## Physical Chemistry Essentials - Hofmann A. 2018

# Thermodynamics

2.2 Free Energy

**2.2.1 The Gibbs Function**

In the previous Sect. (__2.1.13__), we have introduced the Gibbs free energy, by means of its change during a process. Named after Josiah Willard Gibbs (1839—1903), who introduced this function in 1875, the free energy acknowledges that not all energy turned around during a process is intricately linked to molecular and atomic interactions (and thus ’free’ to be harvested); there is a component of the entire energy change that is required to be dissipated into the surroundings.

When we characterised the energy of a system, we found it feasible to distinguish between two types of processes, those that occur at constant volume, and those occur under constant pressure. The former situation is best described by the internal energy *U*, whereas the latter, more frequently occurring situation is best described by the enthalpy *H*. In both cases, we also had to consider that the temperature during the ongoing process remained constant.

Concomitantly, when establishing the free energy, the same pre-requisites will need to be applied. The Gibbs free energy is useful for characterising process that occur under constant pressure and temperature, as it is intimately tied to the enthalpy change d*H* during the process.

For the change of the Gibbs free energy during a process we had obtained:

(2.42)

The definition of the state function *G* in terms of the state functions enthalpy and entropy is thus:

(2.44)

**Differentiation of the Gibbs state function**

Differentiation of Eq. __2.36__ yields:

(2.45)

For the second term on the right hand side of the equation, the product rule (see Appendix A.2.3) needs to be applied:

If we consider processes at constant pressure and temperature:

which yields the relationship we had established earlier:

(2.42)

and we emphasise that this is true only for *T*, *p* = const., since we made this assumption during the above calculation.

**2.2.2 Gibbs Free Energy and the Entropy of the Universe**

Any process that is accompanied by a change in entropy ultimately also changes the entropy in the universe. The entropy change in the universe with respect to a process comprises two components, the entropy change in the system and the entropy change in the surroundings:

(2.46)

We derived in Sect. __2.1.10__ that

(2.31)

and therefore obtain from the above equation:

(2.47)

Using the relationship that links the change in the Gibbs free energy with the change in entropy (Eq. __2.42__) one obtains

(2.48)

This can be substituted into Eq. __2.47__ which yields:

(2.49)

The above equation delivers a fundamental paradigm. From the 2nd law of thermodynamics, we know that any process that happens spontaneously needs to increase the entropy in the universe, i.e.

(2.50)

Therefore, the Gibbs free energy change of a spontaneous process *has* to be negative.

For a reversible process, we have discussed earlier that an equilibrium can be considered as a state of the system where there is cross-over from the forward to the reverse reaction, both of which occur spontaneously. Reversible processes therefore occur while the system remains at equilibrium at all times. For such processes, we know from Eq. __2.33__ that d*S* = 0; therefore:

(2.51)

**2.2.3 The Helmholtz Free Energy**

For processes that occur at constant volume, the internal energy *U* is used to describe the energy and energy changes of a system. One can then define a free energy that linked to internal energy changes, and thus useful for characterising process that occur under constant volume and temperature. This free energy is named after Hermann von Helmholtz (1821—1894), who introduced this function in 1882 independently of Gibbs. This function is thus called the Helmholtz free energy *A*. The state function *A* is defined as

(2.52)

**Differentiation of the Helmholtz free energy function**

The differential of the state function *A* is obtained as:

For the second term on the right hand side of the equation, the product rule (see Appendix A.2.3) needs to be applied:

If we consider processes at constant volume and temperature:

(2.53)

The requirement for *V* = const. is a condition of using the internal energy *U* which describes the energy of systems at conditions of constant volume.

**2.2.4 Free Energy Available to Do Useful Work**

For practical applications, it is useful to obtain an expression that describes the free energy under conditions of thermodynamic equilibrium, i.e. reversible processes.

**Gibbs Free Energy**

From the definition of entropy we know that the heat exchange at constant temperature is

(2.54)

Using the definition of the Gibbs free energy (Eq. __2.42__), we therefore obtain:

The value obtained for d*G* in above equation describes the amount of energy that originates from the process and could be used to perform ’useful’ work, i.e. work that is not compression or expansion. We have already established that the maximum work is done when a process is carried out reversibly (Sect. __2.1.6__); this can easily be understood by recalling that in reversible processes, there is no change of entropy:

Since d*S* = 0, there is no further term subtracted from the enthalpy change d*H*. This shows that the change of the Gibbs free energy of a reversible process equals the enthalpy change, which represents the maximum additional non-expansion work done by a system during this process:

(2.55)

**Helmholtz Free Energy**

In the previous section, we derived the differential of the Helmholtz free energy as

(2.53)

in which we can substitute d*Q* for *T*·d*S*, and therefore obtain

As we discussed above, in reversible processes, there is no change in entropy (d*S* = 0), and therefore there is no heat exchange (d*Q* = 0). Therefore, the Helmholtz free energy is maximised in reversible processes and equals the change in the internal energy for this process:

(2.56)

**2.2.5 Gibbs Free Energy Change for a Reaction (Part 1)**

So far, we have mainly been considering unspecified ’processes’ which comprise physical as well as chemical transformations. We now want to consider a chemical reaction with the stoichiometry coefficients ν_{A}, ν_{B}, ν_{C} and ν_{D}, and the physical states indicated by the subscripts (l) for liquid and (g) for gaseous:

for which we can obtain the macroscopically measurable change in the Gibbs free energy in a generic expression as:

(2.57)

This expression does not take into account that the reaction may be conducted with varying concentrations of the reactants A and B. Once the concentrations are considered, the following expression is obtained:

(2.58)

*Q* is called the reaction coefficient. We will derive this equation later using the chemical potential (Sect. __3.2.4__). For the moment, we just consider the fact that the change of the Gibbs free energy during a reaction is dependent on the concentrations of the individual reactants. Equation __2.58__ also introduces a reference value for the Gibbs free energy of the reaction, Δ*G* ^{—}, and the standard molar concentration c^{—}. The symbol ’—’ (stroked letter O) is used to denote a property or quantity under standard conditions; c^{—} = 1 mol l^{−1}.

