Up to now all of the systems we have been working with were closed systems. That is, no material moved in or out of the system. Now it is time to extend our discussion to open systems, in which material can move in or out of the system.
As usual we begin with the combined first and second laws of thermodynamics, only now we have to take into account that the internal energy, U, will depend on the number of moles of each component present. We write
(1)so, instead of
(2)we must add the contribution of moving material in and out of the system. That is, we write,
The first two terms of Equation 3 must be the same as the combined first and second laws, Equation 2. The remaining terms in Equation 3 are new to us. They are clearly awkward to write so we invent a new symbol for the partial derivative,
(4a, b)and so on.
(Notice that we are using the same symbol, μ , here that we used for G/n previously. That is not an accident, as we shall see. We called μ the chemical potential. In an open system with more than one component μi will be the chemical potential of component i in the mixture.)
With this notation we can rewrite Equation 3 as,
(5)We can carry the terms accounting for the movement of material through to our other "energy" functions, H, A, and G.
From H = U + pV we obtain,
(6a)from A = U - TS we obtain
(6b)and from G = U +pV − TS we obtain,
(6c)From Equations 5 and 6a, b, and c we see that there are four different appearing ways to write the μi's. For example,
(7)All of these definitions are equivalent, but the last one,
(8)will be the most important one because it gives the change in Gibbs free energy that comes from adding or removing material at constant pressure and temperature. For a process a constant temperature and pressure dG becomes
(9a, b)Notice that the quantities, μi , are intensive variables. We can see this because they are an extensive variable, G divided by another extensive variable, ni.
Integration of dU
There are some unique features of the differential, dU, in Equation 5,
(5)which allow us to integrate it in an exceptionally simple manner. (This statement is not true for the differentials dH, dA, and dG.) In Equation 5 all the differentials (dU, dS, dV, dn1, etc.) are differentials of extensive properties. That is, they all depend on the amount of material in the system. In addition, all the coefficients of these differentials (T, p, μ1, etc.) are intensive properties
We can imagine integrating Equation 5 by starting out with the system in one container and transferring it to another container one differential drop at a time while holding all of the intensive variables constant. As we move the system from one container to another all the extensive quantities move to the new container in proportion to the size of the drop. Let's parameterize this process with a variable, x,
where x = 0 means the system is entirely in its old container and x = 1 means the system has been entirely transferred to its new container. The "drop size" is given by the size of dx. Then,
(10)and so on.
We can now rewrite Equation 5 in terms of the differential, dx,
(11a, b)In actual fact we don't have to integrate Equation 11a or 11b to get the desired result, just divide 11b by dx to get
(12)If you insist on moving the system from one container to the other then we can do this by integrating Equation 11b from x = 0 to x = 1,
(13)which gives Equation 12 because the integral is equal to 1.
Equation 12 is called the integrated form of the combined first and second laws for an open system.
We can use Equation 12 to obtain "integrated forms" of H, A, and G,
(14a, b)but the most important one is
(14c)The latter equation is sometimes written
(15)where the summation is over all the components in the system.
The Gibbs-Duhem Equation
We now have (or can get) two different expressions for dG. One expression is Equation 6c,
(6c)and the other can be obtained from Equation 15 as,
(16)Setting Equations 6c and 16 equal to each other, and canceling terms that are the same on both sides we obtain,
(17)Equation 17 is called the Gibbs-Duhem equation. It tells us that the intensive variables in a system can not all be assigned values independently. That is, you can assign virtually any value you desire to all of the intensive variables but one, but the value of that last one will be predetermined by the values of the others.
The Gibbs-Duhem equation is most often used for processes at constant temperature and pressure, whence Equation 17 becomes,
(18)Consider a two-component system at constant T and p. Then,
(19b)which tells us that if we change μ2 then μ1 responds with a change which depends on the ratio of the amounts of the two components.
In a one-component system the Gibbs-Duhem equation takes the form,
(20)We rearrange this equation to give,
(21a, b)which demonstrates that in a one component system the chemical potential is a function only of T and p.
Comment on Legendre Transforms
Recall that we get from Equation 12 to Equations 14a, b, and c by making Legendre transforms. Equation 14c shows that G is a natural function of T, p, and all the ni's. Can we now make more Legendre transforms to obtain a new function which is a function of T, p, and all the μi's as in,
(22)If we plug in Equation 12 for U here we get,
This means that no such function exists. This is a consequence of the fact that the new function, if it existed, would be a function of only intensive variables (all independent) and we have seen from the Gibbs-Duhem equation that the intensive variables are not all independent.
Maxwell's Relations Revisited
Equations 5 and 6a, b, and c open up many new possibilities for Maxwell's relations. For example, from Equation 6c we obtain two equations which may be useful later,
(24)In equations 23 and 24 it is understood that all the other n's are also being held constant. Equations 23 and 24 will be useful when we start worrying about how the chemical potentials depend on temperature and pressure.
Copyright 2004, W. R. Salzman
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