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

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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).

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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

Credit: Extinction Rebellion

On Tuesday 2nd April 2019, environmental activists from the Extinction Rebellion group staged a parliamentary protest in London by glueing their bottoms to the windows of the viewing gallery of the House of Commons during a Brexit debate.

They used this cheeky tactic to call on British politicians to act on the ‘Climate and Ecological Crisis’. On its website, Extinction Rebellion wrote that “Government must act now to halt biodiversity loss and reduce greenhouse gas emissions to net zero by 2025”.

There is an irony here in that the call to immediate action was delivered at the precise moment when the UK government was in a state of complete paralysis over Brexit. Then again, the demonstrators’ demands weren’t meant to be taken entirely at face value. Even the most optimistic environmental biologists and chemical engineers would shake their heads at what was proposed to be accomplished in the space of just six years.

What the demonstration did achieve was to focus attention on the task in hand and the current state of progress. And it prompted CarnotCycle to pen this post on one promising technology for carbon capture and conversion (CCC) in which UK researchers are playing a key role.

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It is a well-known fact, not to mention a subject of serious environmental concern, that cows burp methane as a result of anaerobic fermentation of the grass they eat. And being aerobic respirers they also exhale carbon dioxide. Both these substances are potent greenhouse gases associated with human activity whose atmospheric levels, according to the best available science, must at least be stabilized in order to stand a chance of keeping global warming within manageable limits.

The cow in our picture is asking us an interesting question. What if it were possible to react carbon dioxide and methane together to form products that are not greenhouse gases? Even better, what if the reaction products could be put to useful purposes? Just think how cute that would be!

Putting our physical chemistry hats on for a moment and looking at the above equation, we notice that the oxidation states of carbon in the two molecules are at opposite ends of the scale. Methane has the most reduced form of carbon (-4) while carbon dioxide has the most oxidized form of carbon (+4). A redox reaction between the two looks possible and indeed is possible, albeit at elevated temperatures:

This process – called dry reforming of methane or DRM – was first introduced by Germany’s dynamic duo, Franz Fischer and Hans Tropsch, in 1928 but extensive investigation only started in the 1990s when increasing concerns about the greenhouse effect were raised by the international scientific community.

Notice how DRM simultaneously converts two greenhouse gases into two non-greenhouse gases which together make valuable syngas (1:1), a key industrial intermediate in the production of chemicals and clean fuels.

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Well, sort of. The big difficulty in making DRM viable relates to side reactions. The DRM reaction proceeds above 918K but then so does the thermal decomposition of both methane and carbon monoxide which results in carbon being deposited on the catalyst, clogging up the pores and thereby deactivating it.

This problem can however be mitigated in a very neat way by combining DRM with another methane-reforming process, namely steam reforming (SRM). Coupling reactions in this way not only reutilizes the deposited carbon but also adds a product stream with an H2/CO ratio of 3:1 which enables the syngas ratio to be adjusted for the synthesis of methanol, ethanoic acid or dimethyl ether (DME), which has promise as a sulfur-free diesel fuel, or towards the (2n+1):n H2/CO ratio required for Fischer-Tropsch synthesis of alkane fuels

The other problem is that all these reactions are endothermic (heat requiring). This energy has to be obtained from somewhere, and now here comes the next neat idea. Adding oxygen to the reactant stream allows partial oxidation of methane (POM) and catalytic combustion of methane (CCM) to take place, which are exothermic reactions that can supply the necessary heat

Putting three reforming agents – carbon dioxide, water and oxygen – together in the reactant stream with a methane feedstock creates a sufficiently energy-efficient overall process known as ‘tri-reforming’.

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To complete the conceptual scheme, the tri-reforming process is integrated into a cycle where the syngas output is utilized in the tri-generation of fuels, industrial chemicals and electricity, with the flue gases from these processes being fed back after nitrogen purging to the tri-reforming reactor. Carbon dioxide can also be fed into the cycle from external sources such as power plants and cement works.

