Welcome to Topic 13B: Entropy and Free Energy

In your previous studies (Topic 8: Energetics I), you learned about enthalpy changes—basically, whether a reaction gives out heat or takes it in. But enthalpy doesn't tell the whole story. If "giving out heat" (exothermic) was the only rule, why would ice melt at room temperature? Melting is endothermic, yet it happens spontaneously! In this chapter, we explore Entropy and Free Energy, the "missing pieces" that explain why chemical reactions happen at all.

1. What is Entropy (\(S\))?

Think of entropy as a measure of disorder or randomness. Nature loves chaos! Everything in the universe tends to move from an ordered state to a disordered state.

Key Rule: The more ways the particles and their energy can be arranged, the higher the entropy. Entropy is measured in units of \(J \, K^{-1} \, mol^{-1}\).

Factors that Increase Entropy

  • Change of State: Moving from solid to liquid, or liquid to gas. In a solid, particles are fixed and orderly (low entropy). In a gas, they zoom around randomly (high entropy).
    Example: \(\text{H}_2\text{O}(s) \rightarrow \text{H}_2\text{O}(l)\) involves an increase in entropy.
  • Dissolving a Solid: When you dissolve a crystal (like salt) into water, the rigid structure breaks down, and the ions spread out. This increases disorder.
  • Change in Moles of Gas: This is a big one for exams! If a reaction produces more moles of gas than it starts with, the entropy increases.
    Example: \(\text{CaCO}_3(s) \rightarrow \text{CaO}(s) + \text{CO}_2(g)\). We go from 0 moles of gas to 1 mole of gas. Entropy increases significantly.

Quick Review: High disorder = High Entropy. Gases have much higher entropy than liquids, and liquids have higher entropy than solids.

2. Calculating Entropy Changes

We look at entropy in two parts: the System (the chemicals themselves) and the Surroundings (the air/container around the chemicals).

A. Entropy of the System (\(\Delta S_{system}\))

To find the change in the system, we use the standard entropy values from the Data Booklet:

\(\Delta S_{system} = \sum S_{products} - \sum S_{reactants}\)

B. Entropy of the Surroundings (\(\Delta S_{surroundings}\))

When a reaction is exothermic (\(\Delta H\) is negative), it releases heat into the surroundings. This heat makes the particles in the air move faster, increasing their entropy. The formula is:

\(\Delta S_{surroundings} = -\frac{\Delta H}{T}\)

Don't get caught out!

  1. Temperature (\(T\)) must be in Kelvin (\(K = ^\circ C + 273\)).
  2. Units: \(\Delta H\) is usually given in \(kJ \, mol^{-1}\), but entropy is in \(J \, K^{-1} \, mol^{-1}\). You must multiply \(\Delta H\) by 1000 to convert it to Joules before using this formula.

C. Total Entropy (\(\Delta S_{total}\))

For a reaction to be spontaneous (feasible), the total entropy must be positive.

\(\Delta S_{total} = \Delta S_{system} + \Delta S_{surroundings}\)

3. Gibbs Free Energy (\(\Delta G\))

While \(\Delta S_{total}\) is great, chemists often prefer Gibbs Free Energy because it focuses on the system. It is the ultimate "yes/no" test for a reaction.

The Equation: \(\Delta G = \Delta H - T\Delta S_{system}\)

Is the Reaction Feasible?

  • If \(\Delta G \leq 0\), the reaction is feasible (it can happen spontaneously).
  • If \(\Delta G > 0\), the reaction is not feasible.

Memory Aid: "Gee, I hope it's negative!" If \(\Delta G\) is negative, the reaction is "good to go."

The Temperature of Feasibility

Some reactions are only feasible at high temperatures (like the thermal decomposition of calcium carbonate). You can find the exact temperature where a reaction starts to become feasible by setting \(\Delta G = 0\):

\(0 = \Delta H - T\Delta S_{system}\) \(\implies\) \(T = \frac{\Delta H}{\Delta S_{system}}\)

Key Takeaway: Endothermic reactions (\(\Delta H = +\)) can only be feasible if the entropy change (\(\Delta S\)) is positive and the temperature is high enough to make the \(T\Delta S\) term larger than \(\Delta H\).

4. Free Energy and Equilibrium

There is a direct mathematical link between the feasibility of a reaction (\(\Delta G\)) and the equilibrium constant (\(K\), such as \(K_c\) or \(K_p\)).

The Equation: \(\Delta G = -RT \ln K\)

  • \(R\) is the Gas Constant (\(8.31 \, J \, mol^{-1} \, K^{-1}\)).
  • \(T\) is the temperature in Kelvin.
  • \(K\) is the equilibrium constant.

What this tells us:

  1. If \(\Delta G\) is very negative, \(K\) will be very large (the reaction goes almost to completion).
  2. If \(\Delta G\) is very positive, \(K\) will be very small (the reaction barely happens).
  3. If \(\Delta G = 0\), then \(K = 1\).

5. Why don't all "Feasible" reactions happen? (Kinetic Inhibition)

Don't worry if you find a reaction that has a negative \(\Delta G\) but doesn't seem to happen in real life. Thermodynamics (\(\Delta G\)) tells us if a reaction can happen, but Kinetics tells us how fast it happens.

Kinetic Inhibition: A reaction might be thermodynamically feasible (\(\Delta G < 0\)), but it has a very high activation energy (\(E_a\)). This means the reaction is so slow at room temperature that it effectively doesn't happen at all.

Example: The reaction between hydrogen and oxygen to make water has a very negative \(\Delta G\), but you can leave them in a flask together forever and nothing will happen until you provide a spark to overcome the activation energy.

Quick Summary for Revision

  • Entropy (\(S\)) = disorder. Gases > Liquids > Solids.
  • \(\Delta S_{system}\) = Products - Reactants.
  • \(\Delta S_{surroundings} = -\Delta H / T\) (Watch your units! \(J\) vs \(kJ\)).
  • \(\Delta S_{total}\) must be positive for a reaction to occur.
  • \(\Delta G = \Delta H - T\Delta S_{system}\). Must be negative or zero for feasibility.
  • \(\Delta G = -RT \ln K\) links thermodynamics to equilibrium.
  • Kinetic inhibition means a feasible reaction might be too slow to observe.