Animated Solution for Chemistry - Chemical Thermodynamics: For a spontaneous reaction, the $\Delta G$, equilibrium constant ($K$) and $E_{\text{cell}}^{\circ }$ will be respectively

The concept of spontaneity is one of the most profound ideas in all of physical chemistry. It tells us which way the universe is ticking. When we mix chemicals, will they react on their own, or will they just sit there staring at each other?

To answer this, we don't just guess; we rely on the rigorous laws of thermodynamics and electrochemistry. Let's embark on a journey to decode the three master keys of spontaneity: the Gibbs free energy ($\Delta G$), the equilibrium constant ($K$), and the standard cell potential ($E_{\text{cell}}^{\circ }$).

Gibbs Free Energy ($\Delta G$)

Imagine a boulder rolling down a hill. It moves spontaneously because it is going to a state of lower potential energy. Chemical reactions behave similarly, but their "potential energy" is measured by a quantity called Gibbs Free Energy, denoted by $G$.

For any process to occur spontaneously at constant temperature and pressure, the system must release free energy. It must move to a more stable, lower-energy state.

Mathematically, this means the change in Gibbs free energy must be less than zero. Therefore, our first absolute condition for a spontaneous reaction is that $\Delta G$ must be negative ($\Delta G<0$).

Equilibrium Constant ($K$)

Now, let's connect this thermodynamic reality to chemical equilibrium. Every reaction is a two-way street: reactants turn into products, and products turn back into reactants.

The equilibrium constant ($K$) tells us who wins this tug-of-war. If $K$ is large, products dominate. If $K$ is small, reactants dominate. But how does $K$ relate to spontaneity?

Thermodynamics provides a beautiful bridge between the standard free energy change ($\Delta G^{\circ }$) and the equilibrium constant:

Let's dissect this equation. The universal gas constant ($R$) and the absolute temperature ($T$) are always positive values. We already established that for a spontaneous forward reaction, the free energy change must be negative.

For the entire right side of the equation to be negative, the term $\ln K$ must be positive. The natural logarithm of a number is only positive if that number is strictly greater than one.

Therefore, our second condition is that $K$ must be greater than $1$ ($K>1$). This makes perfect physical sense: a spontaneous forward reaction should naturally produce more products than reactants at equilibrium!

Standard Cell Potential ($E_{\text{cell}}^{\circ }$)

Finally, let's step into the realm of electrochemistry. Many spontaneous redox reactions can be harnessed to generate electricity, like the battery powering your phone right now.

The driving force behind this electrical work is the standard cell potential, denoted as $E_{\text{cell}}^{\circ }$. How does this electrical potential connect back to our master key, $\Delta G$?

The relationship is given by another fundamental equation:

In this formula, $n$ represents the number of moles of electrons transferred in the balanced redox reaction, and $F$ is Faraday's constant (the charge of one mole of electrons). Both $n$ and $F$ are strictly positive numbers.

Once again, we know that for a spontaneous process, $\Delta G^{\circ }$ must be negative. Because of the negative sign already present in the formula, the only way for the overall result to be negative is if $E_{\text{cell}}^{\circ }$ is positive.

Therefore, our third and final condition is that $E_{\text{cell}}^{\circ }$ must be positive ($E_{\text{cell}}^{\circ }>0$). A positive cell potential means the cell can do work on the surroundings, which is the hallmark of a spontaneous galvanic cell.

We have successfully decoded the thermodynamic and electrochemical signatures of a spontaneous reaction. Let's summarize our findings:

1. The system must release free energy, so $\Delta G$ is negative. 2. The forward reaction must be favored at equilibrium, so $K>1$. 3. The reaction must be capable of generating a positive voltage, so $E_{\text{cell}}^{\circ }$ is positive.

Looking at our given options, we can confidently select the one that matches this exact profile. The correct sequence is $-\text{ve}$, $>1$, and $+\text{ve}$.

This elegant interplay between energy, equilibrium, and electricity is what makes chemistry so incredibly powerful!