# Entropy and Gibbs Free Energy¶

Definition

Entropy is s measure of how ordered a system is.
The second law of thermodynamics states that any spontaneous process increases the entropy of the universe.

• To decrease the entropy of a system, work needs to be done
• The act of decreasing the entropy of one system inevitably means increasing the entropy of another
• Entropy, like enthalpy is a state function.

## Examples of disorder¶

#### Systems of a lower state of matter have a lower entropy¶

$$\ce{H2O_{(l)}}$$ has a lower entropy than $$\ce{H2O_{(g)}}$$ (96.9 vs 188.7)

#### Atoms of a higher molecular weight have lower entropy¶

$$\ce{He_{(g)}}$$ has a lower entropy than $$\ce{Ne_{(g)}}$$ (126.1 vs 146.2 )

#### Molecules with fewer bonds have less entropy¶

$$\ce{C2H2_{(g)}}$$ has a lower entropy than $$\ce{C2H6_{(g)}}$$ (200.9 vs 229.2)

#### Systems with fewer moles of the same atom have less entropy¶

$$\ce{I2_{(g)}}$$ has a lower entropy than $$\ce{2I_{(g)}}$$

• This goes for reactions too
• $$\ce{2ZnO_{(s)}}$$ has lower entropy than $$\ce{2Zn_{(s)} + O2_{(g)}}$$ (2 moles vs 3 moles)

## Spontaneity¶

In a chemical sense, is a measure of whether or not a reaction is thermodynamically capable of happening without outside energy input

If an equation has a a positive enthalpy ($$\Delta H=+ve$$), it is logically unlikely to occur, since heat is required form the environment. We can, compare the effects of enthalpy and entropy, to determine which change is driving the reaction:

If $$\Delta H>T\Delta S$$, then the reaction is enthalpy driven

if $$\Delta H<T\Delta S$$ then the reaction is entropy driven

E.g.: $$\ce{2NH4Cl_{(s)} + Ba(OH)2.8H2O_{(s)} -> 2NH3_{(g)} + 10H2O_{(l)} + BaCl2_{(s)}}$$

$\begin{gather} \Delta H^\circ=166\:KJ \text{ (endothemric)}\\ \Delta S^\circ=0.594\:KJ\cdot K^{-1}\\ T=298.15\:K\\ \end{gather}$

So when we calculate $$T \Delta S$$ we get; $$298.15\cdot0.594=177\:KJ$$

Since $$T \Delta S$$ is greater than $$\Delta H$$, the endothermic nature of the reaction os overcome by the entropy to make it soontaneous

Note

The reason for the much larger contribution of entropy is that the reaction converts 3 moles of solid into one mole of sold, 10 moles of liquid and 2 moles of gas. This is both a massive increase in the number of moles of matter, but also a massive increase in the entropy from species that have more ability to move around… randomly.

## Gibbs Free Energy¶

• Is also a state function
• Measures the amount of energy available to do useful chemical work
• If $$\Delta G$$ is $$-ve$$, then the reaction is spontaneous
• If $$\Delta G$$ is $$0$$, then the reaction is in equilibrium
$\Delta G^\circ=\Delta H^\circ-T\Delta S^\circ$

E.g.: In the above example, we can calculate the $$\Delta G$$ as:

$\Delta G=166-(298.15\cdot0.594)=-11\:KJ\cdot mol^{-1}$

Which means that ultimately, despite taking in $$166\:KJ\cdot mol^{-1}$$, the system has actually lost $$11\:KJ\cdot mol^{-1}$$ of energy and is more stable as a result.

### Non-standard state spontaneity¶

To calculate spontaneity in non standard state conditions we can use the following equation

Where $$Q=\frac{\text{products}}{\text{reactants}}$$

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

### Determination of equilibrium constants (rate constants)¶

We can use $$\Delta G$$ to calculate a $$k_x$$

Where $$R=$$ Gas Constant ($$8.314\:J\cdot K^{-1}\cdot mol^{-1}$$)

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