Dear Lothar,

I am not sure, what you mean by 'how the state was achieved' but pressure and volume are fixed, if you look at a certain state of your thermodynamical system. Since pressure is a statistical property of your esemble it might even change at certain small areas of your fixed volume without changing the macroscopic thermodynamical state. However, both entities are macroscopically constant at a given time. The thermodynamical work pdV, on the other hand, is proportional to a CHANGE in volume at a constant pressure.

If you look at a changing system, as opposed to a fixed state, it might very well be dependent on the way the parameters change in your p-V diagram.

You may just consider that:

(1/p)dW=dV

(1/T)dQ=dS

and therefore, 1/p and 1/T are the integrating factors for dW and dQ (inexact differentials, process variables) to become dV and dS (exact differentials, state variables).

The product of two state variables is not necessarily a state variable. The sum of two process variables is not necessarily a process variable.

I make a parallel with climbing a mountain. You can go straight to to top or you can walk on a smooth path. In any case the energy variation is the same. Why in a thermodynamic system is it not the same? Because you are neglecting the losses of energy due for example to friction. If the friction is proportional to the path length, lo smooth, but longer, path will need more energy than climbing straight to the top. In thermodynamic it is the energy transformed in heat that makes the difference. Different thermodynamic paths transfer a different quantity of energy to heat.

In fact the energy is the a state variable, and is U=Q-L.

Q-L is a state variable, but Q and L are not.

Dear Gianpiero, dear Adrian,

when I simply draw the conclusion from the 2 answered you gave, the following escapes:

work dW is an inexact differential due to the heat losses / friction (irreversible) along the path??

Is then irreversibility of a process making the difference between process and state variable ?

Regards

Lothar

Irreversibility is not the key, since two reversible processes connecting the same two states A and B (e.g. an isothermal path and an adiabatic path) will have associated different dW and dQ.

State variables exist for a given state, even if the system does not evolve, and they change along the process. Process variables do not exist for a given state, and they appear associated with a given process. Because dW is pdV, because there are more thermodynamic variables, and because there are many paths compatible with reversible processes between states A and B, pdV does not have a single value and, therefore, it depends on the path.

The set of reversible processes compatible with an initial state A and a final state B will be characterized by an internal energy change deltaU = UB - UA, while deltaU = Q - W will partition differently into Q and W for each of these processes.

Of course, dW = pdV in a simple system. In general, dW is the sum of many terms YdX, with Y being an intensive variable and X an extensive conjugated variable.

Thank all of you for the answers!! And have a nice "rest of the weekend".

The first law requires that:

δW= δQ -dU

Then, for the fictive case of a reversible process, dU can be written in terms of exact differentials. One may imagine reversible changes, such that there is at each instant negligible departure from thermodynamic equilibrium within the system. This excludes isochoric work. Then, mechanical work is given by δW = - P dV and the quantity of heat added can be expressed as δQ = T dS. For these conditions

PdV=dU-TdS

T - is temperature ; S - is entropy ; U - is internal energy

While this has been shown here for reversible changes, it is valid in general, as U can be considered as a thermodynamic state function of the defining state variables S and V

PdV=dU-TdS

I think it is the other way around, it is not that thermodynamic variables were combined to get process variables. The processes came first, and they "combined" (albeit in a very complicated way) to state functions.

Work (and let's stick to gasses for the moment, but similar things are valid for other types of work) is given by:

δW = -pdV

This is work done on the gas, and its definition is in fact mechanical, force times distance, so it follows from classical mechanics.

It was Carnot who realised that heat can do two things to a gas: it can change the temperature, and it can change the volume:

δQ = CVdT + β dV = CVdT +(p+π)dV

I did not write pdV for the second term, because in general that term is not equal to pdV, that is only the case for an ideal gas. If a real gas expands the interaction energy between the molecules becomes smaller and this accounts for part of the heat needed. Another easy example is the photon gas where expansion creates more photons. However, I can always write it as the second term, in which case π turns out to be something called the internal pressure.

It is possible using Carnot's assumption that there are no perpetual motion machines (and consequently all media must lead to the same efficiency) to prove that ∮ (CVdT +πdV) =0 for a Carnot contour, and therefore to identify a state function U with

dU = CVdT +πdV = δW+δQ

Carnot knew there were processes, which is what he studied, but had no knowledge of state functions, and although he did (sort of) identify S. U was in fact only found to be a state function much later (had Carnot been smarter and a better mathematician, this could have been the other way around).

