Peter Mander, author and editor of CarnotCycle

“Thank you to everyone who has visited this blog since its inception in August 2012. CarnotCycle’s country statistics show that thermodynamics interests many, many people. They come to this blog from all over the world, and they keep coming.

It’s wonderful to see all this activity, but perhaps not so surprising. After all, thermodynamics is the branch of science that can help us to use energy wisely and to understand and manage the effects of global warming on our atmosphere, our ice caps and our oceans.

Today’s students of physics and physical chemistry, for whom thermodynamics is a core subject, will be at the forefront of future efforts to keep this planet habitable. They must be knowledgeable and competent, and CarnotCycle is one of the many free resources to help them achieve that aim.”

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Exploring the mutual solubility of phenol and water at the Faculty of Pharmacy, National University of Malaysia.

The phenol-water system is a well-studied example of what physical chemists call partially miscible liquids. The extent of miscibility is determined by temperature, as can be seen from the graph below. The inverted U-shaped curve can be regarded as made up of two halves, the one to the left being the solubility curve of phenol in water and the other the solubility curve of water in phenol. The curves meet at the temperature (66°C) where the saturated solutions of water in phenol, and phenol in water, have the same composition.

The thermodynamic forces driving the behavior of the phenol-water system are first visible in the upwardly convex mutual solubility curve, showing that the enthalpy of solution (ΔHs) in the saturated solution is positive i.e. that the system absorbs heat and so solubility increases with temperature in accordance with Le Châtelier’s Principle.

More rigorously, one can ascertain whether solubility of the minor component increases or decreases with temperature by computing

where s is the solubility expressed in mole fraction units, f2 is the corresponding activity coefficient and Ts is the temperature at which the solution is saturated.

The line of the umbrella curve charts the variation in composition of saturated solutions – phenol in water and water in phenol – with temperature. The area to the left of the curve represents unsaturated solutions of phenol in water and the area to the right represents unsaturated solutions of water in phenol, while the area above the curve represents solutions of phenol and water that are fully miscible i.e. miscible in all proportions.

But what about the area inside the curve, which is beyond the saturation limits of water in phenol and phenol in water? In this region, the system exhibits its most striking characteristic – it divides into two coexistent phases, the upper phase being a saturated solution of phenol in water and the lower phase a saturated solution of water in phenol. The curious feature of these phases is that for a given temperature their composition is fixed even though the total amounts of phenol and water composing them may vary.

To analyze how this comes about, consider the dotted line on the diagram below, which represents the composition of the phenol-water system at 50°C.

Starting with a system which consists of water only we gradually dissolve phenol in it, maintaining the temperature at 50°C, until we reach the point Y on the curve at which the phenol-in-water solution becomes saturated.

Now imagine adding to the saturated solution a small additional amount of phenol. It cannot dissolve in the solution and therefore creates a separate coexistent phase. Since this newly-formed phenol phase contains no water, the chemical potential of water in the solution provides the driving force for water to pass from the aqueous solution into the phenol phase. This cannot happen on its own however since water passing out of a phenol-saturated solution would cause the solution to become supersaturated. This would constitute change from a stable state to an unstable state which cannot occur spontaneously.

What can be postulated to occur is that the movement of water from the solution into the phenol phase simultaneously lowers the chemical potential of phenol in that coexistent phase, allowing phenol to move with the water in such proportion that the phenol-in-water phase remains saturated – as it must do since the temperature remains constant. In other words, saturated solution passes spontaneously from the aqueous phase into the phenol phase, diminishing the amount of the former and increasing the amount of the latter. Because water is the major component of the phenol-in-water phase, this bulk movement will continuously increase the proportion of water in the coexistent water-in-phenol phase until it reaches the saturation point whose composition is given by point Z on the mutual solubility curve.

In terms of chemical potential in the two-phase system, equilibrium at a given temperature will be reached when:

upper phase = sat. soln. of phenol in water
lower phase = sat. soln. of water in phenol

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Applying the Phase Rule

F = C – P + 2

First derived by the American mathematical physicist J. Willard Gibbs (1839-1903), the phase rule computes the number of system variables F which can be independently varied for a system of C components and P phases in a state of thermodynamic equilibrium.

Applying the rule to the 1 Phase region of the phenol-water system, F = 2 – 1 + 2 = 3 where the system variables are temperature, pressure and composition. So for a chosen temperature and pressure, e.g. atmospheric pressure, the composition of the phase can also be varied.

In the 2 Phase region of the phenol-water system, F = 2 – 2 + 2 = 2. So for a chosen temperature and pressure, e.g. atmospheric pressure, the compositions of the two phases are invariant.

