How the spontaneous mixing of gases can be made to do work

One of the many statements of the Second Law of Thermodynamics is the following:

Spontaneous changes are those which, if carried out under the proper conditions,
can be made to do work.

The problem with this statement is that there are examples of spontaneous change where it is not immediately obvious what the ‘proper conditions’ might be. The mixing of two perfect gases is a case in point. As the header diagram shows, two gases initially compartmentalized in a container will each expand to fill the space available to them when the partition is removed. The process is spontaneous but no work is done.

There is however another means of achieving the same mixing result if the partition is replaced with a piston equipped with a membrane permeable to gas A but not to gas B. This can be achieved in practice for example with palladium, which is easily permeable to hydrogen but not to other gases.

During this process the piston does PV work of 1 unit, displacing the original volume of the left compartment V = 1 at pressure P = 1. If the piston were connected to an external arrangement for extending a spring or raising a weight, the spontaneous mixing of gases could be made to do mechanical work.

Reversing the above process by doing mechanical work on the system would afford a method of separating the mixed gases. In either case, maximal efficiency would be achieved (see reference below).

– – – –

Postscript

The remarkable diffusion properties of hydrogen

In Munich in 1883, Max Planck conducted an experiment in which a platinum tube originally containing hydrogen at atmospheric pressure was heated until the platinum became permeable to hydrogen, upon which it was found that almost the whole contents diffused out leaving a high vacuum (see reference below)!

Reference: G.H. Bryan, Thermodynamics, published by BG Teubner, Leipzig 1907, page 126

– – – –

Advertisement

Reversible and irreversible change

Reversible change is a key concept in classical thermodynamics. It is important to understand what is meant by the term as it is closely allied to other important concepts such as equilibrium and entropy. But reversible change is not an easy idea to grasp – it helps to be able to visualize it.

Reversibility and mechanical systems

The simple mechanical system pictured above provides a useful starting point. The aim of the experiment is to see how much weight can be lifted by the fixed weight M1. Experience tells us that if a small weight M2 is attached – as shown on the left – then M1 will fall fast while M2 is pulled upwards at the same speed.

Experience also tells us that as the weight of M2 is increased, the lifting speed will decrease until a limit is reached when the weight difference between M2 and M1 becomes vanishingly small and the pulley moves infinitely slowly, as shown on the right.

We now ask the question – Under what circumstances does M1 do the maximum lifting work? Clearly the answer is visualized on the right, where the lifted weight M2 is as close as we can imagine to the weight of M1. In this situation the pulley moves infinitely slowly (like a nanometer in a zillion years!) and is indistinguishable from being at rest.

This state of being as close to equilibrium as we can possibly imagine is the condition of reversible change, where the infinitely slow lifting motion could be reversed by an infinitely small nudge in the opposite direction.

From this simple mechanical experiment we can draw an important conclusion: the work done under reversible conditions is the maximum work that the system can do.

Any other conditions i.e. when the pulley moves with finite, observable speed, are irreversible and the work done is less than the maximum work.

The irreversibility is explained by the fact that observable change inevitably involves some dissipation of energy, making it impossible to reverse the change and exactly restore the initial state of the system and surroundings.

– – – –

Reversibility and thermodynamic systems

The work-producing system so far considered has been purely mechanical – a pulley and weights. Thermodynamic systems produce work through different means such as temperature and pressure differences, but however the work is produced, the work done under reversible conditions is always the maximum work that a system can do.

In thermodynamic systems, heat q and work w are connected by the first law relationship

What this equation tells us is that for a given change in internal energy (ΔU), both the heat absorbed and the work done in a reversible change are the maximum possible. The corresponding irreversible process absorbs less heat and does less work.

It helps to think of this in simple numbers. U is a state function and therefore ΔU is a fixed amount regardless of the way the change is carried out. Say ΔU = 2 units and the reversible work w = 4 units. The heat q absorbed in this reversible change is therefore 6 units. These must be the maximum values of w and q, because ΔU is fixed at 2; for any other change than reversible change, w is less than 4 and so q is less than 6.

