User:David Shear/Chemical thermodynamics

Chemical reactions
In most cases of interest there are internal degrees of freedom and processes, such as chemical reactions and phase transitions, which always create entropy unless they are at equilibrium, or are maintained at a "running equilibrium" through "quasi-static" changes by being coupled to constraining devices, such as pistons or electrodes, to deliver and receive external work. Even for homogeneous "bulk" materials, the free energy functions depend on the composition, as do all the extensive thermodynamic potentials, including the internal energy. If the quantities { Ni } are omitted from the formulae, it is impossible to describe compositional changes.

For a "bulk" (unstructured) system they are the last remaining extensive variables. For an unstructured, homogeneous "bulk" system, there are sill various extensive compositional variables { Ni } that G depends on, which specify the composition, the amounts of each chemical substance, expressed as the numbers of molecules present or (dividing by Avogadro's number), the numbers of moles


 * $$ G = G(T,P,\{N_i\})\,.$$

For the case where only PV work is possible


 * $$ dG = -SdT + VdP + \sum_i \mu_i dN_i \,$$

in which μi is the chemical potential for the i-the component in the system


 * $$ \mu_i = \left( \frac{\partial G}{\partial N_i}\right)_{T,P,N_{j\ne i},etc. } \,.$$

The expression for dG is especially useful at constant T and P, conditions which are easy to achieve experimentally and which approximates the condition in living creatures


 * $$ (dG)_{T,P} = \sum_i \mu_i dN_i\,.$$

While this formulation is mathematically defensible, it is not particularly transparent since one does not simply add or remove molecules from a system. There is always a process involved in changing the composition; e.g., a chemical reaction (or many), or movement of molecules from one phase (liquid) to another (gas or solid). We should find a notation which does not seem to imply that the amounts of the components ( Ni } can be changed independently. All real processes obey conservation of mass, and in addition, conservation of the numbers of atoms of each kind. Whatever molecules are transferred to or from should be considered part of the "system".

Consequently we introduce an explicit variable to represent the degree of advancement of a process, a progress variable ξ for the extent of reaction (Prigogine & Defay, p. 18; Prigogine, pp. 4-7; Guggenheim, p. 37.62), and to the use of the partial derivative ∂G/∂ξ (in place of the widely used "ΔG", since the quantity at issue is not a finite change). The result is an understandable expression for the dependence of dG on chemical reactions (or other processes). If there is just one reaction


 * $$(dG)_{T,P} = \left( \frac{\partial G}{\partial \xi}\right)_{T,P} d\xi\,$$

If we introduce the stoichiometric coefficient for the i-th component in the reaction


 * $$\nu_i = \partial N_i / \partial \xi \,$$

which tells how many molecules of i are produced or consumed, we obtain an algebraic expression for the partial derivative


 * $$ \left( \frac{\partial G}{\partial \xi} \right)_{T,P} = \sum_i \mu_i \nu_i = -\mathbb{A}\,$$

where, (De Donder; Progoine & Defay, p. 69; Guggenheim, pp. 37,240), we introduce a concise (but odd) name for this quantity, the "Affinity", A. The minus sign comes from the fact the Affinity was defined to represent entropy increase rather than free energy decrease. The differential for G takes on a simple form which displays its dependence on compositional change


 * $$(dG)_{T,P} = -\mathbb{A}\, d\xi \,.$$

If there are a number of chemical reactions going on simultaneously, as is usually the case


 * $$(dG)_{T,P} = -\sum_k\mathbb{A}_k\, d\xi_k \,.$$

, a set of reaction coordinates { ξj }, avoiding the notion that the amounts of the components ( Ni } can be changed independently. The expressions above are equal to zero at equilibrium, while in the general case for real systems, they are negative, due to the fact that all chemical reactions proceeding at a finite rate produce entropy. This can be made even more explicit by introducing the reaction rates dξj/dt. For each and every physically independent process (Prigogine & Defay, p. 38; Prigogine, p. 24)


 * $$ \mathbb{A}\ \dot{\xi} \le 0 \,.$$

This is a remarkable result since the chemical potentials are intensive system variables, depending only on the local molecular milieu. They cannot "know" whether the temperature and pressure (or any other system variables) are going to be held constant over time. It is a purely local criterion and must hold regardless of any such constraints. Of course, it could have been obtained by taking partial derivatives of any of the other fundamental state functions, but nonetheless is a general criterion for (&minus;T times) the entropy production from that spontaneous process; or at least any part of it that is not captured as external work. (See Constraints below.)

