3±i TIME STATES: ∆-GAS ∆—±1 FIELDS< ∆ LIQUID ∆±1 WAVES > ∆±1 PARTICLES ∆-CRYSTALS
In the graph, the worldcycle of matter. We an see how the ∆±1 birth and death phase (plasma made of dissociated ions, is the same and closes the cycle.
The same cycle on the lower ∆-1 quantum scale, in which liquid waves, gaseous fields and solid particles play the same roles on quantum systems.
As those states happen in the ‘human scale, ∆o±1, thermodynamics is the referential stience, albeit it must be fully corrected to avoid the traps of its ‘wrong laws’, which define a Universe with only a single arrow of entropy, as if it were merely a ‘gaseous Universe’; due to the complete misunderstanding of time arrows in classic physics.
The importance for humanity of thermodynamics, today somewhat relegated by the quanta world of electronic machines as humans and life become expendable to the new ‘robotic species’, is difficult to stress.
We are NOT quantum beings as machines are, neither Gravitational, cosmological beings, as the galaxy is. We are thermodynamic beings, and this is the key science for the human system and the planet we live in.
Since in geological structures the interplay of gas, liquid and solid cycles creates the conditions for Gaia, to become a super organism and its ∆-1 life scale to flourish.
Thermodynamics (original texts from britannica cd) and its generator equations
ENERGY. As we have said Energy is a present, Max. e x i function, where entropy understood as motion is dominant but there is form, hence there is in-form-ation in the motion. So its equations can be resumed in terms of lineal, expansive motions with a minimal form.
WORK therefore, which is essentially the measure of ‘active energy’; that is, ‘energy in time’, does not exist if there is not an open displacement, a lineal motion.
TEMPERATURE on the other hand is precisely a vibrational mode of energy, which closes its vibration returning almost all systems in which we measure temperature, vibrational solids, gases studied in volumes closed by pressure; so it plays the role of a ‘time clock’ in the ∆-human scale.
So finally we arrive to HEAT a very anthropomorphic concept as all those related to the present understanding of thermodynamics; but less important in ∆st. Heat is indeed just the entropic, scattering expansion of energy as entropy: ST (ENERGY) > S (HEAT); and so a form of entropy, which therefore allow us to write in simple terms, the Generator ‘elements’ of thermodynamics:
Γ. Spe (Entropic Heat) < ST (Energy-Work) > Tiƒ (Temperature clocks)
The laws of thermodynamics.
What subtle correction must we then introduce into the laws of thermodynamics?
Obviously as thermodynamics considers only a single plane of space-time and reduces the 3 arrows of time and its symmetric generator in space to a single entropic arrow, here is where there are huge misunderstandings, essentially the absurd idea that ‘the Universe is dying’ (Helmoth).
Of course it is if it had only the negative time arrow of entropy and was a single plane; but what entropy ‘kills’, information resurrects, and the losses of ‘entropy’ in a given scale de to thermodynamic equilibrium are reversed by the order of Tiƒ systems, the ‘Demons of Maxwell’, which act creating organic order. Plus, the inverse arrows of the 5th dimension allow that entropy losses, when we try to ‘move’ the energy of an ∆-1 scale to an ∆-scale are compensated by the perfect synchronicity without loss of energy when we act from a whole into a smaller part:
In the graph, thermodynamics change completely in its interpretation when we consider multiple scales of time, as the loss of entropy in Simplex Physics (ab. Sx, or Æ) from ∆-1 to ∆ is upset by the inverse growth of in/form/ation from ∆ to ∆-1 since those motions do not have entropy (synchronicity of motion).
The most important laws of thermodynamics are thus transformed:
-0: The zeroth law of thermodynamics. When two systems are each in thermal equilibrium with a third system, the first two systems are in thermal equilibrium with each other. This property makes it meaningful to use thermometers as the “third system” and to define a temperature scale. And it is indeed truth but it does NOT mean that the system is dead because they have thermal equilibrium. It is all more subtle. A system in thermal equilibrium has a higher degree of order. It becomes therefore able to organise itself better with information, as in life systems which requires homeostasis.
So we shall instead generalise it to all systems with a different name (the willy-nilly game of ‘numbers’, for the awe-inspiring-digital groupie do not apply to GST, as other scales of reality do have different languages and we are unifying them):
The Law of homeostasis, and state that:
“A system of fractal points joined by a present ∆-wave of ‘energy≈heat’ tends to distribute ‘democratically’ the energy and form of the wave to all its components, to ensure the internal balance of them’. And put as customary in the Homologic Method of GST, a minimum of 3 examples from physical, biological and social sciences:
In other terms, systems tend to establish a just, distributive balance between the points of the network, which will receive a minimal amount of energy, call it a blood networks,
-1st: The first law of thermodynamics, or the law of conservation of energy.
The change in a system’s internal energy is equal to the difference between heat added to the system from its surroundings and work done by the system on its surroundings. It is a trivial laws as it is explained today without understanding what is energy. And the most important fact of it: that there is no work or energy expenditure in closed time-cycles.Essentially it means that Time does NOT work, meaning time cycles are closed cycles which do NOT spend work-energy; and this amazingly important fact, hardly understood philosophically by the wannabe gurus of the absolute means ultimately that time is eternal, and so it is the Universe, which has no manifest destiny, no lineal goal. Lineal time indeed would exhaust itself as it would spend the energy of the Universe.
Again we must expand this law to include information and at the same time, diminish its range by applying it only to exchanges within a single plane of existence, which not being the general rule as always there are leaks of energy and information up and down those scales, restrict the accuracy of our measures.
‘Energy never dies, it constantly transforms back and forth into information, through the repetition of body-wave across ∆±1 planes of existence: Spe < ExI>Tiƒ
-2nd: The second law of thermodynamics. Heat does not flow spontaneously from a colder region to a hotter region, or, equivalently, heat at a given temperature cannot be converted entirely into work.
Yet while this is truth it only applies to the transmission of energy from lower, ∆-1 ensembles into an ∆-whole, not viceversa: ∆-wholes synchronise the simultaneous motions of all its ∆-1 parts, achieving with it ‘lesser information’ and ‘simpler, larger motions’ a loss of zero entropy when they move all its parts, according to the direction of future set up by the whole.
So the error of physicists is as usual to generalise a local phenomena of the ∆±1 thermodynamic scale, and forget the balances obtained from non-entropic motions handled by the whole/mind, which restaures the balance of the system.
By ignoring the ∆@ elements of reality thermodynamics looses any value as a philosophy of general global laws.
Such laws apply only to the entropy of a closed single ∆-scale system, which tends toward an equilibrium state in which entropy – the scattering and equality of heat among all its ∆-1 elements, is at a maximum and no energy is available to do useful work at the ∆-level.
This asymmetry between forward and backward processes gives rise to what is known as the “arrow of time” in classic thermodynamics, which as we said is a simplification of the 3 arrows of time, due to the error of a single space-time continuum, converted into a huge global error by extending it to every system.
Indeed, ‘a single entropic arrow of time’ for all phenomena deduced of the study of expansive heat=entropy in steam machines, is a very local reductionist simplex analysis of time arrows. And to expand it to include all the planes of the Universe, all the beings, by reducing the 7 motions of reality to ‘heat, entropic motions’ is plainly bogus. The law merely becomes a single arrow of dying entropic time, when we eliminate all other scales and st, T, ELEMENTS of the system. And we will return to that.
Even when considering the single arrow between atoms and heat-human scale, the comprehension is completely blurred by the language – what entropy measures is not motions backwards and forwards in ‘absolute time’ but in the 5th dimension of social scalar evolution, and backwards in such dimension of dissolution of wholes and its order.
Such as when we want to use and exhaust the motion of ∆-1 systems, which are NOT organised by complex networks but just merely as humans do, with ‘heating machines’, entropic ‘fire’, and some other brutish systems, obviously the molecules and atoms of the lower scale have the same interest to order perfectly and give up its motion to that brutish wholes a mass of humans have to be herded by a military thug.
