Animated Solution for Chemistry - Chemical Thermodynamics: For a given chemical reaction, $A⟶B$ at $300\text{ K}$ the free energy change is $-49.4{\text{ kJ mol}}^{-1}$ and the enthalpy of reaction is $51.4{\text{ kJ mol}}^{-1}$. The entropy change of the reaction is ...... ${\text{JK}}^{-1}{\text{mol}}^{-1}$.

A Journey Through the Gibbs Free Energy Equation

Imagine you are an accountant, but instead of managing money, you are managing the energy of a chemical reaction. Every reaction has an energy budget, and to understand whether a reaction will happen spontaneously, we must look at three key players: Enthalpy ($\Delta H$), Entropy ($\Delta S$), and Gibbs Free Energy ($\Delta G$).

In this problem, we are given a reaction where reactant $A$ transforms into product $B$ at a constant temperature of $300\text{ K}$. We are handed two crucial pieces of information: the change in Gibbs free energy is $\Delta G=-49.4{\text{ kJ mol}}^{-1}$, and the change in enthalpy is $\Delta H=51.4{\text{ kJ mol}}^{-1}$. Our mission is to uncover the hidden third player: the entropy change ($\Delta S$).

Gibbs Free Energy

To solve this mystery, we call upon the master equation of chemical thermodynamics:

This elegant equation tells us that the free energy available to do work ($\Delta G$) is equal to the total heat energy exchanged ($\Delta H$) minus the energy lost to the universe's inherent chaos ($T\Delta S$). Since our goal is to find $\Delta S$, we need to rearrange this equation. By moving terms across the equals sign, we isolate $\Delta S$:

Units and Signs

Now comes the critical phase: substitution. This is where many students fall into a classic trap. We must be incredibly careful with our signs. We substitute $\Delta H=51.4$ and $\Delta G=-49.4$ into our rearranged equation:

Watch out for the double negative! Subtracting a negative value is mathematically identical to adding a positive value. Therefore, the numerator becomes $51.4+49.4$, which equals $100.8{\text{ kJ mol}}^{-1}$.

But wait, there is another trap! The question explicitly asks for the answer in Joules ($\text{J}$), not kilojoules ($\text{kJ}$). To convert kilojoules to Joules, we must multiply our result by $1000$:

Calculating Entropy

With our units aligned and our signs correct, we execute the final calculation. Multiplying $100.8$ by $1000$ gives us $100800$. Dividing this by the temperature ($300\text{ K}$) yields our final answer:

What does this number tell us physically? Because $\Delta S$ is a positive value, it means the randomness or disorder of the system has increased as $A$ converted to $B$. The universe loves chaos, and this reaction is happily contributing to it!