Match Column (I) with Column (II). Column I (i) Equilibrium (ii) Spontaneous reaction (iii) Non spontaneous re

NCERT Exemplar · Q43

Q. Match Column (I) with Column (II). Column I

(i) Equilibrium

(ii) Spontaneous reaction

(iii) Non spontaneous reaction
Column II

(a) ΔG > 0, K < 1

(b) ΔG = 0

(d) ΔG < 0, K > 1

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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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The spontaneity of a reaction is determined by the change in Gibbs free energy ( $\Delta G$ ). At equilibrium, $\Delta G = 0$ . A spontaneous reaction has $\Delta G < 0$ and an equilibrium constant $K > 1$ . A non-spontaneous reaction has $\Delta G > 0$ and $K < 1$ .

The correct matches are: (i) $\to$ (b) and (c), (ii) $\to$ (d), (iii) $\to$ (a). Equilibrium always means $\Delta G = 0$ (b); the special case $\Delta G^\circ = 0$ (c) corresponds to $K = 1$ — an equilibrium mixture in which products and reactants are equally favoured under standard conditions.

The core concept behind matching these columns is understanding the role of Gibbs free energy ( $\Delta G$ ) in determining the spontaneity and equilibrium state of a chemical reaction. Gibbs free energy represents the maximum reversible work that can be performed by a thermodynamic system at constant temperature and pressure. Its change, $\Delta G$ , tells us whether a reaction will proceed on its own, require energy input, or is at a state of balance.

Here's how $\Delta G$ relates to reaction spontaneity and the equilibrium constant ( $K$ ):

$\Delta G$ (Gibbs Free Energy Change): This value indicates the spontaneity of a reaction under any given set of conditions (temperature, pressure, concentrations).
- If $\Delta G < 0$ : The reaction is spontaneous in the forward direction.
- If $\Delta G = 0$ : The reaction is at equilibrium; there is no net change.
- If $\Delta G > 0$ : The reaction is non-spontaneous in the forward direction (the reverse reaction is spontaneous).
$\Delta G^\circ$ (Standard Gibbs Free Energy Change): This is the Gibbs free energy change when the reaction occurs under standard conditions (typically $298 \text{ K}$ , $1 \text{ atm}$ pressure for gases, $1 \text{ M}$ concentration for solutions). $\Delta G^\circ$ is directly related to the equilibrium constant $K$ .
Equilibrium Constant ( $K$ ): This value describes the ratio of products to reactants at equilibrium. It indicates the extent to which a reaction proceeds to completion.
- If $K > 1$ : Products are favored at equilibrium.
- If $K < 1$ : Reactants are favored at equilibrium.
- If $K = 1$ : Products and reactants are equally favored at equilibrium (or more precisely, their activities/concentrations are such that $K=1$ ).

The fundamental relationship connecting $\Delta G^\circ$ and $K$ is:

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

This equation shows that a negative $\Delta G^\circ$ corresponds to $K > 1$ (spontaneous under standard conditions, products favored), and a positive $\Delta G^\circ$ corresponds to $K < 1$ (non-spontaneous under standard conditions, reactants favored).

Now, let's match the items in Column I with the conditions in Column II.

Equilibrium

By definition, a system is at equilibrium when there is no net change in the concentrations of reactants and products over time. Thermodynamically, this corresponds to the Gibbs free energy change being zero.
- Therefore, (i) Equilibrium matches with (b) $\Delta G = 0$ .
Spontaneous reaction

A spontaneous reaction is one that proceeds in the forward direction without external intervention. This occurs when the Gibbs free energy of the system decreases.
- Thus, a spontaneous reaction is characterized by $\Delta G < 0$ . …

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