3.012 Fundamentals of Materials Science Fall 2005 Lecture 11: 10.15.05 Models of the chemical potential Today: LAST TIME........................................................................................................................................... 2�MORE ON THE RELATIONSHIP BETWEEN CHEMICAL POTENTIAL AND GIBBS FREE ENERGY....................... 3�Chemical potentials in multicomponent systems are linked: the Gibbs-Duhem equation......................................................3�THE CHEMICAL POTENTIAL OF THE IDEAL GAS ...................................................................................... 5�Models of the chemical potential in materials...........................................................................................................................5�The ideal gas model.....................................................................................................................................................................5�Chemical potential of ideal gas mixtures...................................................................................................................................7�THE GENERAL SOLUTION MODEL .......................................................................................................... 8�IDEAL SOLUTIONS .............................................................................................................................. 10�REFERENCES ..................................................................................................................................... 11�Reading: Engel and Reid 6.3, 6.4, 6.5, 9.1, 9.2, 9.5 Lecture 11 – chemical potential models 1 of Supplementary Reading: -11�10/17/05�3.012 Fundamentals of Materials Science Fall 2005 Last time Lecture 11 – chemical potential models 2 of 11�10/17/05�3.012 Fundamentals of Materials Science�Fall 2005 More on the relationship between chemical potential and Gibbs free energy •� Last time, we introduced partial molar quantities: the analog of molar quantities for multicomponent systems. Remember that partial molar quantities are directly related to the total thermodynamic function for the system. Thus we have two ways to write out the ‘integrated’ form of the Gibbs free energy, for example: Chemical potentials in multicomponent systems are linked: the Gibbs-Duhem equation Lecture 11 – chemical potential models 3 of 11 10/17/053.012 Fundamentals of Materials Science Fall 2005 Lecture 11 – chemical potential models 4 of 11�10/17/05�3.012 Fundamentals of Materials Science�Fall 2005 The chemical potential of the ideal gas Models of the chemical potential in materials •� So far we’ve derived rules that the chemical potential must obey in the approach toward equilibrium, but we haven’t provided any equations that tell us what goes into the chemical potential. Because the chemical potential embodies chemical forces within materials, it is different for every material. However, several models for the chemical potential have been developed, which are the basis of many first-order calculations in materials science & engineering. Note that in some cases, we can calculate chemical potentials from first-principles quantum mechanics calculations (for example, this can even be done by hand for simple gases). These models will allow us to make predictions for the behavior of multi-component materials. The ideal gas model •� A starting point for models of the chemical potential is to consider perhaps the theoretically simplest form of matter-the ideal gas. We stated that the ideal gas is modeled by two fundamental relationships: •� Using only the ideal gas equation of state, we can derive the chemical potential of the ideal gas. Consider that we have a quantity of a pure ideal gas in a closed system. As we showed above, the chemical potential is the molar free energy of the gas: •� Now, for any state function, we can integrate a differential: ! dX " dXinitialfinal#= $X = Xfinal% Xinitial•� If we integrate our differential for the chemical potential, we can calculate the chemical potential in some arbitrary state relative to that in some other defined reference state. We will proceed in the following way: We want to know the chemical potential at a temperature T and pressure P, relative to a reference state of the gas at the same temperature (T) and a reference pressure, Po. Integrating the expression above: Lecture 11 – chemical potential models 5 of 11 10/17/053.012 Fundamentals of Materials Science Fall 2005 • Very commonly, the reference state pressure is taken as Po = 1 atm. Lecture 11 – chemical potential models 6 of 11�10/17/05�3.012 Fundamentals of Materials Science�Fall 2005 Chemical potential of ideal gas mixtures •� A straightforward extension of the calculation we have just performed is to determine the chemical potential in an ideal gas mixture. The situation is illustrated by the cartoon below: • Pressure is proportional to concentration for gases (P/RT = n/V for ideal gases). To describe ideal gas mixtures, we introduce the concept of partial pressure: Lecture 11 – chemical potential models 7 of 11 10/17/053.012 Fundamentals of Materials Science�Fall 2005 The general solution model •� The simple ideal gas model of the chemical potential above can be extended for use in describing general solid, liquid, or gaseous solutions. •� In the general solution model, the chemical potential is given by: o� The chemical potential of component i is a function of its mole�fraction, the pressure, and the temperature.�o� To calibrate the value of the chemical potential in a given state, it is standard practice to refer to the chemical potential of that state relative to the pure component i in some reference state (typically atmospheric pressure). The first term is the chemical potential of the component in its reference state: � …which emphasizes that the term RT log ai quantifies the difference between the chemical potential of i in its current state from that of pure i in its reference state. o� In addition to the reference state, we’ve also introduced a new quantity ai-the activity of component i. The activity is further defined by introducing a second new quantity γi, the activity coefficient: •� When we introduce statistical mechanics in the second half of the course, we will see how simple
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