Entropy appears with minimal processes of organisation such as heat is to extract the motion of individual ∆-1 elements.
However when a system is fully organised, according to 5D organic laws entropy greatly diminishes, as in Crystals, which have basically zero entropy, or organisms, which increase the order of beings and diminish its entropy.
TO deny the existence of fractal points that gauge information and order the Universe, gravitational forces that balance expansive, electromagnetic entropy, is just a dogma of astounding arrogance on the part of physicists, which just want to construct reality with their ‘restricted matter laws’ and will not accept there is something more out there.
So the law should be rephrased regarding the conservation of motion and information:
‘In the whole Universe entropy does not exist. As the Universe is made of motion and curvature, which balance each other through all its ∆-scales. So when a system becomes disordered and expands its entropy in an ∆-1 scale; the order is restored by the simultaneous, informative order of larger ∆+1 wholes which contract and synchronise the ∆-systems.
So the ∆-n quantum scales of physical, electromagnetic entropy are balanced by the ∆+n scales of gravitational, only-attractive information. And we have to asses the total order of a system, studying at least ∆±n scales together.
-The third law of thermodynamics. The entropy of a perfect crystal of an element in its most stable form tends to zero as the temperature approaches absolute zero. This allows an absolute scale for entropy to be established that, from a statistical point of view, determines the degree of randomness or disorder in a system.
So we come to the third law, which establishes that absolute zero, where thermodynamic effects of entropic disorder by heat, no longer apply. We can say that ‘organic physical matter’ and its most important effects of order creation happen when thermodynamic entropy is minimal – phenomena such as superconductivity, superfluidity, bosons, etc. We talk then of zero as the relative ‘temperature’ of a mind, which creates a still map of reality it will then project on that reality diminishing its entropy. In other words:
‘The Universe is filled with Maxwell’s demons’
Zero cyrstal entropy tells us also several things: a crystal, or perceptive physical, Tiƒ mind that maps out an ‘intelligible’ mirror image of the Universe inside its mind is cold, tends to total order, and minimal motion, so it is the 3rd of physical matter. But the crystal does move the Universe, as it becomes focus of smaller ∆-2 pixels of information below it. So again, when we consider several scales motion never stops.
Latter on we shall therefore redefine those laws in terms of ∆@st, as the ∆@ elements do not exist in physics, and they are the ∆ dual arrows of order of the previous pyramid and the ‘Maxwell demon @minds’ that tell the system where to be and go.
How then including those ∆-scales beyond, modify those laws. Obviously: creating order. I.e. gravitation is a force of order as it only attracts and ‘contracts’, in-form-ing a system; so it balances the tendency to entropy of ¥-rays.
So we must talk of a Universe which rescues the world from entropic death through ∆• elements.
So because thermodynamics developed rapidly during the 19th century in response to the need to optimize the performance of steam engines, it is wrong to expand by dogma to the whole universe the laws of thermodynamics, as previously written, which makes them applicable only to physical, molecular matter, the ∆±1 scales and biological systems, without understanding how order is restored in biological systems by its minds, in ∆±1 by crystals and future arrows of social evolution. As the system either has motion or form, often switching between both states; a fact only recognised by the pioneers of fractal physico-chemistry such as Mr. Mehaute, which clearly proves that when a thermodynamic system stops moving externally, it then subtly starts to create further internal order.
So, the ‘particular’ laws of thermodynamics give a complete description of all changes in the energy state of any system and its ability to perform useful work on its surroundings, but only when we make a ‘ceteris paribus’ analysis of that plane and its internal phenomena, discharging the interchanges of energy and information with the ∆±1 scales.
In that sense thermodynamics also has an scalar structure, and so epistemologically we talk of 2 branches:
∆o: Thermodynamics or Classical thermodynamics which does not involve the consideration of individual atoms or molecules.
∆-1>∆: Such concerns are the focus of the branch of thermodynamics known as statistical thermodynamics, or statistical mechanics, which expresses macroscopic thermodynamic properties in terms of the behaviour of individual particles and their interactions.
It has its roots in the latter part of the 19th century, when atomic and molecular theories of matter began to be generally accepted. And so a clear form to study the ‘errors’ of a single space-time analysis of entropy and one which includes at least a couple of ∆-scales is to see the differences between both scalar approaches. Since the key to understand properly thermodynamics is to analyse how disorder in one scale ∆-heat in fact merely means that the ∆-1 scales wishes to create its proper order and motions.
So we need to include besides the study of open and closed states of a thermodynamic system in a single ∆-scale, the ‘whole picture’ by adding the study of ‘closed and opened’ systems in 3 ∆±1 scales, asking questions such as:
‘This event study transfers energy or information to the ∆±1 scales of the system and if so, which arrow is dominant, ∆+1>∆ order or ∆-1<∆ entropy?
The classic example being, the beta decay equations of disorder of a particle, which seems to miss spin (cyclical order) and entropy (energetic motions) and so, the order was restored by including the ∆-1, neutrino quanta which the process of death and devolution of the nucleon transfers as all entropic phenomena do, to its lower ∆-1 faster/larger scale.
It is this type of actions of balancing the ‘books’ which physicists stubbornly do not do for entropic heat, adding to the mix gravitational in-form-ative forces and ‘@minds’, Maxwell demons, what explains the errors of a dying universe of its science philosophies.
Or in other words, what we call ‘dead heat’ balance or thermodynamic equilibrium means merely the ∆-1 scale, ‘refuses’ to become ‘organised’ and synchronised by the ∆-scale, and prefers to remain in a herd, wave, ‘relatively free’ scale. It is only entropy from the human point of view.
The application of thermodynamic principles begins by defining a system that is in some sense distinct from its surroundings. For example, the system could be a sample of gas inside a cylinder with a movable piston, an entire steam engine, a marathon runner, the planet Earth, a neutron star, a black hole, or even the entire universe. In general, systems are free to exchange heat, work, and other forms of energy with their surroundings.
However this approach of classic thermodynamics is limited in its scope, as we must first define the system, more precisely in its 3 x 3 + 0 dimensions, to know if it is a complete system (with 3 co-existing scales, 3 topologies, 3 time ages and 1 mind-point), in which case there will not be entropy but rather information-creation by the mind on a closed system, growing its information at expenses of the external world – case of life systems (x, x, x in the next graph):
When the system looses the mind-point, there is not an internal element of order, and so the system is ‘ready’ to give up part of its information to an external agent. Yet if the enclosing membrane remains (a fact not easily obtained through time, as it is managed and maintained in isolation by the invagination and connection with its @mind-point), the system will remain in balance with no exchange of e or i (x,x,√) at the ∆+1 ‘mechanical level, leaking though disordered, entropic heat at the ∆-1 level. And a similar case, when the systems only thermally isolated at ∆-1 level happens: the system can be used to cause work, as the mechanical, motion, ∆-level is not isolated.
So what we know as classic thermodynamics apply to such systems in which either or both of the ∆-1 and ∆+1 quantum and mechanical scales are NOT isolated and transfer of energy and motion in inverse faction happens between them (open and closed system); and to those systems the laws of thermodynamic apply when we do a ceteris paribus analysis from the point of view of a single scale – disregarding the order or entropy ∆ or ∇ effects on those 2 other scales.
Only such systems we shall call ‘thermodynamic engines’, classic laws apply. Then those laws will refer basically to the Conservation of present energy of a system, where the present energy is given by the temperature and its composite elements will be volume or expansive entropy and pressure or increasing order, given us the 1st key equation of thermodynamics:
P (t) x V (s) = nK T (exi content)
For example a gas in a cylinder with a movable piston, the state of the system is identified by the temperature, pressure, and volume of the gas. Where the volume is proportional to the spatial expansion OR ENTROPY PROPER using the concept of ENTROPY as expansive motion, defined in GST; the pressure is its inverse ‘pure form’ function, or arrow of order. And the nKT element the present, energy x information parameter of the system, ‘its existential force’ calculated multiplying its quanta of entropy-space, nK and its speed of time, its ‘frequency of time vibration’, T.