Note that in principle, carbon dioxide can be fed into the cycle from established carbon capture and storage (CCS) processes. In this way underground reservoirs of anthropogenic carbon dioxide can be utilized as a feedstock for additional tri-generation.

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Almost. I mentioned at the start that the UK was involved in CCC research so I should say a few words about that. Scientists at Oxford and Cambridge are working with the King Abdulaziz City for Science and Technology in Saudi Arabia and the National Natural Science Foundation of China on the tri-reforming/tri-generation technology detailed above.

Since China is the world’s largest CO2 emitter and Saudi Arabia is the world’s largest oil producer, the Anglo-Sino-Saudi initiative seems a sensible geoscientific cluster. The news was announced on 28 January 2018, more than a year before the Extinction Rebellion demonstrations in London.

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Suggested further reading

A mini-review on CO2 reforming of methane

Published in June 2018 this is a useful and easily readable grounder covering the thermodynamic, kinetic, catalysis and commercial aspects of the subject.

Turning carbon dioxide into fuel

A paper co-written by the Oxbridge scientists involved in the Anglo-Sino-Saudi initiative. It was published in 2010, which shows that these guys have been on the case a while. Climate activists take note, and read their stuff.

Turning carbon dioxide into fuel – a new UK-China-Saudi Arabia initiative

The January 2018 press release referred to above. I reckon my poly-alliterative Anglo-Sino-Saudi sounds better.

Tri-reforming: a new process for reducing CO2 emissions

A bedrock paper (January 2001) from Chunshan Song at Penn State. The process diagram featured above is taken from this paper. If you don’t read anything else, read this one.

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P Mander May 2019

Taking a break from thermodynamics, a subject which deeply fascinated Einstein and to which he made significant contributions.

Just to be clear, the headline does not refer to any of Einstein’s academic papers in which he presented the theory of relativity. The error occurs in the popular account Einstein later wrote under the title Über die spezielle und die allgemeine Relativitätstheorie; it then reappears in curiously modified form in the authorised English translation Relativity – the special and the general theory.

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The German original

Part 1 of Einstein’s book deals with Special Relativity. Having discussed coordinate systems, the Principle of Relativity (in the restricted sense), the Lorentz Transformation and the behavior of measuring rods and clocks in motion, Einstein presents in §15 the general results of the theory.

It is at this point that he gives the classical expression for kinetic energy, then the new expression for it according to relativity theory, and then employs a binomial expansion to extend this new expression into a series

From the 24th edition, 2009

And this is where the error occurs.

The binomial expansion of the first term in the parentheses is

Einstein seems to have neglected the “–1” term in the parentheses which cancels out the first term in the expansion. The actual result is

This is what you would expect the relativistic kinetic energy expression to look like, since when v<<c all terms other than the first can be neglected and it reduces to the classical expression ½mv2.

Einstein’s result, on the other hand, reduces to mc2 + ½mv2 (*). Since this includes the non-zero rest-mass energy it cannot be a purely kinetic energy expression which he states it to be.

(*) the mass m in these expressions indicates the rest mass m0

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The English translation

In 1920 an authorised English translation of Einstein’s popular book was published. The curious feature of this translation is that the parenthetic “–1” term in the relativistic kinetic energy expression has been removed, as can be seen below

From the 15th edition, reprinted 1979

This does not rectify the original error, but simply transfers it from one expression to the other. For while the series expression is a correct extension, the relativistic kinetic energy expression from which it is obtained is incorrect as it now contains the non-zero rest-mass energy.

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osm01

I dare say most of you will remember this classroom demonstration, in which water passes through a semi-permeable membrane and causes the liquid level to rise in the stem of the thistle funnel. The phenomenon is called osmosis, and at equilibrium the osmotic pressure is equal to the hydrostatic pressure.