Carnot made an interesting mistake, since he believed that measurements had shown CV to be dependent on the volume. However, then π cannot be zero (he used the ideal gas law), dU being a total differential implies that the volume derivative of CV must be equal to the temperature derivative of π, and π=0 contradicts that.

Here is my proof (and something that Carnot could have done) that U is a state function.

Hello,

I saw that some more answers and comments are flying in. Thank you all so much!

Lothar

Thermodynamics can be built: 1) empirically, the "traditional" way, starting from engines and processes, from which the whole formalism is elaborated; or 2) axiomatically, the "modern" way, starting from few definitions and principles from which the whole formalism is elaborated, and engines and processes are examples of general results.

Of course, the second way has been made possible thanks to the concepts, developments and ideas that have been developed historically, in which initially Thermodynamics was considered as an applied science or as an engineering discipline (Carnot was an engineer).

Therefore, we may consider the historical development of Thermodynamics, and this was a remarkable accomplishment. But the axiomatic formalism in Thermodynamics represents also a remarkably elegant and powerful theoretical framework.

Agreed, but don't you think that in both cases the process is the important aspect? In both approaches it is the starting point. For the axiomatic approach it is the concept of adiabatic accessibility, which relates to processes in the neighborhood of a given point. Caratheodory derived the existence of entropy from the fact that not all states are adiabatically accessible from a given state. See also the Lieb and Yngvason's monograph (Phys. Rep. 310: 1–96) , or Andre Thess' The Entropy Principle. They both start with defining an order relation which tells us whether there is a process possible from one state to another.

I agree, partially, because the axiomatic formalism is not unique, not necessarily based on processes (see H.B. Callen). Of course, the final goal in Thermodynamics is to find out the final equilibrium state in a system along a given process, as well as the changes in state variables and values of process variables along such state.

Isn't Callen's work more related to Statistical Mechanics? Or is that what you meant by the theoretical framework?

Callen's work is related to both. In his book "Thermodynamics" you will find his axiomatic formalism for Thermodynamics

@Lothar Schuh,

Generally speaking, we may have to use the fitting method to get the specific function W=fw(P, V T).

(In general, two of the three variables(P, V T) are independent in W=fw(P, V T) ), e.g., W=fw(P, V).

W and Q are process quantities which vary with paths, W (or Q) has innumerable forms between the same original and terminal states, and has a unique form for fixed reversible process path.

When the given path is fixed, W and Q are system state variable.

A system goes from initial state 1 through a path to final state 2, the default value W0 of the initial state is W0=0, then, between state 1 and state 2, there is W=fw(P, V, T)

( and Q=fq(P, V, T), E=Ee(P, V, T) ).

(In general, two of the three variables(P, V T) are independent.)

Here, E is internal energy, different from Q and W, it always is system State Variable.

Although we know there is W=fw(P, V T)

( and Q=fq(P, V, T), E=Ee(P, V, T) ),

it is very difficult to write the specific function form of an arbitrary path, in fact, we may have to use the fitting method to get the specific function.

That is, e.g., using Piecewise approximation method, measureing Wi and its corresponding state variable(Pi, Vi) in section i of the process,

using the fitting method to get the approximation function Wi=fi(P, V), In general, it's a piecewise function.

@Adrian. Reading Callen's book (thanks for bringing it to my attention) I noticed that he also starts with defining the fluxes, although he does use energy U as a starting point. His first postulate is that there are states, called equilibrium states, which are completely characterized by energy U and volume V (I leave out composition here, to many letters to write). He then tells us that there are things like adiabatic walls, and that energy of a system in an adiabatic container can be changed by work, which he identifies as -pdV. He realizes that the energy of a system can also be changed by adding heat, (stirring a glass of water will heat it, but also leaving it out in the sun). His next step is then to define the heat flux dQ as the difference between dU (which he shows can be measured) and the work (of which you also know what it is).

So indeed: dU = dQ +dW. (since I can't do barred d's, the difference between exact and inexact differentials is implied).

Getting from dQ to TdS is a bit involved but he basically postulates the way to do it. There exists for every system another function S(U,V) which is single valued, monotonic, and continuously differentiable so that U(S,V) exists, then define the temperature as the derivative of U with respect to S at constant volume, and get dQ = TdS, so that dU = TdS -pdV.

So I maintain my point of view that it is the fluxes that come first, and that there is no need for a "physical rationale that combining 2 thermodynamic state variables" (as the original question was) to get process variables. There is no such rationale needed, although in this case you could argue that the postulates (the existence and properties of U and S) provide the rationale.

