Which quantity out of ΔrG and ΔrG° will be zero at equilibrium?

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Q. Which quantity out of ΔrG and ΔrG° will be zero at equilibrium?

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Concept understanding — Gibbs Free Energy from K

Gibbs Free Energy from $K$ — The Bridge Between Thermodynamics and Equilibrium

Imagine you're pushing a heavy box across a rough floor. You push hard, but the box barely moves. The potential to move is there — you're applying force — but the actual motion is tiny. That's the difference between thermodynamic spontaneity (the push) and equilibrium (where the box sits, barely budging).

Gibbs Free Energy ( $\Delta G$ ) tells you the push — whether a reaction can happen. The equilibrium constant $K$ tells you how far it actually goes before stopping. The equation that links them is one of the most powerful in chemistry:

\Delta G^\circ = -RT \ln K

Let's unpack this from the ground up.

Step 1: What is $\Delta G$ ?

Gibbs Free Energy change ( $\Delta G$ ) measures the maximum useful work a reaction can do at constant temperature and pressure. More practically:

If $\Delta G < 0$ : the reaction is spontaneous (it can happen on its own).
If $\Delta G > 0$ : the reaction is non-spontaneous (it needs energy input).
If $\Delta G = 0$ : the system is at equilibrium — no net change.

But here's the catch: $\Delta G$ depends on how much reactant and product you have at any moment. It's not a fixed number.

Step 2: Standard vs. Non-standard Conditions

Chemists define a standard state (pure substances at 1 bar, 1 M concentration for solutions, 25°C usually). Under those conditions, the free energy change is called $\Delta G^\circ$ (standard Gibbs free energy change).

But real reactions rarely start at standard conditions. So we have:

\Delta G = \Delta G^\circ + RT \ln Q

where $Q$ is the reaction quotient (ratio of products to reactants at that instant, raised to their stoichiometric coefficients).

Note

$R$ is the gas constant (8.314 J/mol·K), $T$ is temperature in Kelvin. The $\ln$ is natural log.

Step 3: At Equilibrium — The Key Insight

At equilibrium, the reaction has no net tendency to go forward or backward. That means:

\Delta G = 0

And the reaction quotient $Q$ becomes exactly the equilibrium constant $K$ .

So plug into the equation:

0 = \Delta G^\circ + RT \ln K

Rearrange:

\Delta G^\circ = -RT \ln K

That's it. This single equation connects a thermodynamic property ( $\Delta G^\circ$ ) with a concentration-based constant ( $K$ ).

Step 4: What This Tells You

$\Delta G^\circ$ value	$K$ value	Meaning
Negative ( $<0$ )	$K > 1$	Products favoured at equilibrium
Zero ( $=0$ )	$K = 1$	Equal amounts at equilibrium
Positive ( $>0$ )	$K < 1$	Reactants favoured at equilibrium

Watch out

A negative $\Delta G^\circ$ does not mean the reaction is fast — only that it's thermodynamically favourable. Kinetics (activation energy) is a separate story.

Step 5: A Concrete Example

Consider the reaction: $\text{N}_2(g) + 3\text{H}_2(g) \rightleftharpoons 2\text{NH}_3(g)$

At 25°C, $\Delta G^\circ = -33.3 \text{ kJ/mol}$ . Using $R = 8.314 \text{ J/mol·K}$ :

$-33,300 = -(8.314)(298) \ln K$ …

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At equilibrium, the actual change in Gibbs free energy for the reaction, $\Delta_r G$ , becomes zero, indicating no net driving force for the reaction. The standard Gibbs free energy change, $\Delta_r G^\circ$ , is generally not zero at equilibrium, as it refers to a hypothetical state where all species are in their standard conditions.

In chemical thermodynamics, Gibbs free energy ( $G$ ) is a crucial state function that helps predict the spontaneity of a process and determine the position of equilibrium under constant temperature and pressure. The change in Gibbs free energy for a reaction, $\Delta_r G$ , represents the maximum non-expansion work that can be extracted from a reaction, or the minimum work required to drive a non-spontaneous reaction.

The key to understanding which quantity is zero at equilibrium lies in distinguishing between $\Delta_r G$ and $\Delta_r G^\circ$ .

$\Delta_r G$ (Gibbs Free Energy Change): This is the actual change in Gibbs free energy for a reaction under any given set of conditions (i.e., specific concentrations or partial pressures of reactants and products). It tells us the spontaneity and direction of a reaction under those specific, current conditions.
$\Delta_r G^\circ$ (Standard Gibbs Free Energy Change): This is the change in Gibbs free energy for a reaction when all reactants and products are in their standard states. Standard states are defined as:
- For gases: 1 bar (or 1 atm, depending on convention) partial pressure.
- For solutions: 1 M concentration.
- For pure solids and liquids: The pure substance in its most stable form at 1 bar and the specified temperature. $\Delta_r G^\circ$ is a constant value for a given reaction at a specific temperature, as it refers to a fixed, hypothetical set of conditions.

Let's break down their roles at equilibrium.

Understanding Equilibrium:

Equilibrium is a dynamic state where the rates of the forward and reverse reactions are equal, resulting in no net change in the concentrations of reactants and products over time. At equilibrium, the system has no further tendency to change spontaneously in either direction.
The Role of $\Delta_r G$ :

The sign of $\Delta_r G$ dictates the spontaneity and direction of a reaction:
- If $\Delta_r G < 0$ , the reaction is spontaneous in the forward direction.
- If $\Delta_r G > 0$ , the reaction is spontaneous in the reverse direction.
- If $\Delta_r G = 0$ , the system is at equilibrium, and there is no net tendency for the reaction to proceed in either direction. Therefore, by definition, at equilibrium, the actual Gibbs free energy change for the reaction, $\Delta_r G$ , must be zero. It represents the point where the driving force for the reaction has been completely expended.
The Role of $\Delta_r G^\circ$ :

$\Delta_r G^\circ$ is a fixed value for a given reaction at a specific temperature. It tells us about the spontaneity of a reaction if all species were in their standard states. It does not change as the reaction proceeds towards equilibrium. The system reaches equilibrium by adjusting the concentrations (or partial pressures) of reactants and products, which in turn changes $\Delta_r G$ , but not $\Delta_r G^\circ$ .

Watch out

A common misconception is to assume that $\Delta_r G^\circ$ is zero at equilibrium. Remember, $\Delta_r G^\circ$ is a constant reference value, while $\Delta_r G$ is the actual, changing value that reaches zero at equilibrium.
Relationship between $\Delta_r G$ and $\Delta_r G^\circ$ :

The relationship between $\Delta_r G$ and $\Delta_r G^\circ$ is given by the equation:
$\Delta_r G = \Delta_r G^\circ + RT \ln Q$

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Q.Which quantity out of ΔrG and ΔrG° will be zero at equilibrium?

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