Lecture 10: 10.08.03 Free Energy: Applying the Second Law in Laboratory Conditions

3.012 Fundamentals of Materials ScienceFall 2003

Lecture 10: 10.08.03 Free energy: applying the second law in laboratory conditions

Today:

Last time......

Two new thermodynamic functions useful for laboratory conditions......

Why we need another thermodynamic function......

Equilibrium at constant temperature and volume: Helmholtz Free Energy1

Equilibrium at constant temperature and pressure: Gibbs Free Energy2

Defining the Gibbs free energy......

Determining equilibrium for experiments in the lab......

References......

Reading:Dill and Bromberg, Ch. 8 ‘Laboratory Conditions and Free Energies’ pp. 131-150

Supplementary Reading:-

Planning Notes:

HOMEWORK PROBLEMS:

Last time

Two new thermodynamic functions useful for laboratory conditions

Why we need another thermodynamic function

In the calculations of equilibrium temperature and pressure we made using the second law in the last lecture, we considered isolated systems where transfer of heat, work, or molecules at the boundaries of the system were prevented. (In our block A/block B heat transfer experiment, the blocks did no work and were not allowed to exchange heat with their surroundings- only with one another). In such constant internal energy systems, it is straightforward to directly apply the fundamental equation for the entropy and the second law to calculate equilibrium properties of the system.

Though the experiments described in those calculations are physically possible, controlling heat and work flow at the boundaries of a real system is often not a simple task- and is unnecessary for most experimental procedures you would be interested in performing. Instead, what we would like to control at the boundaries of our system (are the temperature and pressure (imagine a solution of a material in a test tube, as illustrated below). This is experimentally straightforward- we can place our test tube in a heating or cooling bath to maintain a constant temperature, and if the system is open to the atmosphere, it is maintained at a constant pressure (1 atm). Alternatively, if we seal the test tube with an inflexible barrier, the contents of the tube may be held at constant temperature and volume (what would you worry about in the second case?).

(Dill and Bromberg1)

Thus, instead of keeping the internal energy of the system constant, we would like to keep the temperature and pressure or temperature and volume constant, and predict how the system changes when various thermodynamic driving forces are applied (for our test tube example, perhaps we want to carry out a reaction on the material in the solution). Under such conditions, the internal energy of the system can fluctuate. If we have our test tube in a heating bath, internal energy in the form of heat is being transferred into and out of the system to maintain a constant temperature, and we can no longer apply the fundamental equation for the entropy to the system alone to determine the equilibrium state.
Because we are exchanging heat with the surroundings, we have to worry about the entropy change in the surroundings as well. The second law only dictates the behavior of the total entropy of the universe. Ack!

So how do we get around this problem? We’d much rather make calculations for the system alone, rather than having to understand the thermodynamic behavior of the system and its surroundings. The solution is to define a fundamental equation designed for the conditions at hand.

Mathematically, it is most straightforward to make equilibrium calculations using a thermodynamic function whose independent variables are the variables controlled at the boundaries of the system. Thus for isolated systems as described in our earlier examples, where U, V, and N were controlled at the boundaries, equilibrium is readily calculated by maximizing the fundamental equation for the entropy since S = S(U,V,N).

We will now introduce two new functions, the Helmholtz free energy (F) and Gibbs free energy (G), which are useful in different ways for determining equilibrium behavior under laboratory conditions. The Gibbs free energy in particular is useful for common laboratory conditions because G = G(T,P,N). The Helmholtz free energy is useful for conditions of constant temperature and volume because F = F(T,V,N).

Equilibrium at constant temperature and volume: Helmholtz Free Energy1

Our objective is to define a thermodynamic function for a system at constant temperature and pressure that has an extremum principle at equilibrium- that is, a function that is maximized or minimized in order to satisfy the second law for this system.
This can be derived by considering the system of interest and applying the second law. Imagine for concreteness that we have a test tube in a heat bath as depicted in the left-hand figure above. The bath and test tube are isolated from their surroundings and do not exchange heat, work, or atoms with the atmosphere/benchtop/etc. The test tube is the system of interest, but because the tube is exchanging heat with the bath, we must also consider the entropy of the heat bath (the surroundings). Together, the heat bath + test tube do form an isolated system, so the second law can be applied to the pair (nothing else in the universe is being affected by the test tube and bath):

(Eqn 1)second law for the tube + heat bath

Since the test tube + heat bath form an isolated system, we know that the total energy of the pair must remain constant:

(Eqn 2)

Now we expand the second law for the surroundings and system above using the fundamental equation for the entropy, using the conservation relation given in (Eqn 2):

(Eqn 3)(constant volume and N)

The second equality is a result of the conservation of internal energy, and the fact that the bath maintains the system at constant temperature (Tsurr = Tsystem).

