# Algebra of conserved quantities for chemical reactions

So there is the standard formation enthalpy $\Delta H^0_f$ and there is also formation entropy.

Are there more (independent) quantities? What about the heat capacity? Is there a general theoretical structure where one can show one got all the conserved quantities associated with a chemical reaction? It seems to me that one could this on a quite rigid structure and invesigate the properties of chemical processes from that point of view.

And to what extend are the procedures like using Hess's law just rules of thumb or what are some insights about he deeper structure where they come from.

• Oh yes of course! Ever read about Lavoisier? Jul 17 '12 at 10:20
• No really algebra, but electron spin is conserved, this is why singlet oxygen is very reactive.
– Dale
Dec 21 '15 at 20:06

What I think you are asking about is the number of thermodynamic parameters which characterize a chemical reaction. Those parameters are used to make quantitative predictions of the final state of the reactive system (at least in the context I'm familiar with, i.e. chemical engineering); for example, the amount of heat exchanged with the reactor's environment, the composition of the mixture at equilibrium, it's material phase, overall density, etc.

In classical thermodynamics, the way to address this problem is through the potentials defined on the system of interest, and studying equilibrium as the extremum of one these. The chosen potential depends on the particular configuration of the system under study; and in all cases their relation with equilibrium is a consequence of the Second Law. For example, if the system is restricted to evolve at constant volume and entropy, the minimized potential is the internal energy; if it evolves at constant energy and volume (an isolated simple compressible system) it maximizes entropy; etc.

From a practical point of view of natural systems and industrial processes -which evolve at nearly constant temperature and pressure fixed by the environment- the involved minimized potential is the Gibbs free energy. That the minimization of $G$ yields an equilibrium state for a closed system at constant pressure and temperature is straightforward (it can be found in any physical chemistry textbook). Moreover, when we say here that the system at it's minum $G$ is at equilibrium, we are saying it's at overall equilibrium: it's simultaneously at mechanical equilibrium (because of null pressure gradients), thermal equilibrium (because of null temperature gradients) and in chemical equilibrium (because of null chemical potential gradients); so there's no "force" (in a generalized sense) to drive any further process.

So, going back to your question, which are the independent variables of a reactive system as such? Well, suppose that in your system you have $P$ chemical species which can be both reactives or products of $Q$ reactions taking place in a monophase. For every species, the balance on the number of moles of each one yields:

$n_{i}=n_{0i}+\sum_{j=1}^Q{\nu_{i}^j\cdot \xi_{j}}$

Where $\xi_{j}$ is the extent of the j-th reaction.

Then, we can define the chemical potential as the partial molar Gibbs free energy per species:

$\mu_{i}=\displaystyle\frac{\partial G}{\partial n_{i}}$

And also, given that $G$ is an extensive property (or better phrased, a homogeneous function of first degree in mass) by Euler's Theorem it follows that:

$G=\sum_{i=1}^P\mu_{i}n_{i}=\sum_{i=1}^P\mu_{i}\cdot (n_{0i}+\sum_{j=1}^Q{\nu_{i}^j\cdot \xi_{j}})$

Every chemical potential can be expressed as the sum of it's specific Gibbs free energy of it's pure form, at the temperature and pressure of the mixture, plus a term which ponderates the influence of concentration and chemical affinity of each particular species towards the rest of the present compounds which constitute the mixture (for example, the solvent matrix, if one species is so concentrated that it can be considered so). The latter term contains an adimensional parameter known as the activity of the species:

$\mu_{i}=\mu_{0i}(P,T)+RTln(a_{i})$

The zero index in the first term just means that it's referred to a pure system. This specific free energies can be related to the formation enthalpies and entropies which are tabulated through:

$\mu_{0i}=h_{0i}(P,T)-T\cdot s_{0i}(P,T)\\\mu_{0i}=\left\{{h_{fi}^0(P_{0},T_{0})+\displaystyle\int_{(P_{0},T_{0})}^{(P,T)}dh_{0i}}\right\}-T\cdot \left\{{s_{fi}^0(P_{0},T_{0})+\displaystyle\int_{(P_{0},T_{0})}^{(P,T)}ds_{0i}}\right\}$

Where $h_{f}^0(P_{0},T_{0})$ and $s_{f}^0(P_{0},T_{0})$ are the standard formation enthalpy and entropy, respectively. The integrals evaluate the variation in enthalpy and entropy of the pure compounds due to a difference in temperature and pressure in reference to standard values (25 ºC and 1 bar). The evaluation of the integrals in terms of measurable properties must be done using Maxwell relations and definitions of potentials:

$dh_{0i}=\displaystyle\frac{\partial h_{0i}}{\partial {T}}dT+\displaystyle\frac{\partial h_{0i}}{\partial {P}}dP=C_{p}dT+\frac{dP}{\rho}\\ds_{0i}=\displaystyle\frac{\partial s_{0i}}{\partial {T}}dT+\displaystyle\frac{\partial s_{0i}}{\partial {P}}dP=Cp\frac{dT}{T}-\frac{\alpha}{\rho}dP$

What about the activities? These are non-linear functions of the composition, temperature and pressure of the mixture, and usually that non-linearity is expressed through an activity coefficient (it may be the case that further coefficients are introduced theoretically, but all they do is to "partition" the contributions to non-ideality into a product of these, and so here I talk about a unique coefficient without loss of generality). So:

$a_{i}=x_i\cdot\gamma_{i}(P,T,x_1,x_2,...,x_P)$

And:

$x_i=\displaystyle\frac{n_i}{\sum_{k=1}^P n_k}=\displaystyle\frac{n_{0i}+\sum_{j=1}^Q{\nu_{i}^j\cdot \xi_{j}}}{\sum_{k=1}^P\left\{{n_{0k}+\sum_{j=1}^Q{\nu_{k}^j\cdot \xi_{j}}}\right\} }$

Finally, the (enormously complicated) expression for G turns out to be dependent of the variables:

$G=G(P,T,h_{fi}^0,s_{fi}^0,Cp_{i},\rho_{i},\alpha_{i},n_{0i},\xi_j) \ \ \ \ \ \ i: 1 < i < P \ \ \textrm{and}\ \ \ j: 1 < j < Q$

Those are, a priori, the "independent quantities" involved in the overall reaction's equilibrium. Once you've fixed the temperature, pressure, the original amount of substances and get the intensive properties stated for all the chemical species present in the system along it's evolution, G becomes only a function of the extents of the reactions:

$G=G(\xi_j) \ \ \ \ \ \ j: 1 < j < Q$

And the equilibrium is reached when $G$ is at minimum; that's a problem solvable through optimization. The values of the extent of every reaction at equilibrium is enough to calculate every other useful parameter of the system: it's composition, the heat involved, the change in overall volume, etc.

Summary: the conserved quantities in a chemical reaction are physical constraints (conservation of total mass, balance of moles according to stecheometry, conservation of energy exchanged with the environment, strict increase in entropy, etc) but that's not enough information to uniquely evaluate the equilibrium parameters of a general reaction (or reactions). Most of the expressions I left in an undefined closed form in this answer (such as the activity coefficients, all of them functions of compositions) must be experimentally correlated, or modelled with some theoretical formalisms, generally highly complex. The standard thermodynamic potentials might be tabulated though, as it's the case with the intensive properties of pure substances (specific heats, thermal expansion coefficients, etc).