## 4.3 Driving Force

What drives a reaction forward? What force opposes the driving force to result in equilibrium mixtures of reactants and products? This “force” is a result of chemically reactive species and exists today as a topic of debate and study. However, here we focus on the thermodynamic perspective of this question.

Keep in mind the following key ideas:

- Spontaneous reactions or processes are those that occur without needing an outside source of energy.
- Non-spontaneous reactions or processes do not occur unless an external source of energy is supplied.
- Heat spontaneously flows from hot to cold.
- Energy tends to uniformly disperse.
- Matter tends to uniformly disperse.
- The universe tends toward a state of maximum entropy.

##
**Some Spontaneous Processes**

- Heat flow from hot to cold (thermal potential)
- Mass free falls or free flows from high to low elevation (gravitational potential)
- Space expands from high to low pressure
- Electricity flows from higher to lower electrical potential (voltage)
- Solutes disperse from high to low concentration (chemical potential)

These processes can be carried out in reverse by supplying an external force. Some examples are:

- Heat flow from cold to hot (refrigeration)
- Mass transfer from low to high elevation (lifts or pumps)
- Transfer electricity from low to higher voltage (transformers)
- Create more concentrated systems (remove solvent from solution)

List from *Entropy*, 2020^{10}.

### 4.3.1 Enthalpy

The enthalpy of reaction (Δ*H*; endothermic or exothermic) can affect the
spontaneity of reaction and depends greatly on the temperature.
For example, the melting of ice is an endothermic process.
Ice will spontaneously melt at room temperature (under normal conditions)
because the ice is colder than the surroundings and heat will spontaneously flow
from the surroundings into the ice. The ice heats up and melts.

\[\mathrm{H_2O}(s) \longrightarrow \mathrm{H_2O}(l) \qquad \mathrm{spontaneous~at~>0~^{\circ}C}\]

However, ice will not spontaneously melt at temperatures below 0 °C. Heat must be absorbed by the ice to melt but the surroundings are colder than the ice and cannot provide heat to melt the ice. Ice will lose heat to the surroundings and the ice will become colder.

\[\mathrm{H_2O}(s) \longrightarrow \mathrm{H_2O}(l) \qquad \mathrm{not~spontaneous~at~<~0~^{\circ}C}\]

These examples illustrate how energy wants to be uniformly dispersed. Either the ice will lose heat to the colder surroundings or ice will absorb heat from the hotter surroundings. In both cases, heat flows from hot to cold and is uniformly dispersed.

### 4.3.2 Entropy

Thermodynamic **entropy** is measure of how dispersed energy
(or matter) is and is a *thermal property*.
It is often referred to as a measure of disorder (thermal disorder or randomness).
That said, the concept of entropy can sometimes be extended to the disorder or
randomness of matter.

Take H_{2}O for example.
At 0 K, water will be in a state of the lowest possible energy.
The particles can be arranged such that it adopts a perfect crystal without any
defects (giving rise to a minimum possible energy). This configuration of water
particles at 0 K is defined as having zero entropy. Note: Entropy typically
has units of J K^{–1}.

\[S = 0~~\mathrm{J~K^{-1}} ~~\mathrm{at~0~K}\]
Now we heat the ice up (*T* > 0 K). Heat is transferred to the ice and
we begin to see thermal disorder or randomness. The heat disperses itself
throughout the system of ice and the water molecules absorb some amount
of this heat.
The water molecules begin to vibrate more rapidly, though their translational
motions are still frozen since the particles are locked into rigid positions
(hence ice is a solid). Entropy is increasing. *S* > 0 and Δ *S* > 0.

Continue to heat ice up (e.g. displace more heat into the ice) and eventually the ice melts into a liquid. More thermal energy is displaced into the system (entropy is increasing) and the molecules begin to gain translational motion (they begin to move around). The vibrational motions increase. The thermal energy is dispersed throughout the sample of water.