For an infinitesimal change, the inequality in relation to q can be written

and so for a change at temperature T

The term on the left defines the change in the state function entropy

Since reversible conditions equate to equilibrium and irreversible conditions equate to observable change, it follows that

These criteria are fundamental. They are true for all thermodynamic processes, subject only to the restriction that the system is a closed one i.e. there is no mass transfer between system and surroundings. It is from these expressions that the conclusion can be drawn – as famously stated by Clausius – that entropy increases towards a maximum in isolated systems.

Rudolf Clausius (1822-1888)

– – – –

Die Entropie der Welt strebt einem Maximum zu

Consider an adiabatic change in a closed system: dq = 0 so the above criteria for equilibrium and observable change become dS = 0 and dS > 0 respectively. If the volume is also kept constant during the change, it follows from the first law that dU = 0. In other words the volume and internal energy of the system are constant and so the system is isolated, with no energy or mass transfer between system and surroundings.

Under these circumstances the direction of observable change is such that entropy increases towards a maximum; when there is equilibrium, the entropy is constant. The criteria for these conditions may be expressed as follows

Note:
The assertion that entropy increases towards a maximum is true only under the restricted conditions of constant U and V. Such statements as “the entropy of the universe tends to a maximum” therefore depend on assumptions, such as a non-expanding universe, that are not known to be fulfilled.

– – – –

P Mander March 2015

Spontaneity and the performance of work

Imagine a perfect gas contained by a rigid-walled cylinder equipped with a frictionless piston held in position by a removable external agency such as a magnet. There are finite differences in the pressure (P1>P2) and volume (V2>V1) of the gas in the two compartments, while the temperature can be regarded as constant.

If the constraint on the piston is removed, will the piston move? And if so, in which direction?

Common sense, otherwise known as dimensional analysis, tells us that differences in volume (dimensions L³) cannot give rise to a force. But differences in pressure (dimensions ML^-1T^-2) certainly can. There will be a net force of P1–P2 per unit area of piston, driving it to the right.

– – – –

The driving force

In thermodynamics, there exists a set of variables which act as “generalized forces” driving a system from one state to another. Pressure is one such variable, temperature is another and chemical potential is yet a third. What they all have in common is that they are intensive variables – those which are independent of the quantity of a phase.

Each intensive variable has a conjugate extensive variable – those which are dependent on the quantity of a phase – and together they form a generalized force-displacement pair which has the dimensions of work (= energy ML²T^-2). Examples include pressure × volume, temperature × entropy, and voltage × charge.

But back to those intensive variables. Experience confirms that it is these generalized forces which are the agents of change. And spontaneous change results when there are finite differences in intensive variables between one system and another – the direction of change being determined by their relative values.

– – – –

The result is work

In the illustrated example above, the freed piston moves spontaneously to the right because of finite differences in pressure (P1>P2). As a result, the system consisting of the left hand compartment does PV work on the system consisting of the right hand compartment. Likewise, finite differences in other intensive variables such as temperature and chemical potential will act through their respective force-displacement pairs to perform work.

We can thus conclude that finite differences in intensive variables drive spontaneous change, and that due to the dimensionality of their respective conjugate extensive variables, spontaneous change results in the performance of work. It should however be noted that this work is not always useful work. The spontaneous diffusion of two gases into each other is a classic case, where it is difficult to imagine how the work could be usefully obtained.

– – – –

P Mander March 2015

James Joule and the end of electrical euphoria

St. Peter’s Parish Church, Bradford

The year is 1845. The location is the vicarage of St. Peter’s Parish Church, Bradford in Northern England. Inside the vicarage two men, one a 55 year-old retired sea captain and the other a 26 year-old brewery manager, have been conducting a series of experiments using a powerful magneto-electric apparatus. The results that they would publish the following year in Philosophical Magazine would prove beyond doubt that the electric motor was not the perpetual motion miracle that many believed, and put an end to the ‘electrical euphoria’ that had swept through Europe and the United States for a decade.

The retired sea captain, now the vicar of the parish and the owner of the massive magneto, was William Scoresby Jnr, a renowned Arctic explorer and an authority on whale fishery and compass navigation.

The young brewery manager, who had already published a number of scientific articles in the Annals of Electricity and who represented the ‘brains’ behind the experiments at the vicarage, was James Joule.