If all reactants are present at a concentration of 1 mol l^{−1} each, then the reaction coefficient *Q* equals 1. The term R ⋅ *T* ⋅ ln *Q* = R ⋅ *T* ⋅ ln 1 then becomes zero, as all logarithmic functions are zero when their argument is 1 (see Fig. __2.4__). In this case, the Gibbs free energy change for the reaction, Δ*G*, equals the reference value Δ*G* ^{—}; it is therefore called the standard Gibbs free energy change for this reaction.

*Fig. 2.4*

All logarithmic functions assume the value of zero when their argument is 1

If the reaction has reached equilibrium, all reactants exist with particular concentrations that characterise that state of equilibrium, and there is no net change:

We derived earlier in Sect. __2.2.2__ that there is no change in the Gibbs free energy at equilibrium, Δ*G* = 0 (Eq. __2.51__). The relationship that introduced the reaction coefficient *Q* (Eq. __2.58__) therefore yields for the equilibrium state:

and thus

(2.59)

In the state of equilibrium, the reaction quotient *Q* = *Q* _{eq} is termed the equilibrium constant *K*. From the above relationship it is obvious that knowledge of the value of the equilibrium constant allows calculation of the standard free energy Δ*G* ^{—} of a reaction. Vice versa, if the value of the standard free energy of a reaction is known, the equilibrium constant of the reaction at any temperature can be calculated.

**2.2.6 Temperature Dependence of the Equilibrium Constant**

If we are interested in the temperature dependence of the equilibrium constant, we can use Eq. __2.59__ to evaluate the variation of *K* with temperature. This variation is expressed as the differential of ln *K* with *T*:

We denote the differential with ’δ’ to indicate that *K* is dependent on several parameters (*T*, *p* and *V*). Using Eq. __2.59__ one then obtains:

where the gas constant R can be excluded from the differential as it is a constant. The same is true for Δ*G* ^{—} which is the standard free energy of the reaction and thus a characteristic constant for the process. The temperature differential on the right hand side of the above equation can easily be resolved by setting *x* = *T* and . The derivative of *f*(*x*) is . Therefore, resolves to :

With knowledge of the following relationship between enthalpy and free energy

we obtain the van’t Hoff equation for an isobaric process:

(2.60)

which describes the reaction isobar, i.e. the variation of the equilibrium constant with temperature under these conditions.

For a reaction that occurs under constant volume and temperature—an isochoric process—the van’t Hoff equation is analogous to Eq. __2.60__, with the enthalpy being replaced by the internal energy:

(2.61)

Finally, a reaction that occurs under constant temperature, the van Laar-Planck reaction isotherm delivers the variation of the equilibrium constant with pressure:

(2.62)

The three Eqs. __2.60__—__2.62__ describe numerically what is known as the principle of Le Châtelier (see also Sect. __6.1.2__):

A system at equilibrium, when subject to a disturbance, responds in a way that tends to minimise the effect of the disturbance.

**2.2.7 Pressure Dependence of the Gibbs Free Energy**

We are now interested in the pressure dependence of the Gibbs free energy. To evaluate the variation of the Gibbs function with pressure, we attempt to find a direct relationship between *G* and *p*, using the set of thermodynamic functions and relationships we have encountered so far.

Starting with the definition of the Gibbs free energy, we recall that

(2.44)

We also remember that

(2.13)

and therefore

Since we are interested in a variation of *G* with *p*, we differentiate the above equation, and obtain

The differentials of the products, d(*p* ⋅ *V*) and d(*T* ⋅ *S*), need to be resolved with the product rule (Appendix A.2.3), and yield *p* ⋅ d*V* + *V* ⋅ d*p* and *T* ⋅ d*S* + *S* ⋅ d*T*, respectively. Therefore:

(2.63)

From the 1st law of thermodynamics we know that

(2.15)

The heat exchange d*Q* can be expressed in terms of the entropy change d*Q* = *T* ⋅ d*S*, and the work done by the system at constant pressure is the volume work d*W* = *p* ⋅ d*V*. Equation __2.15__ then becomes

which we can use to substitute in Eq. __2.63__:

This simplifies to:

(2.64)

At constant temperature, d*T* = 0 which renders the expression for the Gibbs free energy change as

(2.65)

For two distinct pressure values, *p* _{initial} and *p* _{final}, the Gibbs free energy change then yields

Importantly, the volume is itself dependent on the pressure and thus cannot be isolated from the integral! Instead, we will attempt to find an expression for the volume of the system in terms of the pressure. For an ideal gas this can conveniently be achieved by using the ideal gas equation

(2.6)

which yields for the Gibbs free energy change:

Since *n*, R and *T* are not dependent on the pressure, they can be isolated from the integral, yielding:

The integral is of the type and resolves to ln*x* (see Table A.2). One thus obtains:

The standard state of a gas is defined as the gas at a pressure of p^{—} = 1 bar. If we define any change of the system with respect to standard conditions, the initial pressure becomes the standard pressure: *p* _{initial} = p^{—}. Above equation then reads:

and resolves to the general form of

(2.66)

with the standard Gibbs free energy *G* ^{—} = *G*(1 bar).

Equation __2.66__ allows us to calculate the Gibbs free energy of an ideal gas at any given pressure, provided that we have a value for the standard Gibbs free energy of that gas.