Continue to heat the water up and eventually the water turns into a gas. More thermal energy is displaced into the system and entropy increases. The gaseous water molecules are now moving through space without restriction of neighboring molecules and intermolecular forces. The particles are free to traverse space and are vibrating in all sorts of manner. Matter is dispersed and as such, the thermal energy they carry.

As we move from solid → liquid → gas, the thermal energy in these phases increase as does the thermal disorder or randomness. Think about it this way. Entropy is an amount of energy that is required to have some substance exist in some state. Ice at 0 °C requires less energy to exist at that state than water vapor at 100 °C.

*Changes in entropy* can be measured as a change in heat, Δ*q* (or Δ*H*
if the process is reversible),
at a certain temperature, *T*.

\[\Delta S = \dfrac{\Delta q}{T} \equiv \dfrac{\Delta H}{T}\]
The temperature, *T*, is the temperature of the system. Let us do a thought experiment.
Imagine displacing 50 kJ worth of heat (Δ*q*) into an object. If the object was
cold (at low *T*), the object will experience a larger entropy change than
an object that was hot (at high *T*).

Recall the heat of fusion for ice (6.01 kJ mol^{–1}). Transferring 6.01 kJ
worth of heat to 1 mole of ice (18.02 g) at 0 °C (273.15 K) and 1 atm
melts the ice and a positive entropy change is experienced.

\[\begin{align*} \Delta S &= \dfrac{\Delta H}{T} \\[1.5ex] &= \dfrac{6.01~\mathrm{kJ~mol^{-1}}}{273.15~\mathrm{K}} \left ( 1~\mathrm{mol} \right ) \left ( \dfrac{10^3~\mathrm{J}}{\mathrm{kJ}}\right ) \\[1.5ex] &= 22.00~\mathrm{J~K^{-1}} \end{align*}\]

The ice experiences a positive entropy change as it converts to a liquid at its normal melting point. It absorbed 22 J worth of heat per Kelvin.

Another way to state this: “A change in entropy is the amount of thermal energy that is displaced by a process at some temperature.” We can rearrange this equation as

\[\Delta H = T\Delta S\]

which says “the thermal energy displaced by a process is equal to the change in entropy at some temperature”. For our melting of ice example, if 22 J per Kelvin is required to melt 1 mol of ice, it would require 6.01 kJ worth of heat.

Of course, ice also freezes at 0 °C where the heat of freezing is -6.01 kJ mol_{–1}.
In fact, if the ice is kept at exactly 0 °C, the rate of melting and the rate of fusion
will be equal (in a closed system) and will be in a state of equilibrium.

\[\mathrm{H_2O}(s) \rightleftharpoons \mathrm{H_2O}(l)\]

We will realize later on down the page that Δ*G* = 0 for this state.

Any reaction or process is spontaneous if the **entropy of the universe**, Δ*S*_{univ},
increases as a result of that process.

\[\Delta S_{\mathrm{univ}} = \Delta S_{\mathrm{sys}} + \Delta S_{\mathrm{surr}}\]

So, the spontaneous process of ice melting at 25 °C is a process that increases the entropy of the universe. A thermal displacement occurs where the surroundings loses energy and the system absorbs the energy. The positive entropy change of the ice (i.e. the system) is larger than the negative entropy change of the surroundings

\[\lvert \Delta S_{\mathrm{sys}} \rvert > \lvert \Delta S_{\mathrm{surr}} \rvert\]

and the entropy change of the universe is positive.

How?
Well, solid water is an ordered system where water molecules are held together in fixed
positions. Liquid water has water molecules that are loosely held together and the
water molecules themselves are more “spread out” (i.e. more dispersed). The liquid
water has a *higher entropy* than solid water. Therefore, the melting of ice
is a process that increases the entropy of the system.

\[\mathrm{H_2O}(s) \longrightarrow \mathrm{H_2O}(l) \qquad \Delta S_{\mathrm{system}} > 0\]

Additionally, the ice is absorbing heat at some temperature, Therefore, Δ*q* is positive giving rise
to a positive change in entropy.