We now relax the requirement of a homogeneous “bulk” system by letting the chemical potentials and the Affinity apply to any locality in which a chemical reaction (or any other process) is occurring. By accounting for the entropy production due to irreversible processes, the inequality for dG is now replace by an equality


 * $$ dG = - SdT + VdP -\sum_k\mathbb{A}_k\, d\xi_k + W'\,$$

or


 * $$ dG_{T,P} = -\sum_k\mathbb{A}_k\, d\xi_k + W'\,$$



Any decrease in the Gibbs function of a system is the upper limit for any isothermal, isobaric work that can be captured in the surroundings, or it may simply be dissipated, appearing as T times a corresponding increase in the entropy of the system and/or its surrounding. Or it may go partly toward doing external work and partly toward creating entropy. The important point is that the extent of reaction for a chemical reaction may be coupled to the displacement of some external mechanical or electrical quantity in such a way that one can advance only if the other one also does. The coupling may occasionally be rigid, but it is often flexible and variable.

In solution chemistry and biochemistry, the Gibbs free energy decrease (∂G/∂ξ, in molar units, denoted cryptically by ΔG) is commonly used as a surrogate for (&minus;T times) the entropy produced by spontaneous chemical reactions in situations where there is no work being done; or at least no "useful" work; i.e., other than perhaps some ± PdV. The assertion that all spontaneous reactions have a negative ΔG is merely a restatement of the second law of thermodynamics, giving it the physical dimensions of energy and somewhat obscuring its significance in terms of entropy. It tends to lend credence to the mistaken impression that there is a principle of minimum energy, while in fact there is no such law of nature. When there is no useful work being done, it would be less misleading to use the Legendre transforms of the entropy appropriate for constant T, or for constant T and P, the Massieu functions &minus;F/T and &minus;G/T respectively.

Constraints
In this regard, it is crucial to understand the role of walls and other constraints, and the distinction between independent processes and coupling. Contrary to the clear implications of many reference sources, the previous analysis is not restricted to homogenous, isotropic bulk systems which can deliver only PdV work to the outside world, but applies even to the most structured systems. There are complex systems with many chemical "reactions" going on at the same time, some of which are really only parts of the same, overall process. An independent process is one that could proceed even if all others were unaccountably stopped in their tracks. Understanding this is perhaps a “thought experiment” in chemical kinetics, but actual examples exist.

A gas reaction which results in an increase in the number of molecules will lead to an increase in volume at constant external pressure. If it occurs inside a cylinder closed with a piston, the equilibrated reaction can proceed only by doing work against an external force on the piston. The extent variable for the reaction can increase only if the piston moves, and conversely, if the piston is pushed inward, the reaction is driven backwards.

Similarly, a redox reaction might occur in an electrochemical cell with the passage of current in wires connecting the electrodes. The half-cell reactions at the electrodes are constrained if no current is allowed to flow. The current might be dissipated as joule heating, or it might in turn run an electrical device like a motor doing mechanical work. An automobile lead-acid battery can be recharged, driving the chemical reaction backwards. In this case as well, the reaction is not an independent process. Some, perhaps most, of the Gibbs free energy of reaction may be delivered as external work.

The hydrolysis of ATP to ADP and phosphate can drive the force times distance work delivered by living muscles, and synthesis of ATP is in turn driven by a redox chain in mitochondria and chloroplasts, which involves the transport of ions across these cellular organelles. The coupling of processes here, and in the previous examples, is often not complete. Gas can leak slowly past a piston, just as it can slowly leak out of a rubber balloon. Some reaction may occur in a battery even if no external current is flowing. There is usually a coupling coefficient, which may depend on relative rates, which determines what percentage of the driving free energy is turned into external work, or captured as "chemical work", a misnomer for the free energy of another chemical process.