Spontaneity and Equilibrium: Why “ΔG < 0 Denotes a Spontaneous Process” and “ΔG = 0 Means the System Is at Equilibrium” Are Incorrect

Spontaneity and Equilibrium: Why “ΔG < 0 Denotes a Spontaneous Process” and “ΔG = 0 Means the System Is at Equilibrium” Are Incorrect

Lionel M. Raff *

Department of Chemistry, Oklahoma State University, Stillwater, Oklahoma 74078, United States

J. Chem. Educ., 2014, 91 (3), pp 386–395

DOI: 10.1021/ed400453s

Publication Date (Web): January 2, 2014

Copyright © 2014 The American Chemical Society and Division of Chemical Education, Inc.

Abstract: http://pubs.acs.org/doi/abs/10.1021/ed400453s

In this article, the author seeks to illustrate a correct and precise presentation of thermodynamic criteria for spontaneity and the equilibrium state.  He makes the argument that the way it is traditionally presented in current General Chemistry textbooks is incorrect and presents a brief review of the thermodynamic equations used as a basis for spontaneity and equilibrium using calculus math.  In this review section, the author presents the mathematical equations that allow quantitative analysis of reaction spontaneity and equilibrium for one reaction systems.  The shortened derivations of the relevant equations are clearly beyond the scope of freshman General Chemistry course even students with a calculus background as this mathematical manipulations are usually covered in upper division courses.

He then illustrates a simple application for a gas-phase reaction involving 2 reactants A and B and 2 products C and D with the assumption of low pressure and ideal gas conditions. A notable result is his generalization of the ultimate requirement for spontaneity (and its direction) and equilibrium:

the force (F) acting in the direction of increasing process coordinate (x) is given by the negative of the partial derivative of the potential energy (F) with respect to the process coordinate:

A process coordinate is any variable whose value measures the progress of the process (in chemical reactions, the process is the reaction).  If F is negative, the process is spontaneous in the direction of decreasing x.  For chemical reactions:

A is the thermodynamic potential if T and V are constant

G is the thermodynamic potential if P and T are constant

Therefore,

(∂A/∂ξ)T,V < 0 or dA < 0 is the criterion for reaction spontaneity at a specific composition if dT = 0 and dV = 0, and

(∂G/∂ξ)T,p < 0 or dG < 0 is the criterion for reaction spontaneity if dT = 0 and dp = 0.

The correct criteria for equilibrium are that these quantities are equal to zero.

The main point of all of this is that the “criteria for reaction spontaneity and equilibrium, regardless of how stated, always involve differential changes dA or dG. They do not involve changes in the thermodynamic potentials over a finite range.  That is, they do not involve ΔA or ΔG. Stated in different terms, the determination of reaction spontaneity and equilibrium at a given temperature requires the evaluation of dA or dG or their derivatives with respect to a reaction coordinate at a specific point in the (V, p, composition) space of the system.”

It is important to remember that if there are N coupled reactions, then there are N reaction coordinates.  In this case, G and A consist of N + 1 hypersurfaces and dG and dA will have contributions from N partial derivatives.  The criteria for spontaneity becomes much more complex.

In the next sections, the author presents two examples showing why using DG  and DA in the criteria for spontaneity and equilibrium is a “fallacy”.

The graph below shows the region where the reaction is spontaneous (dA < 0, A as a function of the reaction coordinate n (product) has a negative slope or dA/dn < 0).  Point C denotes equilibrium.  Beyond point C, n (product b) > 0.82507 mol, dA > 0, non-spontaneous.  DA from 0 à 1 is not negative in the entire range and therefore DArxn = Aproduct – Areactant < 0 is not sufficient to declare the reaction spontaneous because it is not <0 from point C below. As well, DArxn = Aproduct – Areactant > 0 is not sufficient condition to infer that the reaction is non-spontaneous at some arbitrary finite interval.  In the case below, because we know that the reaction must be non-spontaneous toward the end of the interval, DA > 0 is sufficient to indicate non-spontaneity of a FINITE transformation from initial to final but not a necessary condition.  Moreover, if one calculates DA from point D to E below, DA = 0 leading the student to conclude that the reaction is at equilibrium when it is clearly not.  As the author notes, “First, the system is far from its equilibrium point, C. More importantly, the concept that any system can be in equilibrium over any finite range is incorrect. When the temperature is constant, equilibrium exists at a single point, not over a finite range.”

The gist of this is that spontaneity and equilibrium are conditions that exist at single points along the coordinate system and cannot be evaluated over some finite range. dA or dG are differentials, slope values that change at every point (not an average over some finite range).

The same argument given above applies to dG versus DG when P and T are constant.

The only thing that may be safely inferred if only DG > 0 is calculated is that the reaction is nonspontaneous in an undetermined region near the end of the interval. “Again, because the reaction is nonspontaneous at the end of the interval, the transformation process between initial and final states of the interval must be nonspontaneous.”

The author provides a good summary of the main points of this paper to the end.  He also offers some recommendation to address the concern that most General Chemistry students have not been exposed to calculus and therefore the calculations shown here cannot be done at that level.  He does recommend using the plot below for G and discussing it at a more qualitative level.  He also suggest using the gravitational potential energy as a simpler model or analogy to discuss the idea of a potential.