To melt, the system (the ice), absorbs heat from the surroundings. The surroundings, therefore, loses a bit of heat and the entropy of the surroundings decreases.

\[\mathrm{H_2O}(s) \longrightarrow \mathrm{H_2O}(l) \qquad \Delta S_{\mathrm{surr}} < 0\]

The heat that the system (the ice) absorbs causes a greater increase in the entropy gain of the system than the entropy that the surroundings lost.

\[\lvert \Delta S_{\mathrm{sys}} \rvert > \lvert \Delta S_{\mathrm{surr}} \rvert\]

Therefore, the entropy of the universe is positive for this process.
We can rationalize this because heat was more uniformly *dispersed* when the hotter
surroundings distributed some heat to the colder system.

Sign | Spontaneity |
---|---|

ΔS_{univ} > 0 |
spontaneous |

ΔS_{univ} < 0 |
non-spontaneous |

### 4.3.3 Gibbs Energy

Imagine combusting gasoline. Gasoline is a liquid and when combusted, a gas
is formed. To simplify the example, we will consider octane (C_{8}H_{18})
instead of gasoline which is a complex mixture of molecules.
When octane undergoes combustion, the entropy change is relatively large
(≈ 500 J K^{–1}).
The enthalpy of the reaction is, however, much larger (≈ -5400 kJ mol^{–1}).
So, the combustion of octane produces a enormous amount of energy per mole, but not all
of that energy is available for us to extract! *Some* amount of that energy is
required/used to allow the products of the reaction to exist in their states.

It is very inconvenient, if not impossible, to take a measurement of the surroundings. It is much easier to take a measurement of the system. Boil a pot of water on the stove and try to figure out the temperature of the water by measuring the surroundings. It would be easier to just stick a thermometer in the water! Gibbs energy does just this! It it a thermodynamic property that allows us to measure the change in entropy of the universe by only having to measuring the system!

**Gibbs free energy** (Δ*G*), or just Gibbs energy,
according to thermodynamics, represents the maximum amount of non-expansion work
that can be extracted from a thermodynamically closed system.
The relationship between enthalpy, temperature, and
entropy is given below. *All terms* are with respect to the system.

\[\Delta G = \Delta H - T\Delta S\]

Let us tackle three thermodynamic principles here.

A reaction or process is spontaneous if the Gibbs free energy for that process is negative.

This principle is due to the following relationship

\[-\Delta G_{\mathrm{sys}} \propto \Delta S_{\mathrm{univ}}\]

As stated earlier, any process is spontaneous if it increases the entropy of the universe. Therefore, the change in free energy for that process must be negative.

A reaction or process is non-spontaneous if the Gibbs free energy for that process is positive.

\[\Delta G_{\mathrm{sys}} \propto -\Delta S_{\mathrm{univ}}\] Any process that is non-spontaneous has a positive change in Gibbs energy which correlates to a decrease in the entropy of the universe.

A reaction or process is at equilibrium if the Gibbs free energy for that process is zero.

\[\Delta G_{\mathrm{sys}} = \Delta S_{\mathrm{univ}} = 0 ~~\mathrm{at~equilibrium}\]

##
**Δ***G*_{sys} and Δ*S*_{univ} relation

*G*

_{sys}and Δ

*S*

_{univ}relation

Begin with the entropy change of the universe.

\[\Delta S_{\mathrm{univ}} = \Delta S_{\mathrm{surr}} + \Delta S_{\mathrm{sys}}\]

We can express the entropy of the surroundings as a change in heat per unit temperature.

\[\Delta S_{\mathrm{univ}} = \dfrac{\Delta q_{\mathrm{surr}}}{T} + \Delta S_{\mathrm{sys}}\]

Furthermore, we can express the change in heat of surroundings as the change in heat of the system.

\[\Delta q_{\mathrm{surr}} = -\Delta q_{sys}\]

Therefore,

\[\Delta S_{\mathrm{univ}} = \dfrac{-\Delta q_{\mathrm{sys}}}{T} + \Delta S_{\mathrm{sys}}\]

We express heat (*q*) as enthalpy (Δ*H*).