In the diagram below, the compositions of the upper and lower phases remain invariant along the line joining Y and Z, the pressure being atmospheric and the temperature being maintained at 50°C. As we have seen, the upper layer will be a saturated solution of phenol in water where the point Y determines the % weight of phenol (= 11%). Correspondingly, the lower layer will be a saturated solution of water in phenol where the point Z determines the % weight of phenol (= 63%).

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Relationship between X, Y and Z

If a mixture of phenol and water is prepared containing X% by weight of phenol where X is between the points Y and Z as indicated on the above diagram, the mixture will form two phases whose phenol content at equilibrium is Y% by weight in the upper phase and Z% by weight in the lower phase.

Let the mass of the upper phase be M1 and that of the lower phase be M2. The mass of phenol in these two phases is therefore Y% of M1 + Z% of M2. Conservation of mass dictates that this must also equal X% of M1 + M2. Therefore

The relative masses of the upper and lower phases change according to the position of X along the line Y-Z. As X approaches Y the upper phase increases as the lower phase diminishes, becoming one phase of saturated phenol-in-water at point Y. Conversely as X approaches Z the lower phase increases as the upper phase diminishes, becoming one phase of saturated water-in-phenol at point Z.

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Further reading

Logan S.R. Journal of Chemical Education 1998

This paper uses the well-known thermodynamic equation ΔG = ΔH + TΔS as a theoretical basis for determining the circumstances under which spontaneous mixing occurs when two partially miscible liquids are brought together at constant temperature and pressure.

The approach involves the construction of equations for estimating both the enthalpy and entropy of mixing in terms of the mole fraction x of one component, and graphing the change in Gibbs free energy ΔG against x to determine the position of any minimum/minima. The paper goes on to examine the criteria for the existence of two phases on the basis of determining the circumstances under which a system of two phases will have a combined ΔG value that is lower than the corresponding ΔG for a single phase.

The conclusion is reached that the assessment of ΔG on mixing two liquids can provide a qualitative explanation of some of the phenomena observed in relation to the miscibility of two liquids.

The paper is available from the link below (pay to view)
https://pubs.acs.org/doi/abs/10.1021/ed075p339?src=recsys&#

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P Mander November 2018

wil01

Wilhelmy’s birthplace – Stargard, Pomerania – in less happy times

The mid-point of the 19th century – 1850 – was a milestone year for the neophyte science of thermodynamics. In that year, Rudolf Clausius in Germany gave the first clear joint statement of the first and second laws, upon which Josiah Willard Gibbs in America would develop chemical thermodynamics. 1850 was also the year that the allied discipline of chemical kinetics was born, thanks to the pioneering work of Ludwig Wilhelmy.

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Ludwig Ferdinand Wilhelmy was born on Christmas Day 1812 in Stargard, Pomerania (now Poland). After completing his schooling, he studied pharmacy and subsequently bought an apothecary shop. In 1843, at the age of 31, he sold the shop in order to pursue research interests at university, where he made the acquaintance of Rudolf Clausius and Hermann von Helmholtz. In 1846, Wilhelmy received his doctorate from Heidelberg University, and it was here in 1850 that he conducted the first quantitative experiments in chemical kinetics, using a polarimeter to study the rate of inversion of sucrose by acid-mediated hydrolysis.

Wilhelmy’s work had a seminal quality to it because – apart from being a talented individual – he observed that great guiding principle when commencing an exploration of the unknown: he kept things simple.

He chose a monomolecular decomposition reaction, used a large volume of water to keep the acid concentration unchanged during the experiment, maintained constant temperature and followed the inversion process with a polarimeter, which did not physically disturb the conditions of the system under study. By rigorously limiting system variables, Wilhelmy discovered a simple truth: the rate of change of sucrose concentration at any moment is proportional to the sucrose concentration at that moment.

