Reacting systems of ideal gases: Gibbs free energy, equilibrium, and Nernst equation | MIT OpenCourseWare

MIT OpenCourseWare presents a focused case study on reacting systems of ideal gases, deriving Gibbs free energy as a function of composition for the A + B -> 2C reaction, and applying it to a hydrogen plus chlorine example that forms hydrogen chloride. The lecture develops the univariate description of a reacting system, expresses Gibbs free energy in terms of NFC, and interprets Gibbs energy plots to reveal equilibrium composition, curvature due to the ideal entropy of mixing, and the driving force for reaction direction at a given temperature. It then discusses the temperature dependence of equilibrium and uses a simple Van’t Hoff analysis to estimate the reaction enthalpy from two temperature data points. The session closes with a quick primer on the Nernst equation in electrochemistry, connecting thermodynamics to electrochemical potential in a Daniel cell.

Introduction and objective

The lecture tackles reacting systems of ideal gases with a focus on Gibbs free energy as a function of composition. The example chosen is the gas-phase reaction A + B -> 2C, with a concrete hydrogen plus chlorine case H2 + Cl2 -> 2HCl used to illustrate the concepts. The goal is to learn what can be inferred from Gibbs free energy plots as the system composition changes, especially under fixed temperature and pressure, and with a fixed total mole number.

"Gibbs free energy is the sum of partial molar Gibbs free energy weighted by the moles" - MIT OpenCourseWare

The thermodynamic framework

Gibbs free energy for a reacting system is constructed as a sum of contributions from the pure components plus an ideal entropy of mixing term. The partial molar Gibbs energies are weighted by the component mole numbers, and the total pressure is referenced to the pure components. In this univariate case, the evolution of the system can be described by a single reaction coordinate NFC. For A + B -> 2C, the changes in A and B relate to the change in C by DNA = DNB = -1/2 DNC, ensuring the total number of moles can be fixed for this particular reaction, which simplifies the algebra and plotting of G as a function of NFC.

Hydrogen and chlorine example and interpretation of G versus NFC

The instructor sets up the Gibbs free energy expression in terms of initial moles of A and B, the current NFC, and reference chemical potentials. The expression includes the standard chemical potentials of A, B, and C and the ideal entropy of mixing RT sum over the components. With the H2 + Cl2 -> 2HCl example, a plot of G against NFC is generated for 298 K using standard thermodynamic data, albeit with a small adjustment to the formation enthalpy of HCl to aid readability. The minimum of G occurs at an equilibrium C fraction near 1.5 moles, indicating the reaction tends to convert reactants to products up to that point, then curve back due to mixing entropy. The curvature near the minimum reflects the logarithmic term x log x that prevents complete conversion at equilibrium.

"The reaction will proceed to the right until the Gibbs free energy is minimized" - MIT OpenCourseWare

Temperature effects and curve reshaping

To study how temperature affects the equilibrium, the same calculation is repeated at 328 K. The professor explains that the change in G with temperature is governed by the term -S, so increasing temperature generally lowers G and shifts the equilibrium behavior. The two curves are scaled to a common reference to compare their shapes without being overwhelmed by the absolute energies. Zooming in around the minimum reveals a subtle but observable shift of the equilibrium point with temperature, consistent with the thermodynamic intuition that higher temperature can move the equilibrium toward the reactants for an exothermic reaction.

"Gibbs tends to go down as the temperature rises, and entropy is strictly positive" - MIT OpenCourseWare

Van’t Hoff analysis: estimating ΔH° from a temperature shift

Two temperatures and two equilibrium points (298 K and 328 K) are used to estimate the temperature dependence of the equilibrium constant via the van’t Hoff relation. The equilibrium moles of C at the two temperatures are read off the plots (approximately 1.55 and 1.53), from which A and B extents follow by stoichiometry. Partial pressures are inferred using Dalton’s law, P_i = n_i / N_total, and the equilibrium constants K_p at the two temperatures are computed. The slope d(ln K_p)/dT is then approximated by the finite difference, yielding roughly -0.0038 K^-1. Using the Van’t Hoff relationship ΔH° ≈ R T^2 d(ln K_p)/dT with a mid-temperature around 313 K gives an enthalpy ΔH° of about -3 kJ/mol, consistent with the qualitative exothermic/exothermic behavior inferred from Le Chatelier’s principle. The instructor notes that a slightly different ΔH° value was used to generate the plotted curves, but the order of magnitude and sign align with expectations for an exothermic gas-phase reaction forming HCl.

"Estimating the heat capacity difference from a temperature series of G of M of C is an interesting idea" - MIT OpenCourseWare

Nernst in a nutshell: linking thermodynamics to electrochemistry

With the thermodynamics basics established, the lecturer transitions to Nernst and electrochemical cells. A Daniel cell example is introduced, featuring zinc and copper electrodes in sulfate solutions connected through a salt bridge. The discussion emphasizes the electrostatic work of moving charge across a potential difference, and how the generalized work theorem ties reversible work to changes in Gibbs free energy. The Nernst equation is derived by combining electrostatic work (F and E) with thermochemistry, yielding a relation of the form ΔG = ΔG° + RT ln Q and E = -ΔG/(nF) = -ΔG°/(nF) - RT ln Q/(nF). A specific zinc-copper example is mentioned, with activity expressions for the redox couple and a reaction quotient Q that depends on concentrations. The main takeaway is that the cell potential E is a measurable expression of the thermodynamics of the redox reaction, varying with concentrations and temperature, and that this perspective elegantly merges chemical thermodynamics with electrostatics.

"Nernst equation couples electrostatics and chemistry to produce a cell voltage that you can measure" - MIT OpenCourseWare

Closing connections and broader context

The lesson closes by noting how these ideas connect to broader topics in physical chemistry and to OT3 in the course, illustrating the power of combining thermodynamic analyses with concrete, calculable examples. The instructor hints at further exploration of temperature dependence and heat capacities to extract more subtle thermodynamic information, and invites further questions on how to handle more general mole-number changes in real systems.

"This is a nice reaction form to analyze, to just understand the math and the fundamentals of this" - MIT OpenCourseWare