Historical background

This experiment has its origins way back in the mid-18th century, when a French clergyman named Jean-Antoine Nollet tied a piece of pig’s bladder over the mouth of a jar containing alcohol and immersed the whole thing in a vat of water. What prompted him to do this is not known, but we do know the result of his experiment. The bladder swelled up and ultimately burst from the internal pressure.

osm02

Jean-Antoine Nollet 1700-1770

Nollet published his findings in Recherches sur les causes de Bouillonement des Liquides (1748) in which he gave a correct interpretation of the phenomenon, which arises from the much more marked permeability of the bladder to water as compared with alcohol.

osm03

Moritz Traube 1826-1894

The actual measurement of osmotic pressure had to wait for over a century, until the German chemist Moritz Traube showed in 1867 that artificial semipermeable membranes could be made using gelatin tannate or copper ferrocyanide. Traube’s compatriot Wilhelm Pfeffer, a botanist, succeded in depositing the latter in the walls of a porous jar, which when filled with a sugar solution, connected to a mercury manometer and then plunged into pure water, provided a means of measuring osmotic pressures.

osm04

Wilhelm Pfeffer 1845-1920

Following Pfeffer’s osmotic pressure measurements using sucrose solutions, on which JH van ‘t Hoff based his famously flawed gaseous theory of solutions, there were two notable teams of experimentalists – one on each side of the Atlantic – which provided high quality osmotic pressure data to test the ideas of theoreticians. In the USA, Harmon Northrop Morse and Joseph Christie Whitney Frazer led a team at Johns Hopkins University, Baltimore, Maryland from 1901 to 1923. In Britain meanwhile, the aristocrat-turned-scientist Lord Berkeley and co-worker Ernald Hartley set up a private research laboratory near Oxford which operated (with gaps due to war service) from 1904 to 1928.

osm05

Ernald Hartley (1875-1947) Besides being a research chemist, Hartley was an amateur clarinetist who played in the Oxford orchestra for many years. The photo dates from 1925.

While Morse and Frazer used the same principle as Pfeffer, albeit with a more advanced electrochemical method of depositing the membrane in the pores, Berkeley and Hartley reversed the arrangement of solvent and solution, applying measured pressure to the latter to attain equilibrium.

osm06

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Theoretical development in Europe

In Europe, the rigorous application of thermodynamics to the phenomenon of osmosis started in 1887 with Lord Rayleigh, who combined the use of the ideal gas law PV = nRT with the idea of a reversible isothermal cycle of operations in which the sum of work in the complete cycle is zero.

osm07

Lord Rayleigh 1842-1919

Being essentially an attempt to provide hypothesis-free support to van ‘t Hoff’s troubled gaseous theory of solutions, the solute in Rayleigh’s cycle was a mole of ideal gas, which was first dissolved in the solution by applied pressure and then recovered from the solution by osmotic pressure to return the system to its original state.

Rayleigh’s approach, using a zero-sum cycle of operations, was thermodynamically sound and continued to form the basis of theoretical development in its next phase, which in Europe focused on vapor pressure following the influential papers of Alfred Porter in 1907 and Hugh Callendar in 1908.

osm08

Alfred Porter 1863-1939

By 1928, the theoretical model in JAV Butler’s popular textbook The Fundamentals of Chemical Thermodynamics was close to the familiar classroom demonstration of osmosis shown at the head of this post, in which the hydrostatic pressure acting on the solution counteracts the tendency of the solvent to pass through the semi-permeable membrane. At equilibrium, the hydrostatic pressure P is equal to the osmotic pressure.

osm09

JAV Butler 1899-1977

To obtain a thermodynamic relation for osmotic pressure in terms of vapor pressures, Butler uses Rayleigh’s idea of a reversible isothermal cycle of operations together with a semipermeable membrane in the form of a movable piston between the solution and the solvent:

osm10

The diagram shows a solution under hydrostatic pressure P which is equal to the osmotic pressure. Below the semi-permeable piston is pure solvent. Butler then applies the following argumentation:

1] Vaporize 1 mole of the pure solvent at its vapor pressure p0, and expand it reversibly so that the vapor pressure falls to p equal to the partial pressure of the solvent in the solution (Butler assumes that p is not affected by P applied to the solution). Condense the vapor into the solution. Since the work of vaporization and condensation cancel out, the only work done is the work of expansion. Assuming the vapor obeys the ideal gas law, the work (w) done is given by the textbook isothermal expansion formula

2] Now move the semi-permeable piston up against the pressure P until a quantity of solvent equivalent to 1 mole of vapor has passed through it. If the decrease in the volume of the solution is ΔV, the work done is PΔV.