Now it just so happened that in Wilhelmy’s earlier doctoral studies, he had become familiar with utilizing differential equations. So it was a straightforward task for him to model his new discovery as an initial value problem, which he wrote as

where Z is the concentration of sucrose, T is time, S is the acid concentration (presumed unchanging throughout the reaction), and M is a constant today called the reaction velocity constant. Wilhelmy integrated this equation to

where C is the constant of integration. Recognising that when T = 0 the sucrose concentration is its initial value Z0, he wrote

or

He then proceded to show that this equation was consistent with his experimental results, and thus became the first to put chemical kinetics on a theoretical foundation.

wil06

A page from Wilhelmy’s pioneering work “Ueber das Gesetz, nach welchem die Einwirkung der Säuren auf den Rohrzucker stattfindet”, published in Annalen der Physik und Chemie 81 (1850), 413–433, 499–526

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Inversion of sucrose

wil07

Sucrose has a dextrorotatory effect on polarized light, but on acid hydrolysis the resulting mixture of glucose and fructose is levorotatory, because the levorotatory fructose has a greater molar rotation than the dextrorotatory glucose. As the sucrose is used up and the glucose-fructose mixture is formed, the angle of rotation to the right (as the observer looks into the polarimeter tube) becomes less and less. It can be demonstrated that the angle of rotation is directly proportional to the sucrose concentration at any moment during the inversion process.

wil08

A Laurent polarimeter from around 1900.

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A neglected pioneer

Ludwig Wilhelmy’s groundbreaking research into the kinetics of sucrose inversion was published in Annalen der Physik und Chemie in 1850, but it failed to garner attention and in 1854 he left Heidelberg, retiring to private life in Berlin at the age of 42. He died ten years later in 1864, his pioneering work still unrecognized.

It was not until 1884, twenty years after Wilhelmy’s death and thirty four years after his great work, that Wilhelm Ostwald – one of the founding fathers of physical chemisty – called attention to Wilhelmy’s paper. Among those who took notice was the talented Dutch theoretian JH van ‘t Hoff, who in 1884 was engaged on kinetic studies of his own, soon to be published in the milestone monograph Études de dynamique chimique (Studies in Chemical Dynamics). In this book, van ‘t Hoff extended and generalized the mathematical analysis that had originally been given by Wilhelmy.

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A digression on half-life

wil09

In his study of sucrose inversion, Ludwig Wilhelmy showed that the instantaneous reaction rate was proportional to the sucrose concentration at that moment, a result he expressed mathematically as

In his published paper, Wilhelmy did not mention the fact that the fraction of sucrose consumed in a given time is independent of the initial amount. But he might well have noticed that by expressing Z as a fraction of Z0, the left hand side of the equation simply becomes the logarithm of a dimensionless number.

For example the half-life time (T0.5) i.e. the time at which half of the substance present at T0 has been consumed

JH van ‘t Hoff was well aware of this fact – he derived a half-life expression on page 3 of the Études. And in all likelihood he was also aware that this kinetic truth produced a conflict with the thermodynamic necessity for a chemical reaction to reach equilibrium.

Reconciling kinetics and thermodynamics

The starting concentration of sucrose in Wilhelmy’s inversion experiment is Z0. So after the half life period T0.5 has elapsed, the sucrose concentration will be Z0/2. After further successive half life periods the concentrations will be Z0/4, Z0/8, Z0/16 and so on. The fraction of sucrose consumed after n half lives is

This is a convergent series whose sum is Z0 – corresponding to the total consumption of the sucrose and the end of the reaction. The problem with this formula is that it implies that Wilhelmy’s sucrose inversion reaction – or any first order reaction – will take an infinitely long time to complete. This is not consistent with the fact that chemical reactions are observed to attain thermodynamic equilibrium in finite timescales.

In the Études, van ‘t Hoff successfully reconciled kinetic truth and thermodynamic necessity by advancing the idea that a chemical reaction can take place in both directions, and that the thermodynamic equilibrium constant Kc is in fact the quotient of the kinetic velocity constants for the forward (k1) and reverse (k-1) reactions

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Wilhelmy’s legacy

Wilhelmy’s pioneering work may not have been recognised in his lifetime, but the science of chemical kinetics which began with him developed into a major branch of physical chemistry, involving many famous names along the way.

In the 1880s, van ‘t Hoff and the Swedish physical chemist Svante Arrhenius – both winners of the Nobel Prize – made important theoretical advances regarding the temperature dependence of reaction rates, which proved a difficult problem to crack.

In 1899, DL Chapman proposed his theory of detonation. The chemical kinetics of explosive reactions was then taken forward by Jens Anton Christiansen, whose idea of chain reactions was further developed by Nikolaj Semyonov and Cyril Norman Hinshelwood, both of whom won the Nobel Prize for their development of the concept of branching chain reactions, and the factors that influence initiation and termination.

Several other Nobel Prize winners have their names associated with chemical kinetics, including Walther Nernst, Irving Langmuir, George Porter and JC Polanyi. The work of all these illustrious men has enriched this important subject.