In such systems it also applies the fundamental law of ‘immortality of time’, which states that:
‘a closed path of time has not done work, hence spent its potential energy, when it returns to its initial space-time condition’
So P, V, and kT are characteristic parameters that have definite values at each state and are independent of the way in which the system arrived at that state. In other words, any change in value of a property depends only on the initial and final states of the system, not on the path followed by the system from one state to another. Such properties are called state functions.
In contrast, when we talk of an open system (√,√,√) the final state is not one of balance, since there is a gain or loss of energy and information which the external system absorbs or gives. So, the final work done as the piston moves and the gas expands and the heat the gas absorbs from its surroundings depend on the detailed way in which the expansion occurs.
The behaviour of a complex thermodynamic system, such as Earth’s atmosphere, can be understood by first applying the principles of states and properties to its component parts—in this case, water, water vapour, and the various gases making up the atmosphere. By isolating samples of material whose states and properties can be controlled and manipulated, properties and their interrelations can be studied as the system changes from state to state. It is though not accurate to consider then ‘ceteris paribus analysis’ of the system to uphold the wrong limited classic laws of entropy. I.e. when we consider the Earth-sun system we wrongly state that the order of the planet grows because the disorder of the star happens giving up ¥-energy. But it does NOT consider the ∆+3 scale of gravitational, increasing order, which in fact will finally dominate collapsing the star. It is then we could talk of the whole solar system in ‘e x i equilibrium’, which classic thermodynamics called a…
Classic thermodynamics is concerned then with the different states of disorder, when there is a dynamic transfer of Pt, Vs and nkT (ST) between systems, which changes the system parameters of e x i, or with the different paths a system can use to get from a state to other; changing only one of those parameters; which are analysis of value for all other studies of single space-time systems at all other scales.
In that regard, a particularly important concept is thermodynamic equilibrium, in which there is no tendency for the state of a system to change spontaneously. For example, the gas in a cylinder with a movable piston will be at equilibrium if the temperature and pressure inside are uniform and if the restraining force on the piston is just sufficient to keep it from moving. The system can then be made to change to a new state only by an externally imposed change in one of the state functions, such as the temperature by adding heat or the volume by moving the piston. A sequence of one or more such steps connecting different states of the system is called a process. In general, a system is not in equilibrium as it adjusts to an abrupt change in its environment. For example, when a balloon bursts, the compressed gas inside is suddenly far from equilibrium, and it rapidly expands until it reaches a new equilibrium state.
However, the same final state could be achieved by placing the same compressed gas in a cylinder with a movable piston and applying a sequence of many small increments in volume (and temperature), with the system being given time to come to equilibrium after each small increment.
Such a process is said to be reversible because the system is at (or near) equilibrium at each step along its path, and the direction of change could be reversed at any point.
This example illustrates how two different paths can connect the same initial and final states.
The first is irreversible (the balloon bursts), and the second is reversible. The concept of reversible processes is something like motion without friction in mechanics.
It represents an idealized limiting case that is very useful in discussing the properties of real systems. Many of the results of thermodynamics are derived from the properties of reversible processes. And the first conclusion we obtain from that duality, is indeed a general law of all systems: When the change is very fast and the system cannot ‘reorganise step by step’ of its frequency motions, the simultaneous location of all its parts, the system becomes disordered. As all entropic, death processes and big-bangs, ∆+1<<∆-1, are by definition fast transformations that erase the disorder of the system.
Yet when those changes are minimal changes, where the parameter of temporal order is higher (slow processes with minute changes) than the parameter of expansive space, the process is fully reversible with no entropy: ∆<∆-1>∆.
The concept of temperature is fundamental then to any discussion of thermodynamics, as it is the parameter of frequency in time, hence of the speed of the system in its time clocks, which will vary all the other parameters of the system to define its existential SxT force, which in all systems grow when S-imultaneity in space and size grow (spatial force) or its temporal frequency and timing of its actions-reactions grow (temperature speed), within the S≈T limits of internal balance allowed by the system (homeostasis), which for any system establishes an interval of temperature of maximal efficiency.
Thus Temperature is a measure of the density of energy and information, of the frequency of motion, of the quantity of temporal form, of the existential momentum of a thermodynamic ensemble of waves, particle and fields of the ∆±1 planes of existence, with ∆o humans on its middle point. As such temperature is the frequency of the activity of a certain environment in those planes, which increases as we increase the parameters which will increase that density such as Pt x Vs = n Ks Tt
Yet, in any assemble of ∆-1 forms, the arrow of social evolution dominates the system. So when one ∆-1 element has a higher frequency of existence it will share it among SIMILAR entities by colliding and transferring them energy. Without the existence of an attractor, @mind, Maxwell’s demon in the system, the energy of the system will remain in a wave, steady state equilibrium, providing the external membrane is closed, or else it will dissipate its entropy if the external ‘pressure membrane’ disappears.
Temperature is therefore a social form of energy ∆-1 systems prefer to shared to equalise its parameter of energy and information with its close clone molecules.
So, when two objects are brought into thermal contact, heat will flow between them until they come into equilibrium with each other. When the flow of heat stops, they are said to be at the same temperature.
The zeroth law of thermodynamics formalizes this by asserting that if an object A is in simultaneous thermal equilibrium with two other objects B and C, then B and C will be in thermal equilibrium with each other if brought into thermal contact.
Object A can then play the role of a thermometer through some change in its physical properties with temperature, such as its volume or its electrical resistance.
With the definition of equality of temperature in hand, it is possible then to establish a temperature scale by assigning numerical values to certain easily reproducible fixed points. And consider in an opposite fashion to thermodynamic classic laws that:
‘When entropy increases in the system as a whole, an equal amount of order created by isotemperature equilibrium happens in the ∆-1 scale of the system’.
Work and energy
Energy in that sense has a precise meaning in classic single ∆-scale physics that does not correspond to the concept of GST.
The word is derived from the Greek word ergon, meaning work, but the term work itself acquired a technical meaning with the advent of Newtonian mechanics. For example, a man pushing on a car may feel that he is doing a lot of work, but no work is actually done unless the car moves. The work done is then the product of the force applied by the man multiplied by the distance through which the car moves. If there is no friction and the surface is level, then the car, once set in motion, will continue rolling indefinitely with constant speed. The rolling car has something that a stationary car does not have—it has kinetic energy of motion equal to the work required to achieve that state of motion.
The introduction of the concept of energy in this way is of great value in mechanics because, in the absence of friction, energy is never lost from the system, although it can be converted from one form to another. What this means in GST is that closed motions, which are time motions do not spend work or energy because the Universe never spends its immortal closed motions, which is ultimately the substance of which it is made. YOU LIVE in a Universe of time in which spaces are just partial parts, open motions that invariably will return as a zero sum, in spatial, topological form, ages, or scales.
For example, if a coasting car comes to a hill, it will roll some distance up the hill before coming to a temporary stop. At that moment its kinetic energy of motion has been converted into its potential energy of position, which is equal to the work required to lift the car through the same vertical distance. After coming to a stop, the car will then begin rolling back down the hill until it has completely recovered its kinetic energy of motion at the bottom. In the absence of friction, such systems are said to be conservative because at any given moment the total amount of energy (kinetic plus potential) remains equal to the initial work done to set the system in motion.
As the science of physics expanded to cover an ever-wider range of phenomena, it became necessary to include additional forms of energy in order to keep the total amount of energy constant for all closed systems (or to account for changes in total energy for open systems).
For example, if work is done to accelerate charged particles, then some of the resultant energy will be stored in the form of electromagnetic fields and carried away from the system as radiation. In turn the electromagnetic energy can be picked up by a remote receiver (antenna) and converted back into an equivalent amount of work. With his theory of special relativity, Albert Einstein realized that energy (E) can also be stored as mass (m) and converted back into energy, as expressed by his famous equation E = mc2, where c is the velocity of light. All of these systems are said to be conservative in the sense that energy can be freely converted from one form to another without limit. Each fundamental advance of physics into new realms has involved a similar extension to the list of the different forms of energy.