– – –

It was Michael Faraday’s discovery of electromagnetic induction in 1831 that started it all. The magneto was invented that same year, and with the additional invention of the commutator by William Sturgeon and others, the electric motor became a reality in 1832. Inventors in Europe and America, sensing its possibilities, flocked to develop it.

William Sturgeon 1783-1850, inventor of the electromagnet and commutator

In 1835, the euphoria began in earnest when Professor Moritz von Jacobi, a German professor of engineering at the Imperial University of Dorpat in Russia, published a paper on the physics of the electric motor in which he pointed out that if frictional forces and the counter-electromotive force (back emf) could be reduced to small proportions by ingenious design, a near-infinite velocity of rotation could be achieved and with it, virtually limitless power.

Moritz von Jacobi 1801-1874, whose 1835 paper triggered electrical euphoria

Jacobi’s paper was translated into English and appeared in 1836 in the Annals of Electricity, a British journal newly founded and edited by William Sturgeon, who besides independently inventing the commutator had also invented the electromagnet (in 1824). Among those who read Jacobi’s paper in the Annals was James Joule, an 18 year old lad from Manchester who had a fascination with magnetism and electricity.

Born into a wealthy family whose business was beer brewing, young James had plenty of free time to follow his favorite pursuit – electrical experiments. In Jacobi’s paper he saw the opportunity to turn his practical skills and ingenuity to good effect by designing and building an electric motor that would realize the promise of unbounded power. He worked for three solid years on this task, and got absolutely nowhere. But it was not a total waste of time, because along the way the teenager from Manchester made a significant discovery.

– – – –

The 1830s were important years for the emerging science of electricity. The magneto, the electric motor and the tangent galvanometer were all invented in this decade. And in 1834, the man who influenced the direction of scientific enquiry in Britain more than any other – Michael Faraday – discovered the laws of electrolysis. Unusually for Faraday, whose genius otherwise expressed itself in conceptual, qualitative relationships between things, his laws of electrolysis were quantitatively exact.

The effect of this on Joule cannot be understated. Although he had no training and little interest in mathematics, the young experimenter from Manchester was a natural hunter for numerical relationships between measurable quantities. With Faraday’s quantitative discoveries to spur him on, and with a home laboratory full of accurate (and expensive) devices for measuring temperature and electric current, Joule was well-equipped for discovery. And it was not long in coming.

In December 1840, at the age of 21, Joule sent to the Royal Society a paper entitled “On the production of heat by Voltaic electricity”. In it, he stated that the heat generated by the passage of an electric current through a wire is proportional to the square of the current and the resistance of the wire. He further claimed that the law held irrespective of the shape, size or form of the circuit or the type of wire used. In other words, he was enunciating a general law. The Royal Society rejected the paper for the prestigious Transactions, but allowed a short abstract of it to appear in the more lowly Proceedings for that month. What we now know as Joule’s first law (ohmic heating) made a modest entry into this world.

– – – –

William Scoresby (1789-1857), owner of the great magneto-electric machine

James Joule first met Captain the Reverend William Scoresby at a scientific meeting in Manchester in 1842. Scoresby was a colorful character by all accounts; having attended Edinburgh University he decided to follow his father’s calling and go to sea, becoming the master of a whaling ship in true pre-mechanized, Moby Dick fashion. Following the death of his wife, Scoresby gave up the sea to take Holy Orders and was now installed as the Vicar of Bradford.

Unlike most whalers of his era, he had the benefit of a scientific education, and his time at sea had given him a strong interest in terrestrial magnetism and its application in compass navigation. It was a common interest in magnetism that brought Scoresby and Joule together, and it was at this meeting that Joule learned of Scoresby’s ‘great magneto-electric machine’ now housed in the vicarage at Bradford. It was a mighty machine with enormous magnets, capable of producing a hefty current. Joule now had an acquaintance who owned the mother of all magnetos.

– – – –

Joule’s discovery of ohmic heating during his futile attempts to build an ingeniously designed electric motor introduced the first seeds of doubt in his mind about the validity of Professor Jacobi’s bold assertion. The unavoidable heating of the circuit during power production clearly implied a waste of zinc in the Daniell cell which Joule had used a source of voltaic electricity. And since zinc cost 60-70 times the same weight of coal, things didn’t look quite so promising, at least from an economic perspective.

