Activation Energy and Nucleation

When the particles of a reacting system (reactant) undergo a transformation to a product system (product) they follow a path of least resistance. This path is energetically favourable but may still require a lot of energy. The energy required by a system to undergo a particular process is called activation energy and it is measured as an excess energy over the one of the ground state of the system. In allusion to the mechanical kinetic and potential energy the energy of a system consists of free energy and energy of bonds between the particles. In this sense the activation energy is minimum amount of free energy necessary to break the bonds between the reactant particles and to form new bonds of the product. The following figure shows a drawing of the reaction path in case of exothermic reaction (reactant energy > product energy). It is accompanied by a release of heat equal to the energy difference between the reactant and the product.

Energy

Reaction pathway

Activation energy

Heat of reaction E_reactant

E_product

Energy

Reaction pathway

Activation energy

Heat of reaction E_reactant

E_product

The reaction path controls the rate of one reaction. One path with high activation energy would lead to a relatively slow reaction. A relatively fast reaction would proceed along a path with small activation energy. One reaction with a pathway, consisting of few steps with different activation energies (Figure 3), is called a multi-step reaction.

Energy

Reaction pathway

Energy

Reaction pathway

Figure 3. Possible pathway of a multi-step exothermic reaction. The third step with the highest activation energy is the rate-determining step.

One sequence of steps through which the reaction proceeds is called reaction mechanism. The reaction step with the smallest rate caused by the highest activation energy controls the overall rate of the process. This step is called rate-determining step. For example when a phase transformation proceeds together with diffusion and the process occurring at a phase interface is rapid in comparison with the diffusion then the rate-determining step is the diffusion. In this case the transformation process is diffusion-controlled.

The reaction rate is also influenced by the temperature. The temperature effect on the reaction rate originates from the temperature effect on the equilibrium constant:

( )

where R is the gas constant and H is the heat of reaction. The equilibrium constant K is defined as , where and is the rate constant for the forward and the reverse reaction, respectively. The total reaction can be schematically written as , where A and B are the reactants and AB is the product. Substituting K, the equation (12) transforms to:

2

Arrhenius (1889) divided Eq. (13) into two parts, each having a form:

( ) ( )

where Ea is the activation energy. Taking Ea as a constant, Eq. (14) can be integrated. As a result:

RT E A k ln _a/

ln = − (15)

where ln A is a constant of integration. Finally this relation can be transformed to:

(

)

A

k= exp− _a/ (16)

known as Arrhenius equation. The term A is a pre-exponential factor. The rate constant of the reaction at each temperature is determined by Eq. (16) resulting from the linear relation between the logarithm of the rate constant and the reciprocal value of the temperature (Eq. (15)). The values of A and Ea can be calculated from the linear fit of minimum three rate constants determined at three different temperatures. Figure 4 shows an example plot that can give the rate constant k0 at any temperature of interest T0 by the Arrhenius equation.

∆ ∆

Figure 4. Logarithmic plot of example rate constants k obtained experimentally (triangles) can be used to calculate the parameters in the Arrhenius equation (linear fit).

One phase transition proceeds through nucleation and growth. The first energetic barrier, which a system has to pass in order to undergo a phase transition process, is the nucleation barrier. It is related to the interface between the different phases in the system. A one-phase system, having free energy larger than the one of a multi-phase system of the same particles, can exist in a fixed volume for a certain time period. Such a state is called metastable and the system tends to leave it in order to reach a stable multi-phase state of a minimum free energy. The change from the metastable to the stable state occurs as a result of fluctuations in the homogeneous medium forming small quantities of the new phase, or nuclei. As a result the Gibbs

causing an increase of the free energy of the system, is energetically unfavourable. As a result the nuclei, which are very small, are unstable and disappear and the origin phase is stable with respect to the internal fluctuations. For each metastable state a minimum size of a nucleus, which is stable within the initial phase, exists. It is called critical size a_cr and the nuclei of this size are called critical nuclei. The critical size of the nucleus is related to the change of the Gibbs energy as it shown schematically in Figure 5.

∆G

a a_cr

∆G

a a_cr

Figure 5. Nuclei of size below the critical one are energetically unfavourable and disappear. Only the bigger nuclei can continue to grow.

A nucleus of size a<a_cr is unstable because its appearance causes an increase of the Gibbs energy of the system. Only a nucleus of size a>a_cr, decreasing the Gibbs energy, is stable and continues to grow spontaneously. The critical value of the nucleus size can be obtained from the condition:

( )

∂

∆

∂

=a_cr

a a

G (17)

This interfacial energy effect, which the system has to overcome in the phase transition process, can be regarded as a nucleation barrier. Once the system passed this barrier the thermodynamically favourable new phase will start to grow.