Talk:Massieu function

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• Massieu function

--Libb Thims (talk) 22:34, 23 August 2010 (UTC)

This page contains the some information found on the eoht page noted in the article. In particular the part concerning conjugate variables. After having searched out and read the original articles by Massieu this treatment seems inadequate and stresses more the work of Gibbs than Massieu. It seems a bit misleading.

Somewhat better citations for the original articles would be as follows. The original article appears in Comptes Rendus, volume 69 for 1869 . A follow-on to this appeared later in the same volume . A later and more carefully written expository article was given in 1876. .

Massieu was seeking a theoretical account of vapors, saturated vapors and especially high temperature vapors. In this he discovered what he called his characteristic functions. His motivations were made plain in the opening of the third article mentioned. To quote him:

Verdet, made a remarkable lecture in 1862 before the Chemical Society of Paris, he spoke about the lack of connection, which has so long existed, between the various properties of the body and the general laws of physics, “Surely nothing is less satisfying to the spirit than the lack of relations between the various properties of the same body, or similar properties of different body. With no relationship between facts, the best observations constitute no more a science than the best cut stones, arranged in rows according to size or to the proportion of their forms constitutes a building."

With regard to mechanical and thermal properties of the body, thermodynamics, or the mechanical theory of heat, filled the gap. Indeed, two general principles which are the basis for this new science, result in relationships that previously could not find a clear and scientifically acceptable expression. Thus, for example, it suffices to know today, on the one hand, the quantities of heat that must be supplied to a body to vaporize at various temperatures, and, on the other, the tension [pressure] of its maximum steam these same temperatures, in order to deduce the corresponding densities of the saturated steam. Similarly, ... if the coefficient of expansion of a body under constant pressure decreases when the temperature increases, its specific heat at constant pressure increases when the pressure becomes greater.

(This is my translation, which may be a bit clumsy. Émile Verdet (1824-1866) was a physicist of the day. There is a brief Wiki page on him.)

The upshot of the approach that was used by Massieu (see first reference) goes something like this. Start with the first law as given by Clausius

dQ = dU + ApdV.

(Sorry, but I don't know how to set equations for Wiki.)

In this we have the usual Q for heat, U for internal energy, p for pressure and V for volume. The A is the thermal equivalent of mechanical work. (The same symbol as used by Clausius.)

With U as a function of T and V this equation becomes

dQ = (dU/dT)dT + ((dU/dV) + Ap)dV

Note that Massieu did not use the Jacobi curly-∂ for partial derivatives as was common in his day. I'll switch to the modern notation.

To get the element of entropy dS we divide the last expression by T to obtain

dS = dQ/T = [(∂U/∂T)/T]dT + [((∂U/∂V) + Ap)/T]dV.

We are assuming a reversible process so we have an equality rather than an inequality.

This equation is exact so that the second partials are equal

∂[(∂U/∂T)/T]/∂V = ∂[((∂U/∂V) + Ap)/T]/∂T.

Developing the derivatives and with a bit of rearranging we obtain

∂[Ap/T]/∂T = ∂[U/T^2]∂V.

Now if we write the following expression

dψ = (U/T^2)dT + (Ap/T)dV

we see the use of this. The previous expression indicates that this equation is exact. Thus, we have a function ψ that completes the equation. This he called the characteristic function of the system.

By noting that

dψ = (∂ψ/∂T)dT + (∂ψ/∂V)dV

we see immediately that

U = T^2 (∂ψ/∂T),

p = (T/V) (∂ψ/∂V).

It is also easy to see that the entropy is

S = T (∂ψ/∂T) + ψ.

Using the above expression for U and rearranging we get

-ψT = U - TS.

This is remarkably close to the Helmholtz free energy F.

Other useful information, e.g., such as the specific heats of vapors, can also be derived from ψ. The whole point of Massieu's work was to address Verdet's lament.

I would suggest a major rewrite of this page to reflect more accurately what Massieu meant by characteristic function of a system. I would be happy to provide some input, but I have no experience with writing Wiki pages. Perhaps a collaboration would work?

Zorpoid Zorpoid (talk) 22:52, 7 January 2015 (UTC)

References