Components in Chemical Thermodynamics Robert A. Alberty Massachusetts Institute of Technology, Cambridge, MA02139 Chemical eauations are actuallv matrix eauations. and this hos important impllra~ionsi ) r their th&modvnmnic treiltment I , . The f n n d a m e n ~ deauation of tht:rmodvnamics for a n open system involvingouly PV work can be written in matrix notation (2)a s follows: dG = -SdT
+ VdP + pdn
(1)
Here p is the 1xN row matrix of chemical potentials of N species and n is the N x l column matrix of amounts ni of species. At chemical equilibrium the fundamental equation for a n open system involving only PV work can be written in terms of the differential amounts nci of C components. dG = S d T + VdP + ficdnc
Thus, we can see t h a t there are two components a t chemical equilibrium, in contrast to four species. The first term in brackets is the differential of the amount of carbon, and the second term in brackets is the differential of the amount of hydrogen. The only way the amounts of the elements in the system can be changed is by addition of one or more of the four substances to the system. Another form of eq 6 is obtained by making a Gaussian reduction of the A matrix before carrying out the matrix multiplications.
(2)
Here p is the 1xC row matrix of chemical potentials of species and nc is the C x l column matrix of amounts of components. Equation 2 is obtained from eq 1by substituting the equilibrium relations Zvjpj = 0, where vi is the stoichiometric number of species i. The number C of components is equal to the number N of species minus the number R of independent reactions occurring in the system; C = N - R. The column matrix of amounts of components in a system is given by where A is the CxN conservation matrix that gives the numbers of atoms of elements in species and coefficients in any other conservation equations that are obeyed by the system. Therefore, eq 2 can be written
The reason for beinp- interested in the fundamental eauation in terms of amounts of component.; is that at chemical equilihrium they are independent variables. hut the amounts of species are not. As a n example of eq 4 let us consider a n open system that contains the gases ethylene, methane, ethane, and propane in the presence of a catalyst that equilibrates the reactions between these four gases. At constant temperature and pressure, eq 4 is
where the first row of the A matrix gives the numbers of carbon atoms in the four species and the second row gives the numbers of hydrogen atoms. Carrying out the matrix multiplications yields
Now the first row in the A matrix gives the number of C2H4 units in each of the species, and the second row gives the number of CH4 units in each of the species. This way of writing the A matrix indicates ( I ) that there are two reactions:
C2H, + CH4 = C3H8 Carrying out the matrix multiplications yields
The first term in brackets is the differential of the amount of CzHa (free and bound) in the system, and the second term in brackets is the differential of the amount of CH4 (free and bound) in the system. Equation 10 shows t h a t we can speak of the "ethylene" component with p(CzH4) and the "methane" component with p(CH4),and so this is in agreement with eq 2. The initial state of a closed reaction system can be described by specifying the number F of intensive variables calculated using the phase rnle. In this case,
and so I: I: and n,(H)ln,(C) or rl: P, nJethylene component)ln,(methane component) can be used, where the n,'s are initial amounts. Equations 6 and 10 show that different choices of components can be made. Note that components are conserved like elements. If the order of columns in the A matrix is changed, chemical potentials of other species become involved. The first columns in the A matrix should be used for species that contain all of the elements in the system. Literature Cited 1. Albert) R. A J Chem. Edue. 1631 68,984. 2. Alberty R. A. J Phys. Chem. 1993 97,6226
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