From a purely scientific perspective the heating effect also set Joule wondering, like Count Rumford before him: Where does the heat come from? If heat were a substance, as the scientific establishment insisted it was, the heating effect could only be explained by transference, i.e. the magneto acting as a heat pump to move heat from the armature to the outside circuit and being necessarily cooled in the process. This was something that could be ascertained by careful experiment, and Joule appointed himself to the task.

The summer of 1842 was a hectic time for Joule. Besides visiting Scoresby in Bradford to experiment with the massive magneto in the vicarage, he was also slaving away at home in Manchester on those careful experiments which would eventually bring him to prominence as a great man of science. But years of struggle lay ahead of him before he achieved that goal.

Suffice to say that in his 1843 paper “On the calorific effects of magneto-electricity and on the mechanical value of heat”, read before the British Association for the advancement of science, Joule demonstrated that heat was not a substance; it was generated by the chemical action of the battery when the magneto-electric machine was operated as a motor, and by mechanical work when operated as a magneto. And he went on to propound the thesis that not only were work and heat interconvertible, but that a precise numerical relationship existed between them – what he called the mechanical value of heat.

In 1843, no-one listened. The scientific establishment dismissed Joule’s youthful paper – he was only 24 – as the work of a misguided and uneducated amateur.

– – – –

While the British Association was busy ignoring Joule, Joule was busy thinking about zinc, the anode metal that was consumed in the Daniell cell which he had used in his experiments. The idea was forming in Joule’s mind that the work obtained from an electric motor must arise from the conversion of heat due to the chemical reaction by which zinc was consumed in the cell. Since a precise numerical relation existed between work and heat, there must be a limit to the work obtainable from the consumption of a given amount of zinc. In Joule’s mind, Professor Jacobi’s golden vision of the electric motor as a source of limitless power was fading fast.

Joule followed up on his new thinking with a truly inspired idea. He reasoned that since the heat released from the complete combustion of zinc in a calorimeter must represent the maximum heat attainable, he could calculate – by means of the mechanical equivalent – the maximum work attainable from the consumption of a given amount of zinc by any means whatsoever. Using this yardstick, he could rate the performance of any design of electric motor.

Joule wasted no time in doing the calorimetry and determining the maximum work using his best estimate of the mechanical equivalent of heat. He was now ready to test his idea on an actual magneto-electric machine, the biggest one he knew of, which sat in the home of Captain the Reverend William Scoresby.

– – – –

I mentioned at the start of this post that Joule represented the ‘brains’ behind the experiments conducted at the vicarage of St. Peter’s Church Bradford in 1845. Without any disrespect to Scoresby’s undoubted competences as a scientific thinker and writer, the work that was accomplished here and published under their joint names the following year was very much a synthesis of all that Joule had learned from his own efforts. Indeed, Joule notes in the introduction to their paper that he wrote it almost entirely himself.

Joule used three Daniell cells to power Scoresby ‘s machine as an electric motor, and his purpose in conducting the experiment was to determine the quantity of heat (arising from the consumption of zinc) that was converted into useful work. His method was simple, but brilliantly conceived.

Joule first connected the battery to the motor at rest i.e. with the rotor prevented from turning, and with a carefully calibrated tangent galvanometer measured the standing current (a). He then allowed the rotor to turn, and when the motor was running steadily, he measured the driving current (b). Joule knew from his own discovery that ohmic heating would be generated by the current in the circuit, and from Faraday’s law that the measured current was in direct proportion to the quantity of zinc consumed in the battery. The difference of the currents, a – b, “will therefore represent the quantity of heat converted by the engine into useful mechanical effect” as Joule put it (see Appendix I for clarification of this statement).

It was now a simple matter to compute the mechanical effect of the motor. Joule had already determined by calorimetry that if all the heat from the combustion of a grain of zinc could be converted into work, it would be equal to raising 158 pounds through a distance of 1 foot.
Since (a – b) represented the heat actually available for conversion, and (a) the amount theoretically available, the work obtainable from the motor per grain of zinc was:

W = 158 x (a – b)/a.