But for now, we must return to Ludwig Wilhelmy in Heidelberg. It is 1854, and having failed to garner any interest in his seminal studies, he has packed his bags at the university, handed in his keys at the porter’s lodge, and is ready to begin the long journey home to Berlin.

wil15

Heidelberg, Germany in the 1850s

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Mouse-over links to works referred to in this post

Jacobus Henricus van ‘t Hoff Studies in Chemical Dynamics

Ludwig Wilhelmy “Ueber das Gesetz, nach welchem die Einwirkung der Säuren auf den Rohrzucker stattfindet” , published in Annalen der Physik und Chemie 81 (1850), 413–433, 499–526

P Mander April 2016

CarnotCycle is a thermodynamics blog but occasionally it takes five for recreational purposes

Poker dice is played with five dice each with playing card images – A K Q J 10 9 – on the six faces. There are a total of 6 × 6 × 6 × 6 × 6 = 7776 outcomes from throwing these dice, of which 94% are scoring hands and only 6% are busts.

Here are the number of outcomes for each hand and their approximate probabilities

And here is the data presented as a pie chart

poker03

The percentage share of 5 of a kind (0.08%) is omitted due to its small size

A noticeable feature of the data is that the number of outcomes for 1 pair is exactly twice that for 2 pairs, and likewise the number of outcomes for Full house is exactly twice that for 4 of a kind. But when outcomes are calculated in the conventional way, it is not obvious why this is so.

Taking the first case, the conventional calculation runs as follows:

1 pair – There are 6C1 ways to choose which number will be a pair and 5C2 ways to choose which of five dice will be a pair, then there are 5C3 × 3! ways to choose the remaining three dice

6C1 × 5C2 × 5C3 × 3! = 3600 outcomes

2 pairs – There are 6C2 ways to choose which two numbers will be pairs, 5C2 ways to choose which of five dice will be the first pair and 3C2 ways to choose which of three dice will be the second pair. Then there are 4C1 ways to choose the last die

6C2 × 5C2 × 3C2 × 4C1 = 1800 outcomes

Conventional calculation gives no obvious indication of why there are twice as many outcomes for a 1-pair hand than a 2-pair hand.

But there is an alternative method of calculation which does make the difference clear.

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A different approach

Instead of starting with component parts, consider the hand as a whole and count the number (n) of different faces on view. The number of ways to choose n from six faces is computed by calculating 6Cn. Now multiply this by the number of distinguishable ways of grouping the faces, which is given by n!/s! where s is the number of face groups sharing the same size. The number of combinations for the hand is 6Cn × n!/s!

Since the dice are independent variables, each combination is subject to permutation taking into account the number of indistinguishable dice in each of the n groups according to the general formula n!/n1! n2! . . nr!

The total number of outcomes for any poker dice hand is therefore

It is easy to see from the table why there are twice as many outcomes for a 1-pair hand than a 2-pair hand. The number of combinations (6Cn × n!/s!) is the same in both cases but there are twice as many permutations for 1 pair. Similarly with Full house and 4 of a kind, the number of combinations is the same but there are twice as many permutations for Full house.

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Note that

is reducible to

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P Mander April 2018

Credit: Microbiology Online Notes

Although the Carbon Cycle is a well-accepted concept illustrated by countless graphics on the internet such as the one shown above, I wonder if it deserves to be called a cycle in the general sense of uninterrupted circular motion. Because the fossil fuels (coal, oil and gas) formed over millions of years from atmospheric CO2 are actually end products. Left to themselves they would remain as coal, oil and gas and the cycle would stop turning. To say that the cycle is completed by human interference sounds somewhat contrived.

But despite this criticism, the Carbon Cycle has an obvious value: it helps us to see a bigger picture. And this broader understanding can be further enhanced by looking at another aspect of carbon which changes during its journey around the cycle – namely its oxidation state.

I have not come across a graphic that includes this, so here is one I drew to illustrate the idea.

Rather than describing carbon in terms of its sequence of physical transformations, this cycle shows how carbon’s oxidation state reflects changes in the nature of its chemical bonds. Oxidation states are conveniently if somewhat abstractly represented by dimensionless numbers, which in the case of carbon range from +4 (most oxidized state) to –4 (most reduced state). The lower the number, the more energy is present in carbon’s chemical bonds. So in the process of carboniferous fuel creation the oxidation state number decreases. Conversely, the process of energy release from carboniferous fuel results in an increase in oxidation state number. The natural abundance of carbon with its wide range of oxidation states centered about zero is what gives carbon its usefulness as both a source and a store of energy.

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Energy in, Energy out

Below is a quantified illustration of how carboniferous fuels store and release more energy as the oxidation state number decreases.

