# Spontaneity, Entropy, & Free Energy

Hello there! If you’re joining me for the first time, I would recommend checking out some of my previous crash course posts as well. My last post was on Thermochemistry.

Today’s lesson is going to be long and it’s going to be hard! It won’t be too bad, because I’m going to try my best to explain it simply. Read on for spontaneity, entropy, and free energy! ### The Second Law of Thermodynamics

The energy of the world is constant. The entropy of the world tends towards a maximum.

According to the original definition of the second law, it’s impossible to build a cyclic machine that converts heat into work with 100% efficiency. How does this translate into chemical reactions?

A process that occurs under the given set of conditions without outside intervention is called a spontaneous process.

For example, gas expanding in an evacuated chamber, or heat flowing from a hot body to a cold body.

Heat flowing from a cold body to a hot body is not forbidden by the first law of thermodynamics (energy not created or destroyed). Exothermicity is neither sufficient nor necessary for spontaneity. Just because molecules seek a state of minimum energy isn’t enough to explain some endothermic changes – such as the melting of ice. Exothermicity favors the spontaneity of a reaction but does not guarantee it.

So what’s the criterion for a spontaneous reaction?

### Entropy is a measure of the dispersal of energy

For now, entropy (S) may be viewed as a measure of the randomness or disorder of a system.

Usually, the greater the disorder of a system, the greater the entropy. For example, the entropy of a gas is greater than entropy of a liquid is greater than entropy of a solid.

Statistically, we see that entropy is a measure of energy dispersal (since “disorder” can be imprecise and subjective). Entropy is the number of ways energy can be distributed – and disorder can be attained in many more ways than order.

So, we conceptualize entropy in terms of probability.

Entropy is a direct measure of the number of microstates accessible to the system.

W is the number of ways of changing the inside without changing the outside. Image Source: Philschatz

Because there are the most microstates associated with two balls on the left chamber and two in the right chamber, (c) is the most probable macrostate.

The Boltzmann formula gives the relationship between entropy (S) and the number of microscopic states associated with a particular macroscopic state.

S = kb lnW

where W is unitless and kb is the Boltzmann constant, 1.381 × 10^-23 JK^-1, or (R/Na).

The direction of spontaneous change is that which results in the macroscopic state with the greatest number of microscopic states.

### Absolute Entropies, & The Third Law of Thermodynamics

In contrast to enthalpy and internal energy, we can assign absolute values for the entropy of a substance because of the third law of thermodynamics: that the entropy of a pure, perfect crystalline substance (element or compound) is zero at 0 K.

• When a substance is heated, entropy always increases and is positive – the number of accessible microstates increases, allowing for more dispersal of energy.
• The standard entropy is the absolute entropy of a substance at 1 bar, and usually 25 C.
• The units of entropy are J/K – the same as the Boltzmann’s constant.
• Entropy is a state function, so that the system’s entropy is dependent on the state and independent of how the system got to that state.
• The entropy change for a process is ΔS = Sf – Si, where Sf and Si are the entropies of the system in the final and initial states, respectively.
• Melting, vaporizing, dissolving, mixing, and heating are all processes that lead to an increase in entropy.

### The Second Law of Thermodynamics

The second law expresses the connection between entropy and the spontaneity of a reaction: the entropy of the universe increases in a spontaneous (irreversible) process and remains unchanged in an equilibrium (reversible) process.

For a spontaneous (irreversible) process,

ΔS univ = ΔS sys + ΔS surr > 0

For an equilibrium (reversible) process,

ΔS univ = ΔS sys + ΔS surr = 0

If “ΔS univ” is negative, the process is not spontaneous in the direction described but it spontaneous in the opposite direction.

To determine if a chemical process is spontaneous, we need to calculate the entropy change of the system and that of the surroundings:

### Entropy Changes in the System

ΔS sys = ΔS rxn = Σ S(products) – Σ S(reactants)

If a reaction produces more gas molecules than it consumes, “ΔS sys” is usually positive.

During an exothermic process, the heat transferred to the surroundings increases the disorder and the entropy of the surroundings, so

ΔS surr ∝ q surr

ΔS surr ∝ -q sys

And for a constant pressure process involving only PV-work:

q sys = ΔH sys ; so that

ΔS surr ∝ -ΔH sys

The change in entropy for a given amount of heat also depends on temperature, because sneezing in a quiet library has a larger effect than sneezing on a busy Manhattan street. So, an exothermic reaction will not cause a large change in the entropy of the surroundings if the temperature is high.

ΔS surr ∝ 1/T

Combining the two proportionalities,

ΔS surr = -ΔH sys / T

Keep in mind, however, that a spontaneous reaction may or may not occur at an observable rate.

### Gibbs Free Energy

The second law of thermodynamics tells us that a spontaneous reaction increases the entropy of the universe (system + surroundings).

A new state function, Gibbs free energy, makes it possible to consider only the system without paying attention to the surroundings when determing whether a reaction is spontaneous.

For a constant temperature process,

ΔG = ΔH – TΔS

• If ΔG < 0, the reaction is spontaneous in the forward direction (exergonic)
• If ΔG = 0, the reaction is at equilibrium
• If ΔG > 0, the reaction is spontaneous in the opposite direction (endergonic)

### Entropy and Enthalpy-Driven Reactions The dissolution of ammonium nitrate in water (endothermic) is ane xample of an entropy-driven reaction.

The synthesis of ammonia is an example of an enthalpy-driven reaction.

Like entropy, free energy depends on pressure and concentration, so we tabulate the standard free-energy change – the free energy change that accompanies the conversion of reactants in their standard state to products in their standard state (1 bar pressure, with a specified temperature).

### Standard Free Energy of Formation

The standard free energy of formation is the free-energy change that occurs when one mole of a compound in its standard is formed from elements in thier most stable in form in their standard states: All the thermodynamic quantities are evaluated at a single temperature.

The standard free energy of formation of all elements in their most stable state is by definition equal to 0.

### Calculating Standard Free Energy Changes

We don’t tabulate standard free energy changes directly, but using one of two methods:

(1) From standard free energy of formation: (2) Why is this useful? Because if the standard free energy change is negative, the reactants in their standard states will be converted spontaneously to products in their standard states.

CONGRATULATIONS, you made it through Spontaneity, Entropy, and Free Energy! Now you are ready to take on the world with your new knowledge.

As usual, if you have any questions or feedback, leave it in the comments and I will respond as soon as I can. Thanks so much for reading! Bye for now, 