Nothing wrong with that of course, although for the uninitiated his postulates could seem rather arbitrary but they are, as you stated, founded in the history of thermodynamics.

For completeness and contemplation here are Callen's postulates.

Postulate 2: There exists a function (called the entropy S) of the extensive parameters of any composite system, defined for all equilibrium states and having the following property: The values assumed by the extensive parameters in the absence of an internal constraint are those that maximize the entropy over the manifold of constrained equilibrium states.

Postulate 3: The entropy of a composite system is additive over the constituent subsystems. The entropy is continuous and differentiable, and is a monotonically increasing function of the energy.

Although Callen indeed nicely shows how to derive all we know and love from this, I keep a certain preference for Carnot's only postulate: There are no perpetual motion machines.

Yes, I agree, there is no need for a physical rationale in that combining two thermodynamic state variables results in a process variable. But for combining a process variable and a state variable to get a state variable (integrating factors).

I do not underestimate Carnot's developments. I just said there are two ways: one thing is to define dQ and dW, and other thing is to consider engines, the maximum efficiency and other constraints. In fact, for many people, Kelvin's and Clausius versions of the Second Law look somewhat awkward, considering that such a law should be formulated in a more general way.

I don't mean to say that there are no things in Carnot's methods that bother me. And it is not just the errors he makes, some of which can be attributed to insufficient experimental data. What I've always found strange that on the basis of a wrong model for heat he can derive results that are correct. Heat in his view is conserved, it is the falling of it from high to low temperature that allows us to take work from it, like getting work from water in a waterfall. Although I now think I have sort of an argument as to why this can actually work. The thing that bothers me most is what you state in your last line: it should be possible to formulate it in a more general way. In all derivations I've seen and tried myself, at one point or another the ideal gas is used. Of course Carnot states himself: "The maximum amount of power that can be obtained by steam must be equal to the maximum power that can be obtained by any means." So finding the maximum for an ideal gas must be the maximum for any substance, and therefore the maximum cannot depend on any property of the medium, we might as well use an ideal gas.

I guess it is that same rationale that is used in the more mathematical approach: finding an integrating factor for dq= dU + pdV. For functions of two variables it is known that such a factor can always be found, and for the ideal gas it is not too hard to show that it must be 1/T, although this result is not unique (Am. J. Phys. 74 (2006), 699). Somehow this means that it is (must be, or could be?) 1/T for any substance. Which leads to a universal relation between internal pressure and specific heat for all possible substances. I've always found it slightly strange that there is not a substance in the world for which the volume derivative of the specific heat is not the same as the temperature derivative of the internal pressure.

I can put restrictions on the function of temperature , based on the fact that the result for efficiency cannot depend on the scale or the zero point used (Fahrenheit and Celcius should give the same efficiency) and the fact that two Carnot engines in tandem (between Th, Tm, and Tc) must have the same efficiency as one engine between Th and Tc but this is not sufficient to get the 1/T either.

Of course if you take the axiomatic approach all these worries disappear. But then again, postulate 2 of Callen does not make me very happy, and with the adiabatic accessibility I always get stuck at the question why you are allowed to use work to stir the system and make it hotter, but not to use the work to run a heat pump, and make it colder. I still think it is much easier to say that heat never flows spontaneously from cold to hot (Clausius) or that you can't take heat from a single reservoir and convert it to work (Kelvin), but I'm not even totally convinced that these are the same thing.

I agree about everything.

But, for example, I do not find comfortable that Clausius and Kelvin statements are in fact two examples of a general law. Thus, Clausius could have also said "molecules never flow from lower to higher chemical potential". There is something underlying, and that is the entropy change in an evolving isolated system.

I just forgot to say that, following Lotar's question, P and V are state variables and their product PV is also a state variable. That is, delta(PV) = (PV)B - (PV)A in a process connecting states A (initial) to B (final) only depends on states A and B. However, the path integral intA-B(pdV) is not a state process because the reversible connection between states A and B can be done in many different ways.

Therefore, the combination of 2 states variables is another state variable. But the path integral of a state variable may not be a state variable.

W= fw(T, V, P) ( and Q= fq(T, V, P)) is a process quantity which varies with path, it has innumerable forms between the same original and terminal states, and has a unique form for fixed reversible process path.

When the path is fixed, W= fw(T, V, P) ( and Q= fq(T, V, P)) is the system state variable.