Energy release in kJ per mole of CO2 formed. Numbers are indicative

The green arrow shows the oxidation state decrease from +4 to 0 associated with photosynthesis in terrestrial plants and marine plankton. The carbon in the repeating molecular unit is reduced by hydrogenation, and combusting this fuel e.g. in the form of cellulose releases 447 kJ per mole of CO2 formed. Further reduction to oxidation state –1 is associated with the creation of coal (not illustrated) which releases about 510 kJ per mole of CO2 formed. Continued reduction to oxidation state –2 is associated with petroleum which releases around 610 kJ per mole of CO2 formed. Finally the formation of natural gas represents the lowest possible oxidation state of carbon, –4. On combustion natural gas releases the maximum energy of 810 kJ per mole of CO2 formed.

These numbers illustrate a general (inverse) relationship between the magnitude of the carbon oxidation state and the amount of energy generated by combustion (see Appendix 1 for more data). For each mole of CO2 released, natural gas (–4) produces nearly twice as much energy as cellulosic biomass (0). That is an appreciable difference, which perhaps deserves more attention in public discourse about greenhouse gas emissions than it receives.

It is also worth noting that although it takes millions of years for natural gas to be formed from atmospheric carbon dioxide in Nature’s Carbon Cycle, the same carbon transformation can be achieved by human beings on a vastly accelerated timescale using a process known as the Sabatier reaction. This has been recently demonstrated in a remarkable Power-to-Gas (P2G) project conducted in Austria.

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Underground storage and conversion

In Pilsbach, Upper Austria, energy company RAG Austria AG conducted a P2G project called Underground Sun Storage in which excess electricity production from wind and solar was converted by electrolysis of water to hydrogen gas which was then pumped down into a depleted natural gas reservoir at a depth of 1 km. Following the successful conclusion of this project, a second P2G project called Underground Sun Conversion was then initiated in which carbon dioxide sourced from biomass combustion or DAC was co-injected with hydrogen into the gas reservoir.

According to RAG, the pores in the matrix of the underground reservoir contain micro-organisms which within a relatively short time convert the hydrogen and carbon dioxide into natural gas, recreating the process by which natural gas originates but shortening the timescale by millions of years. In its project description RAG Austria claims that “this enables the creation of a sustainable carbon cycle”.

How the micro-organisms effect the reaction between H2 and CO2 is not described in the material I have seen. Perhaps microbial enzymes serve as catalysts – the Sabatier reaction is spontaneous and indeed thermodynamically favored under the temperature and pressure conditions of the reservoir (313K, 107 bar).

– – – –

Energy in, Energy out (again)

Ok, so let’s take a look at the thermodynamics of the processes by which RAG Austria turns carbon dioxide into natural gas. Applying Hess’s Law, it can be seen that the sum of the electrolytic process and the Sabatier reaction is equivalent to a reversal of methane combustion and corresponds to the energy stored underground in the C-H bonds of the methane molecule – note that the oxygen is formed above ground during electrolysis and is vented to the atmosphere*

*the sum of reactions bears a curious similarity to the process of photosynthesis: 6CO2 + 6H2O → C6H12O6 + 6O2

Energy loss is intrinsic to both parts of this methane synthesis program. The electrical efficiency of water electrolysis using current best practises is 70–80%, while the Sabatier reaction between H2 and CO2 taking place in the underground reservoir is exothermic and loses around 15% of the energy used to form hydrogen in the initial stage.

On the face of it, underground storage in a natural gas reservoir of hydrogen alone would seem to offer better process economics. But carbon capture and underground conversion can be titrated to achieve a variable quotient between stored hydrogen and converted methane. Both have their economic attractions; what methane lacks in terms of process inefficiencies can be compensated for in several ways in relation to hydrogen. Superior energy density, more efficient transportation as LNG and compatibility with existing energy supply infrastructures are some of them. And then there is the larger issue of the value that society places upon the desire for carbon neutrality in existing energy systems on the one hand, and the promise of carbon-free energy systems on the other.

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Appendix 1

Carbon oxidation state and heat of combustion per mole of CO2 produced


Negative numbers in the last column indicate exothermic reaction i.e. heat release. Units are kJmol-1

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Suggested further reading

EURAKTIV article on energy storage projects June 2019
https://www.euractiv.com/section/energy/news/four-energy-storage-projects-that-could-transform-europe/

RAG Austria AG website – Underground Sun Conversion
https://www.underground-sun-conversion.at/en/project/project-description.html

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P Mander June 2019