\[\Delta S_{\mathrm{univ}} = \dfrac{-\Delta H_{\mathrm{sys}}}{T} + \Delta S_{\mathrm{sys}}\]

Next, we multiply every term by *T* *and* –1 give

\[-T\Delta S_{\mathrm{univ}} = \Delta H_{\mathrm{sys}} - T\Delta S_{\mathrm{sys}}\]

Finally, we define the Gibbs free energy of the system (Δ*G*_{sys}) as –*T*Δ*S*_{univ} giving

\[\Delta G_{\mathrm{sys}} = \Delta H_{\mathrm{sys}} - T\Delta S_{\mathrm{sys}}\]

Since every term is defined from the perspective of the system, we drop the “sys” labels and simply have

\[\Delta G = \Delta H - T\Delta S\] It is important to note/remember that Gibbs energy is related to the change in entropy of the universe.

\[\Delta G = -T\Delta S_{\mathrm{universe}}\]

Let us return to our example of ice melting and freezing at 0 °C and in a state of equilibrium.

\[\begin{align*} \Delta G &= \Delta H - T\Delta S\\ &= 6.01~\mathrm{kJ~mol^{-1}} - 273.15~\mathrm{K} \left ( 22.0~\mathrm{J~K^{-1}} \right ) \left ( \dfrac{\mathrm{kJ}}{10^3~\mathrm{J}} \right ) \\ &= 0.00~\mathrm{kJ~mol^{-1}} \end{align*}\]

We see that Δ*G* = 0 at the equilibrium. What would Δ*G* be
at 1 °C?

\[\begin{align*} \Delta G &= \Delta H - T\Delta S\\ &= 6.01~\mathrm{kJ~mol^{-1}} - 274.15~\mathrm{K} \left ( 22.0~\mathrm{J~K^{-1}} \right ) \left ( \dfrac{\mathrm{kJ}}{10^3~\mathrm{J}} \right ) \\ &= -0.02~\mathrm{kJ~mol^{-1}} \end{align*}\]

Here, Δ*G* is negative and ice will spontaneously melt (it favors products)
at 1 °C! Let us repeat this once more for ice at -1 °C.

\[\begin{align*} \Delta G &= \Delta H - T\Delta S\\ &= 6.01~\mathrm{kJ~mol^{-1}} - 272.15~\mathrm{K} \left ( 22.0~\mathrm{J~K^{-1}} \right ) \left ( \dfrac{\mathrm{kJ}}{10^3~\mathrm{J}} \right ) \\ &= 0.02~\mathrm{kJ~mol^{-1}} \end{align*}\]

Here, Δ*G* is positive and ice will not spontaneously melt (it favors reactants)
at –1 °C!

When Δ*G* < 0, the process is spontaneous.

Spontaneous reactions proceed until the change in Gibbs free energy is zero.

A reaction occurs meaning the initial change in free energy is negative. At some point,
the reaction reaches equilibrium and the driving force behind the reaction is
equally balanced with a reverse driving force. Here, Δ*G* = 0 and the process is at equilibrium.

Standard Gibbs free energy (where all reactants and products are in their standard states) can be related to an equilibrium constant for a system at equilibrium and is given as

\[\Delta G^{\circ} = -RT\ln K\]

For values of *K* < 1 (reactant favored), ln(*K*) < 0 and for values of *K* > 1 (product favored), ln(*K*) > 0.

For a system that is not at equilibrium, we use Δ*G* and *Q* to determine the direction that
the reaction must proceed.

\[\Delta G = RT\ln \dfrac{Q}{K}\]

Here, if *Q* > *K*, then ln(*Q*/*K*) > 0 and Δ G is positive for the forward direction.
The reaction must proceed left to reach equilibrium. If *Q* < *K*, then Δ*G* is negative
and proceeds right to reach equilibrium.

An alternative relation is given as

\[\Delta G = \Delta G^{\circ} + RT\ln Q\]

### References

*Entropy*

**2020**,

*22*(6). https://doi.org/10.3390/e22060648.