Yet because there was not a concept of fractal scales, the relationship between those forms of energy and the relative present states of each scale, its different time clocks and quanta of space, where not clearly defined.
Thermodynamics encompasses all of these forms of energy, with the further addition of heat to the list of different kinds of energy.
However, heat, which is a concept restricted to the subjective human desire to ‘be the focus and attention of all other scales’ is fundamentally different from the others in that the conversion of work (or other forms of energy) into heat is not completely reversible. Since it is not the whole tale, but merely the duality of human scales.
So this specific law of energy-information transfer MEANS merely that part of the motion from ∆+1 < ∑∆-1 becomes absorbed, stolen by the more informative ∆-1 entities, unless there is truly an organic network that establishes the simultaneity of all those motions (as in biological organisms):
∆ +1 ‹ Tiƒ ∆ + Spe ∆-1
That is ∆+1 transfers part of its motion to the well organise, synchronous ∆-particles of the lower scale, but also to its field, ∆-1 which won’t return it back.
(nt.1 When we use also ≤ and ≥ bigger or smaller classic symbols of algebra, GST < expansive entropy an > informative flow, are substituted by ‹ and ›, to avoid confessions).
In the example of the rolling car, some of the work done to set the car in motion is inevitably lost as heat due to friction, and the car eventually comes to a stop on a level surface. Even if all the generated heat were collected and stored in some fashion, it could never be converted entirely back into mechanical energy of motion. This fundamental limitation is expressed quantitatively by the second law of thermodynamics (see below).
The role of friction in degrading the energy of mechanical systems may seem simple and obvious, but the quantitative connection between heat and work, as first discovered by Count Rumford, played a key role in understanding the operation of steam engines in the 19th century and similarly for all energy-conversion processes today.
Total internal energy
Although classical thermodynamics deals exclusively with the macroscopic properties of materials—such as temperature, pressure, and volume—thermal energy from the addition of heat can be understood at the microscopic level as an increase in the kinetic energy of motion of the molecules making up a substance.
This is the proper way to define then the system not as one that looses energy but one which transfers it to ∆-1 scales.
For example, gas molecules have translational kinetic energy that is proportional to the temperature of the gas: the molecules can rotate about their centre of mass, and the constituent atoms can vibrate with respect to each other (like masses connected by springs). Additionally, chemical energy is stored in the bonds holding the molecules together, and weaker long-range interactions between the molecules involve yet more energy. The sum total of all these forms of energy constitutes the total internal energy of the substance in a given thermodynamic state. The total energy of a system includes its internal energy plus any other forms of energy, such as kinetic energy due to motion of the system as a whole (e.g., water flowing through a pipe) and gravitational potential energy due to its elevation.
It is then when the concept of a ‘dying entropic universe’ makes no longer sense, reason why the 2nd law should be abolished in its present format. Let us now study in more depth each of those laws with its added corrections.
The first law of thermodynamics
The laws of thermodynamics are deceptively simple to state, but they are far-reaching in their consequences. The first law asserts that the total energy of a system plus its surroundings is conserved; in other words, the total energy of the universe remains constant as keep extending a nested series of worlds in ∆, S and T scales and symmetries, to infinity.
Yet since energy and its symmetric form information is a combination of entropy and form≈curvature, we deduce that the total curvature and entropy of the Universe is also conserved, on the total cycle of times, in which a world cycle does not make work. Work is non-existent in the long term in the Universe. Work never happens because all cycles are closed so the total energy and information of the five-dimensional block of existences is completed. And yet it never ceases to create its details.
The first law is put into action by considering the flow of energy across the boundary separating a system from its surroundings. Consider the classic example of a gas enclosed in a cylinder with a movable piston. The walls of the cylinder act as the boundary separating the gas inside from the world outside, and the movable piston provides a mechanism for the gas to do work by expanding against the force holding the piston (assumed frictionless) in place. If the gas does work W as it expands, and/or absorbs heat Q from its surroundings through the walls of the cylinder, then this corresponds to a net flow of energy W − Q across the boundary to the surroundings. In order to conserve the total energy U, there must be a counterbalancing change
in the internal energy of the gas. The first law provides a kind of strict energy accounting system in which the change in the energy account (ΔU) equals the difference between deposits (Q) and withdrawals (W).There is an important distinction between the quantity ΔU and the related energy quantities Q and W. Since the internal energy U is characterized entirely by the quantities (or parameters) that uniquely determine the state of the system at equilibrium, it is said to be a state function such that any change in energy is determined entirely by the initial (i) and final (f) states of the system: ΔU = Uf − Ui. However, Q and W are not state functions. Just as in the example of a bursting balloon, the gas inside may do no work at all in reaching its final expanded state, or it could do maximum work by expanding inside a cylinder with a movable piston to reach the same final state. All that is required is that the change in energy (ΔU) remain the same. By analogy, the same change in one’s bank account could be achieved by many different combinations of deposits and withdrawals. Thus, Q and W are not state functions, because their values depend on the particular process (or path) connecting the same initial and final states. Just as it is only meaningful to speak of the balance in one’s bank account and not its deposit or withdrawal content, it is only meaningful to speak of the internal energy of a system and not its heat or work content.
From a formal mathematical point of view, the incremental change dU in the internal energy is an exact differential (see differential equation), while the corresponding incremental changes d′Q and d′W in heat and work are not, because the definite integrals of these quantities are path-dependent. These concepts can be used to great advantage in a precise mathematical formulation of thermodynamics (see below Thermodynamic properties and relations).
The classic example of a heat engine is a steam engine, although all modern engines follow the same principles. Steam engines operate in a cyclic fashion, with the piston moving up and down once for each cycle. Hot high-pressure steam is admitted to the cylinder in the first half of each cycle, and then it is allowed to escape again in the second half. The overall effect is to take heat Q1 generated by burning a fuel to make steam, convert part of it to do work, and exhaust the remaining heat Q2 to the environment at a lower temperature. The net heat energy absorbed is then Q = Q1 − Q2. Since the engine returns to its initial state, its internal energy U does not change (ΔU = 0). Thus, by the first law of thermodynamics, the work done for each complete cycle must be W = Q1 − Q2. In other words, the work done for each complete cycle is just the difference between the heat Q1 absorbed by the engine at a high temperature and the heat Q2 exhausted at a lower temperature. The power of thermodynamics is that this conclusion is completely independent of the detailed working mechanism of the engine. It relies only on the overall conservation of energy, with heat regarded as a form of energy.
In order to save money on fuel and avoid contaminating the environment with waste heat, engines are designed to maximize the conversion of absorbed heat Q1 into useful work and to minimize the waste heat Q2. The Carnot efficiency (η) of an engine is defined as the ratio W/Q1—i.e., the fraction of Q1 that is converted into work. Since W = Q1 − Q2, the efficiency also can be expressed in the form
If there were no waste heat at all, then Q2 = 0 and η = 1, corresponding to 100 percent efficiency. While reducing friction in an engine decreases waste heat, it can never be eliminated; therefore, there is a limit on how small Q2 can be and thus on how large the efficiency can be. This limitation is a fundamental law of nature—in fact, the second law of thermodynamics (see below).
Isothermal and adiabatic processes
Because heat engines may go through a complex sequence of steps, a simplified model is often used to illustrate the principles of thermodynamics. In particular, consider a gas that expands and contracts within a cylinder with a movable piston under a prescribed set of conditions. There are two particularly important sets of conditions. One condition, known as an isothermal expansion, involves keeping the gas at a constant temperature. As the gas does work against the restraining force of the piston, it must absorb heat in order to conserve energy. Otherwise, it would cool as it expands (or conversely heat as it is compressed). This is an example of a process in which the heat absorbed is converted entirely into work with 100 percent efficiency. The process does not violate fundamental limitations on efficiency, however, because a single expansion by itself is not a cyclic process.
