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MASSIEU'S CHARACTERISTIC FUNCTIONS

A Massieu characteristic function is an example of a thermodynamic potential function. This type of function was first discovered by François Jacques Dominique Massieu (1839-1896) in 1869 [5].

Thermodynamic potentials are now found by using the Legendre transformation on the internal energy U or the entropy S. Massieu's original discovery of his functions did not use the Legendre transformation. In this article we will only be concerned with what Massieu did and not with what has been done subsequently.

1. INTRODUCTION

The laws of thermodynamics assert the existence of two state functions associated with every thermodynamic system, namely the internal energy U and the entropy S. While these functions are theoretically important it is often more useful to introduce the additional state functions, now called thermodynamic potentials. These are the enthalpy, Helmholtz free energy and Gibbs free energy.

Often the initial formulation of the thermodynamic potentials is attributed to Herman von Helmholtz, Pierre Duhem or Josiah Willard Gibbs. However, all three of these investigators give credit to François Massieu. Helmholtz gives an incorrect citation [1]. In his well-known work Gibbs [3] did not assign credit, but in a follow-up summary article [4] he gives a clear reference. The term ‘thermodynamic potential’ was first used by Duhem [2] and he also gives Massieu his due.

Gibbs says, in a footnote [4], that:

“M. Massieu appears to have been solve the problem of representing all the properties of a body of invariable composition which are concerned in reversible processes by means of a single function.”

The laws of thermodynamics assert that we have two functions of state, the internal energy U and the entropy, describe all possible thermodynamic systems. What Gibbs is saying here is that Massieu was the first to do the same job, with certain provisos, using only a single function of state.

While this is a very important accomplishment, there is more to it as noted by Massieu [7]:

“	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 constitute 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 of its maximum steam these same temperatures, in order to deduce the corresponding densities of the saturated steam. Similarly, we can say … 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.”

In other words the functions that Massieu finds can be used to extract “relations between the various properties of the same body, or similar properties of different body”

2. MASSIEU'S FIRST FUNCTION

Massieu begins his development [5] with Clausius’ first law of thermodynamics

In this we have the usual quantities pressure p, volume V, temperature T and the heat Q. The internal energyU is a function of V and T. In this A is a conversion factor from mechanical work to thermal units. At the time including such a factor was the usual practice.

With U is a function of T and V we have the total differential

and by substituting this into ($$) and regrouping we get

According to the second law of thermodynamics there is a function of state S(T, V), called the entropy. For any reversible process dS = dQ/T. So we have the following

The second equality results from substituting ($$) for dQ. The next equality follows on developing dU and regrouping terms.

Since S is a state function ($$) is an exact differential equation, thus we necessarily have

Writing out the derivative on the left and the first of the derivatives on the right we obtain

This is just the condition we need to conclude that there is a function ψV so that the following exact differential equation holds

The function ψV is the first such function that Massieu called the characteristic function of the system under consideration.

Now let us see what kind of information we can extract from this function. The total differential of ψV is

Comparing ($$) and ($$) we see that

and

Thus, given V and with the measurement of T and V we obtain the internal energy and the pressure.

We can also determine the heat dQ by using equation ($$) in ($$) to arrive at

When this is divided through by T and integrated we obtain

Let us check this. Dividing ($$) through by T to get the element of entropy thus,

The task is to integrate this in order to determine S. For this we have

For the first integral in ($$) we integrate by parts as follows

For the second integral we have

In the first integral in this we see that since T and V are independent variables the T2 and 1/T can be taken from beneath the integral leaving second expression. In this form we see that the fundamental theorem of calculus applies.

The third integral is just changing the variable of integration, thus

Adding these we obtain

There is no lose of generality if we allow the factor 2 to be included as part of the characteristic function ψV, thus we may write, along with Massieu,

From all this we see that given the characteristic function ψV for the system we can use its partial derivatives along with V and T to obtain U, p and S with little effort.

Let us make one further observation. If we put ($$) into ($$) we see that

or

The first of these is another way to define ψV and the second is yet another way to obtain the internal energy. Moreover, we can write as

We recognize on the right hand side of this the Helmholtz free energy.

3. MASSIEU'S SECOND FUNCTION

For Massieu’s first characteristic functions it was assume that the independent variable were the temperature T and the volume V. His next concern was to use the pressure p rather than the volume. To this end he introduces

This is the Legendre transform of U that changes the variable V to p. (Massieu gives no explicit mention of Legendre. More on this below.) With G as a function of T and p we have

With this the first law of thermodynamics can modified as follows

Dividing this through by T and using ($$) we obtain

This is an exact differential equation so that we necessarily must have

Developing this we arrive at

This is just what is needed to establish that there exists a function ψp so that the following equation is an exact differential equation

The total differential is

Comparing these two equations, it is evident that

and

Using these we can evaluate the entropy to obtain

and so

Substituting for G using ($$) we obtain

The right had side of this is the Gibbs free energy.

4. MASSIEU’S USE OF HIS CHARACTERISTIC FUNCTIONS

Here we would put remarks about how he used his functions in the study of gases, vapors and superheated vapors. This might be a bit off-topic though.

5. USING THE LEGENDRE TRANSFORMATION TO DERIVE MASSIEU’S FUNCTIONS

The functions Massieu has derived can be had by using (in 20th century manner) the Legendre transformation.

Massieu gave no explicit mention of Legendre. However, the use of such contact transformations was often tacitly used in the study of differential equations.