– – – –

It was this formula, and the generalization it implied, that provided the final refutation of Professor Jacobi’s heady suggestion of infinite power and put an end to electrical euphoria. For it showed that the work obtainable from a source of electrical power is finite, and limited by the heat (=energy) in the fuel.

Furthermore, it showed that the maximum amount of work can only be achieved when (b) becomes vanishingly small. But since (b) is the driving current, the power will then also tend to zero, and it would take an infinitely long time to accomplish the work (students of classical thermodynamics will no doubt detect a familiar principle here).

Besides electric motors, Joule and Scoresby also discussed the fuel economy of steam engines and horses in their paper, and came to a firm conclusion on which of them was the most efficient. But for now we must leave them, sipping their brandy by the fireside and contemplating what they had discovered.

For Joule, it would be a further two years before anyone paid any attention at all to the fact that he had discovered an important electrical heating law, laid the foundations for the first law of thermodynamics, revealed the limits imposed on power production by the second law, and saved gullible investors from wasting their life savings on electrical dream machines.

But he was only 26 years old. He had time on his side.

– – – –

8 years later…

William John Macquorn Rankine (1820-1872)

One day in 1853, a letter arrived for the now famous James Joule. It was from William Rankine, a Scottish engineer and physicist, who was at the time playing a central role in laying the theoretical foundations of thermodynamics.

In the letter, Rankine drew attention to the fact that the efficiency of an electric motor, given by the fraction (a – b)/a in the Joule-Scoresby formula

was exactly analogous to the thermodynamic expression for the efficiency of a heat engine

(q₁ – q₂)/q₁

where q₁ is the heat received, q₂ is the heat rejected, and the difference between them is the heat converted into useful work. This expression had been deduced from complex theoretical considerations by Rankine, Rudolf Clausius and William Thomson, all skilled mathematical physicists.

It is a testament to Joule’s abilities as an experimental physicist that, without the benefit of his contemporaries’  mathematical training, he nonetheless found his way to the same mathematical truth.

An engraving from the earliest known portrait of James Joule, painted 18 years after the experiments with William Scoresby in Bradford

– – – –

Links to the original scientific papers cited in this post

Professor Jacobi’s paper, published in the Annals of Electricity in 1836: http://archive.org/stream/annalselectrici01sturgoog#page/n422/mode/2up

James Joule’s first published work, a letter to the Annals of Electricity written in 1838 at the age of 19: http://archive.org/stream/annalselectrici01unkngoog#page/n150/mode/2up

An abstract of Joule’s rejected 1840 paper “On the production of heat by Voltaic electricity”, in which he states the law of ohmic heating: http://archive.org/stream/scientificpapers01joul#page/58/mode/2up

Joule’s 1843 paper “On the calorific effects of magneto-electricity and on the mechanical value of heat”, ignored by science until William Thomson (later Lord Kelvin) suddenly seized on its significance in 1847: http://archive.org/stream/scientificpapers01joul#page/122/mode/2up

Joule & Scoresby’s joint paper, published in Philosophical Magazine in 1846: http://archive.org/stream/scientificpapers00joul#page/n17/mode/2up

– – – –

photo credits

Moritz von Jacobi / wikipedia
William Sturgeon / ssplprints.com
William Scoresby /sciencedirect.com
William Rankine / Wikipedia

– – – –

Appendix I

The formula at the end of the above paragraph from Joule and Scoresby’s paper correctly expresses the theoretical duty of an electric motor produced at the expense of zinc consumed in a Daniell cell. But the reasoning by which it is reached contains an apparent non sequitur which necessitates explanation.

The quantity a – b has the units of electric current (dimensions QT^-1) and so cannot represent a quantity of heat or work whose units are those of energy (dimensions ML^2T^-2). The term a – b in the numerator requires multiplication by the electromotive force E of the battery powering the motor and by running time T to give it the correct units. But because the same multiplication factor applies to the denominator the multiplication factors cancel out so that (a – b)/a represents the same fraction as ET.(a – b)/ET.a which is the dimensionally correct expression for work obtained from heat supplied – in other words the electric motor’s thermodynamic efficiency.

– – – –

P Mander October 2013, additions May 2023