The second condition, known as an adiabatic expansion (from the Greek adiabatos, meaning “impassable”), is one in which the cylinder is assumed to be perfectly insulated so that no heat can flow into or out of the cylinder. In this case the gas cools as it expands, because, by the first law, the work done against the restraining force on the piston can only come from the internal energy of the gas. Thus, the change in the internal energy of the gas must be ΔU = −W, as manifested by a decrease in its temperature. The gas cools, even though there is no heat flow, because it is doing work at the expense of its own internal energy. The exact amount of cooling can be calculated from the heat capacity of the gas.
Many natural phenomena are effectively adiabatic because there is insufficient time for significant heat flow to occur. For example, when warm air rises in the atmosphere, it expands and cools as the pressure drops with altitude, but air is a good thermal insulator, and so there is no significant heat flow from the surrounding air. In this case the surrounding air plays the roles of both the insulated cylinder walls and the movable piston. The warm air does work against the pressure provided by the surrounding air as it expands, and so its temperature must drop. A more-detailed analysis of this adiabatic expansion explains most of the decrease of temperature with altitude, accounting for the familiar fact that it is colder at the top of a mountain than at its base.
The second law of thermodynamics
The first law of thermodynamics asserts that energy must be conserved in any process involving the exchange of heat and work between a system and its surroundings.
A machine that violated the first law would be called a perpetual motion machine of the first kind because it would manufacture its own energy out of nothing and thereby run forever.
Such a machine would be impossible even in theory. However, this impossibility would not prevent the construction of a machine that could extract essentially limitless amounts of heat from its surroundings (earth, air, and sea) and convert it entirely into work. Although such a hypothetical machine would not violate conservation of energy, the total failure of inventors to build such a machine, known as a perpetual motion machine of the second kind, led to the discovery of the second law of thermodynamics. The second law of thermodynamics can be precisely stated in the following two forms, as originally formulated in the 19th century by the Scottish physicist William Thomson (Lord Kelvin) and the German physicist Rudolf Clausius, respectively:
A cyclic transformation whose only final result is to transform heat extracted from a source which is at the same temperature throughout into work is impossible.
A cyclic transformation whose only final result is to transfer heat from a body at a given temperature to a body at a higher temperature is impossible.
The two statements are in fact equivalent because, if the first were possible, then the work obtained could be used, for example, to generate electricity that could then be discharged through an electric heater installed in a body at a higher temperature. The net effect would be a flow of heat from a lower temperature to a higher temperature, thereby violating the second (Clausius) form of the second law. Conversely, if the second form were possible, then the heat transferred to the higher temperature could be used to run a heat engine that would convert part of the heat into work. The final result would be a conversion of heat into work at constant temperature—a violation of the first (Kelvin) form of the second law.
Central to the following discussion of entropy is the concept of a heat reservoir capable of providing essentially limitless amounts of heat at a fixed temperature. This is of course an idealization, but the temperature of a large body of water such as the Atlantic Ocean does not materially change if a small amount of heat is withdrawn to run a heat engine. The essential point is that the heat reservoir is assumed to have a well-defined temperature that does not change as a result of the process being considered.
Entropy and efficiency limits
The concept of entropy was first introduced in 1850 by Clausius as a precise mathematical way of testing whether the second law of thermodynamics is violated by a particular process. The test begins with the definition that if an amount of heat Q flows into a heat reservoir at constant temperature T, then its entropy S increases by ΔS = Q/T. (This equation in effect provides a thermodynamic definition of temperature that can be shown to be identical to the conventional thermometric one.) Assume now that there are two heat reservoirs R1 and R2 at temperatures T1 and T2. If an amount of heat Q flows from R1 to R2, then the net entropy change for the two reservoirs is
(3) ΔS is positive, provided that T1 > T2. Thus, the observation that heat never flows spontaneously from a colder region to a hotter region (the Clausius form of the second law of thermodynamics) is equivalent to requiring the net entropy change to be positive for a spontaneous flow of heat. If T1 = T2, then the reservoirs are in equilibrium and ΔS = 0.The condition ΔS ≥ 0 determines the maximum possible efficiency of heat engines. Suppose that some system capable of doing work in a cyclic fashion (a heat engine) absorbs heat Q1 from R1 and exhausts heat Q2 to R2 for each complete cycle. Because the system returns to its original state at the end of a cycle, its energy does not change. Then, by conservation of energy, the work done per cycle is W = Q1 − Q2, and the net entropy change for the two reservoirs is
To make W as large as possible, Q2 should be kept as small as possible relative to Q1. However, Q2 cannot be zero, because this would make ΔS negative and so violate the second law of thermodynamics. The smallest possible value of Q2 corresponds to the condition ΔS = 0, yielding
This is the fundamental equation limiting the efficiency of all heat engines whose function is to convert heat into work (such as electric power generators). The actual efficiency is defined to be the fraction of Q1 that is converted to work (W/Q1), which is equivalent to equation (2).The maximum efficiency for a given T1 and T2 is thus
A process for which ΔS = 0 is said to be reversible because an infinitesimal change would be sufficient to make the heat engine run backward as a refrigerator.As an example, the properties of materials limit the practical upper temperature for thermal power plants to T1 1,200 K. Taking T2 to be the temperature of the environment (300 K), the maximum efficiency is 1 − 300/1,200 = 0.75. Thus, at least 25 percent of the heat energy produced must be exhausted into the environment as waste heat to avoid violating the second law of thermodynamics. Because of various imperfections, such as friction and imperfect thermal insulation, the actual efficiency of power plants seldom exceeds about 60 percent. However, because of the second law of thermodynamics, no amount of ingenuity or improvements in design can increase the efficiency beyond about 75 percent.
Entropy and heat death
The example of a heat engine illustrates one of the many ways in which the second law of thermodynamics can be applied. One way to generalize the example is to consider the heat engine and its heat reservoir as parts of an isolated (or closed) system—i.e., one that does not exchange heat or work with its surroundings. For example, the heat engine and reservoir could be encased in a rigid container with insulating walls. In this case the second law of thermodynamics (in the simplified form presented here) says that no matter what process takes place inside the container, its entropy must increase or remain the same in the limit of a reversible process. Similarly, if the universe is an isolated system, then its entropy too must increase with time. Indeed, the implication is that the universe must ultimately suffer a “heat death” as its entropy progressively increases toward a maximum value and all parts come into thermal equilibrium at a uniform temperature. After that point, no further changes involving the conversion of heat into useful work would be possible. In general, the equilibrium state for an isolated system is precisely that state of maximum entropy. (This is equivalent to an alternate definition for the term entropy as a measure of the disorder of a system, such that a completely random dispersion of elements corresponds to maximum entropy, or minimum information. )
Entropy and the arrow of time
The inevitable increase of entropy with time for isolated systems plays a fundamental role in determining the direction of the “arrow of time.” Everyday life presents no difficulty in distinguishing the forward flow of time from its reverse. For example, if a film showed a glass of warm water spontaneously changing into hot water with ice floating on top, it would immediately be apparent that the film was running backward because the process of heat flowing from warm water to hot water would violate the second law of thermodynamics. However, this obvious asymmetry between the forward and reverse directions for the flow of time does not persist at the level of fundamental interactions. An observer watching a film showing two water molecules colliding would not be able to tell whether the film was running forward or backward.
So what exactly is the connection between entropy and the second law? Recall that heat at the molecular level is the random kinetic energy of motion of molecules, and collisions between molecules provide the microscopic mechanism for transporting heat energy from one place to another. Because individual collisions are unchanged by reversing the direction of time, heat can flow just as well in one direction as the other. Thus, from the point of view of fundamental interactions, there is nothing to prevent a chance event in which a number of slow-moving (cold) molecules happen to collect together in one place and form ice, while the surrounding water becomes hotter. Such chance events could be expected to occur from time to time in a vessel containing only a few water molecules. However, the same chance events are never observed in a full glass of water, not because they are impossible but because they are exceedingly improbable. This is because even a small glass of water contains an enormous number of interacting molecules (about 1024), making it highly unlikely that, in the course of their random thermal motion, a significant fraction of cold molecules will collect together in one place. Although such a spontaneous violation of the second law of thermodynamics is not impossible, an extremely patient physicist would have to wait many times the age of the universe to see it happen.
