Free-energy Calculations Are Dependent on the Rates of the Reactions

In chemistry and biochemistry, there is often a distinct separation between thermodynamics (will a reaction happen?) and kinetics (how fast will it happen?). Standard Gibbs free energy change ($\Delta G^\circ$) tells us about reaction spontaneity and equilibrium position, but it says nothing about the time it takes to reach that equilibrium.

However, free-energy calculations are strongly dependent on the rates of reactions when we consider the activation energy barrier. According to Transition State Theory, the rate constant of a reaction is directly related to the Gibbs free energy of activation ($\Delta G^\ddagger$). This is the energy difference between the reactants and the highest-energy transition state.

By knowing the reaction rate constant ($k$) at a specific temperature, we can calculate this crucial free-energy barrier using the Eyring equation. This calculator performs that specific conversion, linking kinetic data to thermodynamic activation parameters.

Activation Free Energy Calculator (from Rate Constant)

This tool uses the rearranged Eyring equation to calculate the Gibbs Free Energy of Activation ($\Delta G^\ddagger$) based on observed reaction rate kinetics.

Enter value in s⁻¹ (assuming first order or pseudo-first order).

Enter absolute temperature in Kelvin (K).

Results will appear here…

function calculateActivationEnergy() { // 1. Get Inputs var k_rate = parseFloat(document.getElementById('rateConstantK').value); var T_kelvin = parseFloat(document.getElementById('temperatureT').value); var resultDisplay = document.getElementById('activationResult'); // 2. Define Physical Constants var R = 8.314462618; // Universal gas constant in J/(mol·K) var h = 6.62607015e-34; // Planck constant in J·s var kb = 1.380649e-23; // Boltzmann constant in J/K // 3. Validate Inputs if (isNaN(k_rate) || k_rate <= 0) { resultDisplay.innerHTML = "Please enter a valid, positive reaction rate constant ($k$)."; return; } if (isNaN(T_kelvin) || T_kelvin <= 0) { resultDisplay.innerHTML = "Please enter a valid, positive temperature in Kelvin."; return; } // 4. Calculation Logic (Rearranged Eyring Equation) // The equation is: k = (kb*T / h) * exp(-ΔG‡ / RT) // Rearranging for ΔG‡: ΔG‡ = -RT * ln( (k*h) / (kb*T) ) // Calculate the term inside the natural log: (k * h) / (kb * T) var numerator = k_rate * h; var denominator = kb * T_kelvin; var lnTermRatio = numerator / denominator; // Safety check for log calculation issues if (lnTermRatio <= 0) { resultDisplay.innerHTML = "Calculation error: Resulting ratio for logarithm is non-positive due to extreme inputs."; return; } var naturalLogVal = Math.log(lnTermRatio); // Calculate ΔG‡ in Joules/mol // ΔG‡ = -R * T * ln(…) var deltaG_Joules = -1 * R * T_kelvin * naturalLogVal; // Convert to kilojoules/mol for standard reporting var deltaG_kJ = deltaG_Joules / 1000; // 5. Display Results resultDisplay.innerHTML = "Gibbs Free Energy of Activation ($\Delta G^\ddagger$):" + "" + deltaG_kJ.toFixed(2) + " kJ/mol" + "(" + deltaG_Joules.toFixed(0) + " J/mol)"; }

The Theoretical Basis: Transition State Theory

The connection used in the calculator above is derived from Transition State Theory (TST). TST assumes that there is a quasi-equilibrium between reactants and an activated transition state complex.

The fundamental Eyring equation relates the reaction rate constant $k$ to temperature $T$ and the activation Gibbs energy $\Delta G^\ddagger$:

$k = \frac{k_B T}{h} e^{-\frac{\Delta G^\ddagger}{RT}}$

Where:

• $k_B$: Boltzmann constant
• $h$: Planck constant
• $R$: Universal gas constant

Because the rate constant $k$ is exponentially dependent on the free energy of activation, small changes in this energy barrier lead to significant changes in the observed reaction rate. Conversely, measuring the rate at a known temperature allows us to back-calculate the height of this energy barrier, as performed by the tool above.

Why This Matters

Calculating the $\Delta G^\ddagger$ is crucial for understanding reaction mechanisms. A high free energy of activation indicates a slow reaction that requires significant energy to overcome the barrier, whereas a low $\Delta G^\ddagger$ corresponds to a fast reaction. Catalysts work specifically by lowering this $\Delta G^\ddagger$ barrier, thereby increasing the rate constant $k$ without changing the overall thermodynamic equilibrium of the reactants and products.