Gibbs Free Energy Simulator

1. Real‑World Phenomenon

Gibbs free energy unites the laws of thermodynamics with the spontaneity of chemical and biochemical reactions. When iron rusts, ATP fuels muscle contraction, or ammonia forms from nitrogen and hydrogen, the direction and extent of those transformations depend on the balance of enthalpy and entropy. The calculator above retains the familiar equation $\Delta G=\Delta H-T\Delta S$ but elevates it with an animation. A particle rolls along a reaction coordinate whose slope equals the computed free‑energy change. Energy bars track the absolute magnitudes of enthalpy and the $\mathrm{T\Delta S}$ term so that users can see which contribution dominates. As you edit parameters, the visualization updates immediately, turning abstract state functions into moving energy landscapes.

2. Variables and Assumptions

The model assumes a single-step reaction of unit stoichiometric extent. Enthalpy change $\mathrm{\Delta H}$ enters in kilojoules per mole, entropy change $\mathrm{\Delta S}$ in joules per mole per kelvin, and absolute temperature $T$ in kelvin. The simulation converts $\mathrm{\Delta S}$ to kilojoule units for consistency. Progress of reaction is represented by a dimensionless extent $\xi$ ranging from 0 (pure reactants) to 1 (pure products). At each time step, the algorithm assumes the free energy varies linearly with extent, $\mathrm{G(\xi )}=\xi \mathrm{\Delta G}$. No intermediate metastable states or activation barriers appear; the purpose is to visualize the overall driving force rather than detailed kinetics. All inputs must be finite numbers; negative temperatures or undefined entropies trigger validation errors. Units remain strictly SI, and entering kilocalories or calories would yield incorrect results, so tooltips remind users of the expected dimensions.

3. Governing Equations

The heart of the calculator is the Gibbs relation. For a reaction at constant pressure and temperature, $\mathrm{\Delta G}=\mathrm{\Delta H}-T\mathrm{\Delta S}$. A negative $\mathrm{\Delta G}$ indicates the forward reaction is spontaneous, while a positive value favors the reverse. To animate progress, we prescribe a simple first‑order kinetic model for the extent:

$\frac{\mathrm{d\xi }}{\mathrm{dt}}=k\left(\mathrm{sign}(-\mathrm{\Delta G})\right)\left(1-\xi \right)$.

The rate constant $k$ is set to unity for simplicity. If $\mathrm{\Delta G}$ is negative, the extent grows toward one; if positive, it decays toward zero. This stripped‑down kinetics mirrors how exergonic reactions proceed without external input while endergonic ones regress unless driven. Energies along the path follow $H=\mathrm{\xi \Delta H}$ and $\mathrm{T\Delta S}=\mathrm{\xi T\Delta S}$. Their difference gives $G$, which the simulation uses to draw the energy line and scale the bars.

4. Numerical Scheme

The differential equation for $\xi$ is integrated with a fourth‑order Runge–Kutta (RK4) method. Each step evaluates four slope estimates, combining them as $\mathrm{\xi \_\{n+1\}}=\mathrm{\xi \_n}+\frac{\mathrm{\Delta t}}{6}(\mathrm{k\_1}+2\mathrm{k\_2}+2\mathrm{k\_3}+\mathrm{k\_4})$. RK4’s error scales with ${\mathrm{\Delta t}}^4$, allowing stable animation for time steps between 0.001 and 0.1 s as enforced by the input clamp. Because the underlying kinetics are linear, an analytic solution exists, but using RK4 aligns this simulator with the numerical approaches employed in more complex systems. The code tracks the initial free energy to monitor numerical drift; deviations appear in the caption.

5. Worked Example

Consider the decomposition of calcium carbonate: $\mathrm{CaCO₃}\to \mathrm{CaO}+\mathrm{CO₂}$. Standard enthalpy change is about $178$ kJ/mol and entropy change is $160$ J/mol·K. At $1200$ K, $\mathrm{\Delta G}=178-1200\times 0.160=-14.8$ kJ/mol. Entering these values and pressing Play sets the particle rolling toward products. The bars show a large positive enthalpy opposed by an even larger $\mathrm{T\Delta S}$ term, yielding a small negative free energy. Exporting the CSV reveals time, extent, and energies; plotting them externally confirms the exponential approach to equilibrium predicted by the analytic solution.

6. Comparison Table

The table summarizes three scenarios to highlight how temperature and entropy shift spontaneity.

ΔH (kJ/mol)	ΔS (J/mol·K)	T (K)	ΔG (kJ/mol)
-40	-100	298	-10.2
25	120	350	-17.0
60	-50	400	80.0

The first row corresponds to the default form entries: an exothermic reaction with decreasing entropy remains spontaneous at room temperature. The second row shows how a modestly endothermic process can turn favorable when accompanied by a large entropy gain at elevated temperature. The third illustrates that positive enthalpy and negative entropy reinforce to produce a strongly non‑spontaneous reaction.

7. How to Read the Animation

The canvas portrays the reaction coordinate horizontally, with zero on the left and products on the right. The orange line traces Gibbs free energy from reactants to products; its slope equals $\mathrm{\Delta G}$. A blue dot representing the system’s state glides along this line according to the RK4 integration. The energy bars beneath the canvas use blue stripes for enthalpy magnitude and orange stripes for the $\mathrm{T\Delta S}$ term, relying on texture rather than color alone. The caption narrates current time, extent, and free‑energy contribution, and the hidden text summary mirrors these values for screen readers. Keyboard users can focus the canvas and press the space bar to toggle play and pause, ensuring accessibility.

8. Limitations

The simulator treats $\mathrm{\Delta H}$ and $\mathrm{\Delta S}$ as constant, ignoring their temperature dependence. Real reactions may exhibit nonlinear free‑energy surfaces or multiple intermediates. Activation energies are absent, so endergonic reactions simply run backward rather than stalling behind kinetic barriers. The kinetic model uses an arbitrary rate constant and does not conserve energy or mass explicitly; it merely illustrates directionality. Numerical error grows with large time steps or extremely steep energy slopes, though the RK4 scheme keeps drift small for reasonable parameters. The particle cannot overshoot the interval [0,1]; the code clamps extent to preserve physical meaning.

9. Suggested Extensions

Future versions could incorporate temperature‑dependent heat capacities, enabling adiabatic reaction paths where temperature evolves alongside extent. Adding an activation‑energy barrier would allow exploration of catalysis by letting users reduce the barrier height and watch the particle surmount it more readily. Coupling two reactions could depict how exergonic processes drive endergonic ones, a cornerstone of metabolic networks. A phase‑space plot of extent versus free energy could show relaxation toward equilibrium as a spiral when multiple reactions interact.

10. References and Related Tools

• P. Atkins and J. de Paula, Physical Chemistry, 11th ed., Oxford University Press, 2017.
• D. Alberty, Thermodynamics of Biochemical Reactions, Wiley, 2003.
• Explore energy quanta with the Photon Energy Calculator, model particle motion in gases via the Mean Free Path Calculator, or study electrical energy storage using the RLC Circuit Damping Simulator.