The foregoing demonstrates an important point: the second law of thermodynamics is statistical in nature. It has no meaning at the level of individual molecules, whereas the law becomes essentially exact for the description of large numbers of interacting molecules. In contrast, the first law of thermodynamics, which expresses conservation of energy, remains exactly true even at the molecular level.
The example of ice melting in a glass of hot water also demonstrates the other sense of the term entropy, as an increase in randomness and a parallel loss of information. Initially, the total thermal energy is partitioned in such a way that all of the slow-moving (cold) molecules are located in the ice and all of the fast-moving (hot) molecules are located in the water (or water vapour). After the ice has melted and the system has come to thermal equilibrium, the thermal energy is uniformly distributed throughout the system. The statistical approach provides a great deal of valuable insight into the meaning of the second law of thermodynamics, but, from the point of view of applications, the microscopic structure of matter becomes irrelevant. The great beauty and strength of classical thermodynamics are that its predictions are completely independent of the microscopic structure of matter.
Most real thermodynamic systems are open systems that exchange heat and work with their environment, rather than the closed systems described thus far. For example, living systems are clearly able to achieve a local reduction in their entropy as they grow and develop; they create structures of greater internal energy (i.e., they lower entropy) out of the nutrients they absorb. This does not represent a violation of the second law of thermodynamics, because a living organism does not constitute a closed system.
In order to simplify the application of the laws of thermodynamics to open systems, parameters with the dimensions of energy, known as thermodynamic potentials, are introduced to describe the system. The resulting formulas are expressed in terms of the Helmholtz free energy F and the Gibbs free energy G, named after the 19th-century German physiologist and physicist Hermann von Helmholtz and the contemporaneous American physicist Josiah Willard Gibbs. The key conceptual step is to separate a system from its heat reservoir. A system is thought of as being held at a constant temperature T by a heat reservoir (i.e., the environment), but the heat reservoir is no longer considered to be part of the system. Recall that the internal energy change (ΔU) of a system is given by
where Q is the heat absorbed and W is the work done. In general, Q and W separately are not state functions, because they are path-dependent. However, if the path is specified to be any reversible isothermal process, then the heat associated with the maximum work (Wmax) is Qmax = TΔS. With this substitution the above equation can be rearranged as
Note that here ΔS is the entropy change just of the system being held at constant temperature, such as a battery. Unlike the case of an isolated system as considered previously, it does not include the entropy change of the heat reservoir (i.e., the surroundings) required to keep the temperature constant. If this additional entropy change of the reservoir were included, the total entropy change would be zero, as in the case of an isolated system. Because the quantities U, T, and S on the right-hand side are all state functions, it follows that −Wmax must also be a state function. This leads to the definition of the Helmholtz free energy
such that, for any isothermal change of the system,
is the negative of the maximum work that can be extracted from the system. The actual work extracted could be smaller than the ideal maximum, or even zero, which implies that W ≤ −ΔF, with equality applying in the ideal limiting case of a reversible process. When the Helmholtz free energy reaches its minimum value, the system has reached its equilibrium state, and no further work can be extracted from it. Thus, the equilibrium condition of maximum entropy for isolated systems becomes the condition of minimum Helmholtz free energy for open systems held at constant temperature. The one additional precaution required is that work done against the atmosphere be included if the system expands or contracts in the course of the process being considered. Typically, processes are specified as taking place at constant volume and temperature in order that no correction is needed.Although the Helmholtz free energy is useful in describing processes that take place inside a container with rigid walls, most processes in the real world take place under constant pressure rather than constant volume. For example, chemical reactions in an open test tube—or in the growth of a tomato in a garden—take place under conditions of (nearly) constant atmospheric pressure. It is for the description of these cases that the Gibbs free energy was introduced. As previously established, the quantity
is a state function equal to the change in the Helmholtz free energy. Suppose that the process being considered involves a large change in volume (ΔV), such as happens when water boils to form steam. The work done by the expanding water vapour as it pushes back the surrounding air at pressure P is PΔV. This is the amount of work that is now split out from Wmax by writing it in the form
where W′max is the maximum work that can be extracted from the process taking place at constant temperature T and pressure P, other than the atmospheric work (PΔV). Substituting this partition into the above equation for −Wmax and moving the PΔV term to the right-hand side then yields
This leads to the definition of the Gibbs free energy
such that, for any isothermal change of the system at constant pressure,
is the negative of the maximum work W′max that can be extracted from the system, other than atmospheric work. As before, the actual work extracted could be smaller than the ideal maximum, or even zero, which implies that W′ ≤ −ΔG, with equality applying in the ideal limiting case of a reversible process. As with the Helmholtz case, when the Gibbs free energy reaches its minimum value, the system has reached its equilibrium state, and no further work can be extracted from it. Thus, the equilibrium condition becomes the condition of minimum Gibbs free energy for open systems held at constant temperature and pressure, and the direction of spontaneous change is always toward a state of lower free energy for the system (like a ball rolling downhill into a valley). Notice in particular that the entropy can now spontaneously decrease (i.e., TΔS can be negative), provided that this decrease is more than offset by the ΔU + PΔV terms in the definition of ΔG. As further discussed below, a simple example is the spontaneous condensation of steam into water. Although the entropy of water is much less than the entropy of steam, the process occurs spontaneously provided that enough heat energy is taken away from the system to keep the temperature from rising as the steam condenses.A familiar example of free energy changes is provided by an automobile battery. When the battery is fully charged, its Gibbs free energy is at a maximum, and when it is fully discharged (i.e., dead), its Gibbs free energy is at a minimum. The change between these two states is the maximum amount of electrical work that can be extracted from the battery at constant temperature and pressure. The amount of heat absorbed from the environment in order to keep the temperature of the battery constant (represented by the TΔS term) and any work done against the atmosphere (represented by the PΔV term) are automatically taken into account in the energy balance.
Gibbs free energy and chemical reactions
All batteries depend on some chemical reaction of the form
for the generation of electricity or on the reverse reaction as the battery is recharged. The change in free energy (−ΔG) for a reaction could be determined by measuring directly the amount of electrical work that the battery could do and then using the equation Wmax = −ΔG. However, the power of thermodynamics is that −ΔG can be calculated without having to build every possible battery and measure its performance. If the Gibbs free energies of the individual substances making up a battery are known, then the total free energies of the reactants can be subtracted from the total free energies of the products in order to find the change in Gibbs free energy for the reaction,
Once the free energies are known for a wide variety of substances, the best candidates for actual batteries can be quickly discerned. In fact, a good part of the practice of thermodynamics is concerned with determining the free energies and other thermodynamic properties of individual substances in order that ΔG for reactions can be calculated under different conditions of temperature and pressure.In the above discussion, the term reaction can be interpreted in the broadest possible sense as any transformation of matter from one form to another. In addition to chemical reactions, a reaction could be something as simple as ice (reactants) turning to liquid water (products), the nuclear reactions taking place in the interior of stars, or elementary particle reactions in the early universe. No matter what the process, the direction of spontaneous change (at constant temperature and pressure) is always in the direction of decreasing free energy.
