Introduction

Gibbs free energy calculation is central to predicting reaction spontaneity in thermodynamics and physical chemistry, especially in coordination and solution equilibria.

The key relationship used is the ΔG = ΔH − TΔS equation, which links enthalpy, entropy and temperature to the feasibility of a chemical reaction.

For the reaction Cd²⁺ + 4CH₃NH₂ → Cd(CH₃NH₂)₄²⁺ at 25 °C, ΔH = −13.7 kcal mol⁻¹ and ΔS = −16.0 cal K⁻¹ mol⁻¹, and the goal is to compute ΔG in kcal mol⁻¹ correct to two decimal places.

Step 1: Gibbs Free Energy Equation

For any process at constant temperature and pressure, the Gibbs free energy change is given by the relation ΔG = ΔH − TΔS, where ΔH is enthalpy change, ΔS is entropy change and T is absolute temperature in Kelvin.

This equation predicts reaction spontaneity, because a negative ΔG generally indicates that the process is thermodynamically favorable under the specified conditions.

Step 2: Temperature and Unit Matching

First convert 25 °C to Kelvin using T(K) = 25 + 273, giving T = 298 K as the standard approximation used in most examination problems.

Next match the energy units, because ΔH is expressed in kcal mol⁻¹ while ΔS is given in cal K⁻¹ mol⁻¹, so either ΔH must be converted to cal or the TΔS term must be converted to kcal to combine them correctly.

The convenient choice is to convert the entropy contribution from cal to kcal, using the relation 1 kcal = 1000 cal to maintain consistency in the final ΔG value.

Step 3: Calculating TΔS

Compute TΔS in cal mol⁻¹: TΔS = 298 K × (−16.0 cal K⁻¹ mol⁻¹) = −4768 cal mol⁻¹ for the given reaction conditions.

Convert this result to kcal mol⁻¹ by dividing by 1000, giving TΔS = −4.768 kcal mol⁻¹, which is now compatible with the enthalpy value expressed in kcal mol⁻¹.

At this stage the thermodynamic data in common units are ΔH = −13.7 kcal mol⁻¹ and TΔS = −4.768 kcal mol⁻¹, both negative, indicating that enthalpy and entropy terms oppose each other in the free energy balance.

Step 4: Computing ΔG and Rounding

Substitute the values into the Gibbs equation to obtain ΔG = −13.7 kcal mol⁻¹ − (−4.768 kcal mol⁻¹) = −13.7 kcal mol⁻¹ + 4.768 kcal mol⁻¹ = −8.932 kcal mol⁻¹.

Rounding this numerical value to two decimal places yields ΔG ≈ −8.93 kcal mol⁻¹, which is strongly negative and therefore indicates that the complex‑formation reaction is thermodynamically spontaneous at 25 °C.

Some treatments that use T = 298.15 K instead of 298 K obtain a slightly different numerical value (around −8.47 kcal mol⁻¹), but the sign and spontaneity conclusion remain the same in either case.

Conceptual Recap

The Gibbs relation ΔG = ΔH − TΔS integrates enthalpy and entropy contributions, ensuring that both heat change and disorder change are included when assessing chemical feasibility.

Proper use of Kelvin temperature and consistent energy units is essential for accurate thermodynamic calculations, preventing magnitude errors and misinterpretation of reaction spontaneity.For the Cd²⁺–methylamine complex, the negative ΔH favors formation while the negative ΔS disfavors it at higher temperature, yet at 25 °C the combined effect still yields a negative ΔG and a spontaneous reaction.