(Eqn 4)

(Eqn 5)

(Eqn 5) gives us a thermodynamic function that is mimized to satisfy the second law for a system held at constant temperature and pressure. With this equation we need not know the state of the heat bath- we only need to know the state of the system itself.

The differential function above is defined as a new thermodynamic function: The Helmholtz free energy:

(Eqn 6)DEFINITION OF HELMHOLTZ FREE ENERGY

The total differential of the Helmholtz free energy is given by:

(Eqn 7)

The differential of F at constant temperature reduces to:

(Eqn 8)FUNDAMENTAL EQUATION FOR HELMHOLTZ FREE ENERGY

(constant temperature)

One important application of the Helmholtz free energy is in model calculations, such as lattice models of solids- where it is simpler to make predictions at constant volume (harder to model constant pressure). Thus we will return to the Helmholtz free energy when we introduce statistical mechanics and molecular models of materials.

The Helmholtz free energy is an example of a change of variables- changing from maximizing entropy of a system at constant (U,V,N) to minimizing free energy of a system at constant (T, V, N). There are multiple ways to arrive at this change of variables, known as a Legendre transformation. (Those of you in 3.16 will recognize this).

Equilibrium at constant temperature and pressure: Gibbs Free Energy2

Defining the Gibbs free energy

We defined a useful thermodynamic function (the enthalpy, H) by simply subtracting the ‘mechanical’ component of the internal energy from U.

In a similar manner, we can define a very useful function called the Gibbs free energy (G), by subtracting both the ‘thermal’ energy and the ‘compressive’ energy from U:

(Eqn 9)

DEFINITION OF GIBBS FREE ENERGY

But what is the Gibbs free energy? We’ve taken the total internal energy and subtracted off the thermal energy and the compressive energy- what’s left? The answer is: energy arising from other kinds of work terms. So far we’ve mostly focused on two sources of internal energy, compressive work from an applied pressure and thermal energy due to heat transfer. However we also introduced other types of work: work of polarization, chemical work, magnetic work, etc.
In many practical problems in materials science & engineering, it is these other kinds of work in which we are most interested..

The differential form of the Gibbs free energy (the fundamental equation for the free energy) is useful for many calculations:

(Eqn 10)

…where the sum is over the C components in the system and the P phases in the system. The differential for G reduces to a simple expression under conditions of constant temperature and pressure:

(Eqn 11)at constant temperature and pressure

Thus changes in free energy at constant temperature and pressure are dictated by the chemical potential of the components in a system- will they react? Mix? Unmix?

Determining equilibrium for experiments in the lab

The second law defines equilibrium by the change in entropy in the entire universe. As already discussed, that’s an incredible pain when we are only really interested in what’s happening in our test tube! The Gibbs Free energy gives us a solution to this problem for the case of experiments at constant temperature and pressure. Let’s see how the Gibbs Free energy relates to the second law at equilibrium when we have typical laboratory conditions: a closed system (no exchange of molecules with the surroundings), at constant temperature and pressure.

The second law defines equilibrium by:

(Eqn 12)

dS = 0 mathematically means S is at an extremum (a minimum or a maximum); however, the second law also says:

(Eqn 13)for a spontaneous process

…which implies that the system has to be approaching a maximum when it reaches equilibrium.
Let’s apply the second law to a system at constant temperature and pressure and see what comes out:

(Eqn 14)for a spontaneous process

From the fundamental equation for the entropy, we have:

(Eqn 15)

Plugging into the second law:

(Eqn 16)for a spontaneous process

(Eqn 17)for a spontaneous process

…where we’ve dropped the subscript on T (our system is at constant temperature Tsurr = Tsystem = T). Now, if we compare (Eqn 17) to the fundamental equation for the energy:

(Eqn 18)

(Eqn 19)

We thus reduce our condition for a spontaneous process to:

(Eqn 20)spontaneous process

Recalling the differential form of the Gibbs free energy at constant temperature and pressure, we can make a substitution:

(Eqn 21)spontaneous process at constant temperature and pressure

(Eqn 20) and (Eqn 21) are for practical calculations perhaps the most important in thermodynamics.
In words, the second law demands that the Gibbs free energy must be lowered by the change happening to a system for that change to happen spontaneously at constant temperature and pressure. It also means that the Gibbs free energy at constant temperature and pressure is a minimum at equilibrium.

References

1.Dill, K. & Bromberg, S. Molecular Driving Forces (New York, 2003).

2.Carter, W. C. (2002).

Lecture 10 – Free energies1 of 10 9/2/03