Enthalpy and the heat of reaction
As discussed above, the free energy change Wmax = −ΔG corresponds to the maximum possible useful work that can be extracted from a reaction, such as in an electrochemical battery. This represents one extreme limit of a continuous range of possibilities. At the other extreme, for example, battery terminals can be connected directly by a wire and the reaction allowed to proceed freely without doing any useful work. In this case W′ = 0, and the first law of thermodynamics for the reaction becomes
where Q0 is the heat absorbed when the reaction does no useful work and, as before, PΔV is the atmospheric work term. The key point is that the quantities ΔU and PΔV are exactly the same as in the other limiting case, in which the reaction does maximum work. This follows because these quantities are state functions, which depend only on the initial and final states of a system and not on any path connecting the states. The amount of useful work done just represents different paths connecting the same initial and final states. This leads to the definition of enthalpy (H), or heat content, as
Its significance is that, for a reaction occurring freely (i.e., doing no useful work) at constant temperature and pressure, the heat absorbed is
where ΔH is called the heat of reaction. The heat of reaction is easy to measure because it simply represents the amount of heat that is given off if the reactants are mixed together in a beaker and allowed to react freely without doing any useful work.The above definition for enthalpy and its physical significance allow the equation for ΔG to be written in the particularly illuminating and instructive form
Both terms on the right-hand side represent heats of reaction but under different sets of circumstances. ΔH is the heat of reaction (i.e., the amount of heat absorbed from the surroundings in order to hold the temperature constant) when the reaction does no useful work, and TΔS is the heat of reaction when the reaction does maximum useful work in an electrochemical cell. The (negative) difference between these two heats is exactly the maximum useful work −ΔG that can be extracted from the reaction. Thus, useful work can be obtained by contriving for a system to extract additional heat from the environment and convert it into work. The difference ΔH − TΔS represents the fundamental limitation imposed by the second law of thermodynamics on how much additional heat can be extracted from the environment and converted into useful work for a given reaction mechanism. An electrochemical cell (such as a car battery) is a contrivance by means of which a reaction can be made to do the maximum possible work against an opposing electromotive force, and hence the reaction literally becomes reversible in the sense that a slight increase in the opposing voltage will cause the direction of the reaction to reverse and the cell to start charging up instead of discharging.As a simple example, consider a reaction in which water turns reversibly into steam by boiling. To make the reaction reversible, suppose that the mixture of water and steam is contained in a cylinder with a movable piston and held at the boiling point of 373 K (100 °C) at 1 atmosphere pressure by a heat reservoir. The enthalpy change is ΔH = 40.65 kilojoules per mole, which is the latent heat of vaporization. The entropy change is
representing the higher degree of disorder when water evaporates and turns to steam. The Gibbs free energy change is ΔG = ΔH − TΔS. In this case the Gibbs free energy change is zero because the water and steam are in equilibrium, and no useful work can be extracted from the system (other than work done against the atmosphere). In other words, the Gibbs free energy per molecule of water (also called the chemical potential) is the same for both liquid water and steam, and so water molecules can pass freely from one phase to the other with no change in the total free energy of the system.
Thermodynamic properties and relations
In order to carry through a program of finding the changes in the various thermodynamic functions that accompany reactions—such as entropy, enthalpy, and free energy—it is often useful to know these quantities separately for each of the materials entering into the reaction. For example, if the entropies are known separately for the reactants and products, then the entropy change for the reaction is just the difference
and similarly for the other thermodynamic functions. Furthermore, if the entropy change for a reaction is known under one set of conditions of temperature and pressure, it can be found under other sets of conditions by including the variation of entropy for the reactants and products with temperature or pressure as part of the overall process. For these reasons, scientists and engineers have developed extensive tables of thermodynamic properties for many common substances, together with their rates of change with state variables such as temperature and pressure.The science of thermodynamics provides a rich variety of formulas and techniques that allow the maximum possible amount of information to be extracted from a limited number of laboratory measurements of the properties of materials. However, as the thermodynamic state of a system depends on several variables—such as temperature, pressure, and volume—in practice it is necessary first to decide how many of these are independent and then to specify what variables are allowed to change while others are held constant. For this reason, the mathematical language of partial differential equations is indispensable to the further elucidation of the subject of thermodynamics.
Of especially critical importance in the application of thermodynamics are the amounts of work required to make substances expand or contract and the amounts of heat required to change the temperature of substances. The first is determined by the equation of state of the substance and the second by its heat capacity. Once these physical properties have been fully characterized, they can be used to calculate other thermodynamic properties, such as the free energy of the substance under various conditions of temperature and pressure.
In what follows, it will often be necessary to consider infinitesimal changes in the parameters specifying the state of a system. The first law of thermodynamics then assumes the differential form dU = d′Q − d′W. Because U is a state function, the infinitesimal quantity dU must be an exact differential, which means that its definite integral depends only on the initial and final states of the system. In contrast, the quantities d′Q and d′W are not exact differentials, because their integrals can be evaluated only if the path connecting the initial and final states is specified. The examples to follow will illustrate these rather abstract concepts.
Work of expansion and contraction
The first task in carrying out the above program is to calculate the amount of work done by a single pure substance when it expands at constant temperature. Unlike the case of a chemical reaction, where the volume can change at constant temperature and pressure because of the liberation of gas, the volume of a single pure substance placed in a cylinder cannot change unless either the pressure or the temperature changes. To calculate the work, suppose that a piston moves by an infinitesimal amount dx. Because pressure is force per unit area, the total restraining force exerted by the piston on the gas is PA, where A is the cross-sectional area of the piston. Thus, the incremental amount of work done is d′W = PAdx.
However, Adx can also be identified as the incremental change in the volume (dV) swept out by the head of the piston as it moves. The result is the basic equation d′W = PdV for the incremental work done by a gas when it expands. For a finite change from an initial volume Vi to a final volume Vf, the total work done is given by the integral
Equations of state
The equation of state for a substance provides the additional information required to calculate the amount of work that the substance does in making a transition from one equilibrium state to another along some specified path. The equation of state is expressed as a functional relationship connecting the various parameters needed to specify the state of the system. The basic concepts apply to all thermodynamic systems, but here, in order to make the discussion specific, a simple gas inside a cylinder with a movable piston will be considered.
The equation of state then takes the form of an equation relating P, V, and T, such that if any two are specified, the third is determined. In the limit of low pressures and high temperatures, where the molecules of the gas move almost independently of one another, all gases obey an equation of state known as the ideal gas law: PV = nRT, where n is the number of moles of the gas and R is the universal gas constant, 8.3145 joules per K. In the International System of Units, energy is measured in joules, volume in cubic metres (m3), force in newtons (N), and pressure in pascals (Pa), where 1 Pa = 1 N/m2. A force of one newton moving through a distance of one metre does one joule of work. Thus, both the products PV and RT have the dimensions of work (energy). A P–V diagram would show the equation of state in graphical form for several different temperatures.
To illustrate the path-dependence of the work done, consider three processes connecting the same initial and final states. The temperature is the same for both states, but, in going from state i to state f, the gas expands from Vi to Vf (doing work), and the pressure falls from Pi to Pf. According to the definition of the integral in equation (22), the work done is the area under the curve (or straight line) for each of the three processes. For processes I and III the areas are rectangles, and so the work done is
respectively. Process II is more complicated because P changes continuously as V changes. However, T remains constant, and so one can use the equation of state to substitute P = nRT/V in equation (22) to obtain
for an (ideal gas) isothermal process,
WII is thus the work done in the reversible isothermal expansion of an ideal gas. The amount of work is clearly different in each of the three cases. For a cyclic process the net work done equals the area enclosed by the complete cycle.
Heat capacity and specific heat
As shown originally by Count Rumford, there is an equivalence between heat (measured in calories) and mechanical work (measured in joules) with a definite conversion factor between the two. The conversion factor, known as the mechanical equivalent of heat, is 1 calorie = 4.184 joules. (There are several slightly different definitions in use for the calorie. The calorie used by nutritionists is actually a kilocalorie.) In order to have a consistent set of units, both heat and work will be expressed in the same units of joules.
The amount of heat that a substance absorbs is connected to its temperature change via its molar specific heat c, defined to be the amount of heat required to change the temperature of 1 mole of the substance by 1 K. In other words, c is the constant of proportionality relating the heat absorbed (d′Q) to the temperature change (dT) according to d′Q = ncdT, where n is the number of moles. For example, it takes approximately 1 calorie of heat to increase the temperature of 1 gram of water by 1 K. Since there are 18 grams of water in 1 mole, the molar heat capacity of water is 18 calories per K, or about 75 joules per K. The total heat capacity C for n moles is defined by C = nc.
