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

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.

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.

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.

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 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.