last hundred years or so, in particular now that computers are routinely used, there is certainly more work ahead than behind us. Some molecules are extremely complex, some interact in a very complex fashion, and computers allow us to make ever more complex models. Among systems that are poorly described in general are almost all electrolytes, many systems, aqueous systems at extreme conditions (e.g. supercritical water with other components), other mixtures at extreme conditions (e.g. oil reservoirs at 1000 bar), mixtures with very complex molecules (e.g. asphaltenes). In oil and gas processing we deal with components that have been measured for several decades. Making medicine we probably deal with brand new components where there are no data. Several important systems or products are disperse, e.g. paint, several foods and cosmetics, and these are even more complex.

But whenever we study any system closely, even an assumed simple one, we usually find data are lacking or are inconsistent, or the model may be too crude for our purpose. If we need to do experiments we find that actually obtaining equilibrium is not a simple matter, and that analyzing the composition is very difficult with the accuracy we need. Even without developing more advanced models – which is also needed – there is more than enough work for any foreseeable future.

Finally, this overview of the development of thermodynamics reflects the bias, background (chemical engineering thermodynamics mostly applied to oil and gas processing), knowledge and ignorance of the author. Although the three laws and Gibbs are fixed points, there is plenty of scope for different viewpoints of what should be included and what not in such a short story of such a rich topic. 11

References Cheng, K. C.: Historical Development of the theory of Heat and Thermodynamics: Review and Some Observations, Engineering, 13, 19-37 (1992)

Lambert, F.L.: Disorder- A Cracked Crutch for Supporting Entropy Discussions, J.Chem.Ed., 79, 187-192 (2002)

Tassios, D.P.: Applied Chemical Engineering Thermodynamics, Springer-Verlag, 1993

Wikipedia, the free encyclopedia