The cycle is now complete and the system has returned to its original state. The total work done is zero and we may equate the two terms

(1)

where P is the osmotic pressure, ΔV is the partial molal volume of the solvent in the solution, p0 is the vapor pressure of the pure solvent and p is the vapor pressure of the solvent in the solution. This thermodynamically exact relation, which involves measured vapor pressures, is in good agreement with experimental determinations of osmotic pressure at all concentrations.

There is a great irony here, in that this equation is exactly the one that JH van ‘t Hoff found his way to in Studies in Chemical Dynamics (1884), before he abandoned his good work and went completely off-track with his gaseous theory of solutions.

ae01

JH van ‘t Hoff 1852-1911

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Theoretical development in America

In the US, the theory of the semipermeable membrane and the ‘equilibrium of osmotic forces’ was the work of one supremely gifted man, Josiah Willard Gibbs, who more or less single-handedly laid the theoretical foundations of chemical thermodynamics in his milestone monograph On the Equilibrium of Heterogeneous Substances.

dce03

J Willard Gibbs 1839-1903

But before delving into the powerful idea he introduced, let us return to the subject of equilibrium in a system subject to osmotic pressure with a set-up that is slightly different to that used by Butler. In the diagram below, the piston supplies pressure Psoln to the solution which is just enough to stop solvent passing through the membrane and bring about equilibrium at constant temperature; the osmotic pressure is defined as the excess pressure Psoln – p01.

osm13

The question can now be asked: Does the condition of osmotic equilibrium coincide with equality of a thermodynamic variable on either side of the membrane? Clearly it cannot be pressure or volume, nor can it be temperature since constant temperature does not prevent osmotic disequilibrium.

The P, V, T variables do not provide an affirmative answer, but in his monumental masterwork, Gibbs supplied one of his own invention which did – the chemical potential, symbolized μ. It is an intensive variable which acts as a ‘generalized force’, driving a system from one state to another. In the present context the force drives chemical components, capable of passing through a membrane, from a state of higher potential to a state of lower potential.

So given a membrane dividing solution from solvent and permeable only to the latter, we can understand the osmotic force driving the solvent (designated by subscript 1) through the membrane into the solution in terms of movement to a region of lower potential since

Now the difference in potential can be calculated according to the textbook formula

(2)

where x1 is the mole fraction (<1) of the solvent in the solution. To achieve equilibrium, the chemical potential of the solvent in the solution must be increased by the amount –RTlnx1 (a positive quantity since lnx1 is negative). This can be done by increasing the pressure on the solution since

is always positive (V1 is the partial molar volume of the solvent in the solution).

The osmotic pressure is defined as the excess pressure Psoln – p01. As can be seen from the diagram below, this is the pressure required to raise the chemical potential of the solvent in the solution so that it becomes equal to the chemical potential of the pure solvent.

osm16

Since the slope is V1, it follows that

(3)

Combining (2) and (3) and designating the osmotic pressure by P gives the desired equilibrium relation

This is exactly equivalent to equation (1) derived by Butler, since by his terminology

The two methods of proof are thus shown to be equivalent – we can regard osmotic pressure as the excess pressure required to increase either the chemical potential or the vapor pressure of the solvent in the solution. But Gibbs saw an advantage in using potentials, which he voiced in an 1897 letter to Nature entitled Semi-Permeable Films and Osmotic Pressure:

“The advantage of using such potentials in the theory of semi-permeable diaphragms consists … in the convenient form of the condition of equilibrium, the potential for any substance to which a diaphragm is freely permeable having the same value on both sides of the diaphragm.”

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