However, since d′Q is not an exact differential, the heat absorbed is path-dependent and the path must be specified, especially for gases where the thermal expansion is significant. Two common ways of specifying the path are either the constant-pressure path or the constant-volume path. The two different kinds of specific heat are called cP and cV respectively, where the subscript denotes the quantity that is being held constant. It should not be surprising that cP is always greater than cV, because the substance must do work against the surrounding atmosphere as it expands upon heating at constant pressure but not at constant volume. In fact, this difference was used by the 19th-century German physicist Julius Robert von Mayer to estimate the mechanical equivalent of heat.
Heat capacity and internal energy
The goal in defining heat capacity is to relate changes in the internal energy to measured changes in the variables that characterize the states of the system. For a system consisting of a single pure substance, the only kind of work it can do is atmospheric work, and so the first law reduces to
(28)Suppose now that U is regarded as being a function U(T, V) of the independent pair of variables T and V. The differential quantity dU can always be expanded in terms of its partial derivatives according to
where the subscripts denote the quantity being held constant when calculating derivatives. Substituting this equation into dU = d′Q − PdV then yields the general expression
for the path-dependent heat. The path can now be specified in terms of the independent variables T and V. For a temperature change at constant volume, dV = 0 and, by definition of heat capacity,
The above equation then gives immediately
for the heat capacity at constant volume, showing that the change in internal energy at constant volume is due entirely to the heat absorbed.To find a corresponding expression for CP, one need only change the independent variables to T and P and substitute the expansion
for dV in equation (28) and correspondingly for dU to obtain
For a temperature change at constant pressure, dP = 0, and, by definition of heat capacity, d′Q = CPdT, resulting in
The two additional terms beyond CV have a direct physical meaning. The term
represents the additional atmospheric work that the system does as it undergoes thermal expansion at constant pressure, and the second term involving
represents the internal work that must be done to pull the system apart against the forces of attraction between the molecules of the substance (internal stickiness). Because there is no internal stickiness for an ideal gas, this term is zero, and, from the ideal gas law, the remaining partial derivative is
With these substitutions the equation for CP becomes simply
for the molar specific heats. For example, for a monatomic ideal gas (such as helium), cV = 3R/2 and cP = 5R/2 to a good approximation. cVT represents the amount of translational kinetic energy possessed by the atoms of an ideal gas as they bounce around randomly inside their container. Diatomic molecules (such as oxygen) and polyatomic molecules (such as water) have additional rotational motions that also store thermal energy in their kinetic energy of rotation. Each additional degree of freedom contributes an additional amount R to cV. Because diatomic molecules can rotate about two axes and polyatomic molecules can rotate about three axes, the values of cV increase to 5R/2 and 3R respectively, and cP correspondingly increases to 7R/2 and 4R. (cV and cP increase still further at high temperatures because of vibrational degrees of freedom.) For a real gas such as water vapour, these values are only approximate, but they give the correct order of magnitude. For example, the correct values are cP = 37.468 joules per K (i.e., 4.5R) and cP − cV = 9.443 joules per K (i.e., 1.14R) for water vapour at 100 °C and 1 atmosphere pressure.
Entropy as an exact differential
Because the quantity dS = d′Qmax/T is an exact differential, many other important relationships connecting the thermodynamic properties of substances can be derived. For example, with the substitutions d′Q = TdS and d′W = PdV, the differential form (dU = d′Q − d′W) of the first law of thermodynamics becomes (for a single pure substance)
The advantage gained by the above formula is that dU is now expressed entirely in terms of state functions in place of the path-dependent quantities d′Q and d′W. This change has the very important mathematical implication that the appropriate independent variables are S and V in place of T and V, respectively, for internal energy.
This replacement of T by S as the most appropriate independent variable for the internal energy of substances is the single most valuable insight provided by the combined first and second laws of thermodynamics. With U regarded as a function U(S, V), its differential dU is
A comparison with the preceding equation shows immediately that the partial derivatives are
Furthermore, the cross partial derivatives,
must be equal because the order of differentiation in calculating the second derivatives of U does not matter. Equating the right-hand sides of the above pair of equations then yields
This is one of four Maxwell relations (the others will follow shortly). They are all extremely useful in that the quantity on the right-hand side is virtually impossible to measure directly, while the quantity on the left-hand side is easily measured in the laboratory. For the present case one simply measures the adiabatic variation of temperature with volume in an insulated cylinder so that there is no heat flow (constant S).
The other three Maxwell relations follow by similarly considering the differential expressions for the thermodynamic potentials F(T, V), H(S, P), and G(T, P), with independent variables as indicated. The results are
As an example of the use of these equations, equation (35) for CP − CV contains the partial derivative
which vanishes for an ideal gas and is difficult to evaluate directly from experimental data for real substances. The general properties of partial derivatives can first be used to write it in the form
Combining this with equation (41) for the partial derivatives together with the first of the Maxwell equations from equation (44) then yields the desired result
comes directly from differentiating the equation of state. For an ideal gas
is zero as expected.
The departure of
from zero reveals directly the effects of internal forces between the molecules of the substance and the work that must be done against them as the substance expands at constant temperature.
The Clausius-Clapeyron equation
Phase changes, such as the conversion of liquid water to steam, provide an important example of a system in which there is a large change in internal energy with volume at constant temperature. Suppose that the cylinder contains both water and steam in equilibrium with each other at pressure P, and the cylinder is held at constant temperature T. The pressure remains equal to the vapour pressurePvap as the piston moves up, as long as both phases remain present. All that happens is that more water turns to steam, and the heat reservoir must supply the latent heat of vaporization, λ = 40.65 kilojoules per mole, in order to keep the temperature constant.
The results of the preceding section can be applied now to find the variation of the boiling point of water with pressure. Suppose that as the piston moves up, 1 mole of water turns to steam. The change in volume inside the cylinder is then ΔV = Vgas − Vliquid, where Vgas = 30.143 litres is the volume of 1 mole of steam at 100 °C, and Vliquid = 0.0188 litre is the volume of 1 mole of water. By the first law of thermodynamics, the change in internal energy ΔU for the finite process at constant P and T is ΔU = λ − PΔV.
The variation of U with volume at constant T for the complete system of water plus steam is thus
A comparison with equation (46) then yields the equation:
However, for the present problem, P is the vapour pressure Pvapour, which depends only on T and is independent of V. The partial derivative is then identical to the total derivative
(50) giving the Clausius-Clapeyron equation
This equation is very useful because it gives the variation with temperature of the pressure at which water and steam are in equilibrium—i.e., the boiling temperature. An approximate but even more useful version of it can be obtained by neglecting Vliquid in comparison with Vgas and using
from the ideal gas law. The resulting differential equation can be integrated to give
For example, at the top of Mount Everest, atmospheric pressure is about 30 percent of its value at sea level. Using the values R = 8.3145 joules per K and λ = 40.65 kilojoules per mole, the above equation gives T = 342 K (69 °C) for the boiling temperature of water, which is barely enough to make tea.
The sweeping generality of the constraints imposed by the laws of thermodynamics makes the number of potential applications so large that it is impractical to catalog every possible formula that might come into use, even in detailed textbooks on the subject. For this reason, students and practitioners in the field must be proficient in mathematical manipulations involving partial derivatives and in understanding their physical content.
One of the great strengths of classical thermodynamics is that the predictions for the direction of spontaneous change are completely independent of the microscopic structure of matter, but this also represents a limitation in that no predictions are made about the rate at which a system approaches equilibrium. In fact, the rate can be exceedingly slow, such as the spontaneous transition of diamonds into graphite. Statistical thermodynamics provides information on the rates of processes, as well as important insights into the statistical nature of entropy and the second law of thermodynamics.
The 20th-century English scientist C.P. Snow explained the first three laws of thermodynamics, respectively, as:
- You cannot win (i.e., one cannot get something for nothing, because of the conservation of matter and energy).
- You cannot break even (i.e., one cannot return to the same energy state, because entropy, or disorder, always increases).
- You cannot get out of the game (i.e., absolute zero is unattainable because no perfectly pure substance exists).