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,

(3)

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 *μ _{i}*
will be the chemical potential of component

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,

From *H* = *U* + *pV* we obtain,

(6a)from

(6b)and from

(6c)From Equations 5 and 6a, b, and c we see that there are four different appearing ways to write the μ

(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

(9a, b)Notice that the quantities,

**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

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*,

(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

(12)If you insist on moving the system from one container to the other then we can do this by integrating Equation 11b from

(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

(19a)or

(19b)which tells us that if we change

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

**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 *n _{i}*'s. Can we now make
more Legendre transforms to obtain a new function which is a function of

(22)If we plug in Equation 12 for

**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,

(23)and

(24)In equations 23 and 24 it is understood that all the other

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