# Electrochemical and electronic basics for sensor technology

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## Derivation of the equilibrium concentration of the point defects

The change in the Gibbs energy of a crystal due to the formation of vacancies is assumed to be $ΔG$ and can be split into enthalpy and entropy terms:

$ΔGV=ΔHV−TΔS.V$

We are now looking for the number or concentration of vacancies in a given crystal for which $ΔGV$ adopts a minimum. To do this you have to $ΔHV$ and $ΔS.V$ as a function of the number of point defects $NV$ be known.

The formation of point defects in the ideal crystal is always endothermic, $ΔHV$ so is positive. That $ΔGV$ becomes negative at all, is therefore entirely on the entropy term $−TΔS.V$ and its dependence on the point defect concentration.

$ΔS.V$ contains two posts. On the one hand, the entropy of the crystal changes because the bond strengths and thus the oscillation frequencies of the atoms in the vicinity of a point defect change. This contribution corresponds to a change in the vibrational entropy of the entire crystal by $ΔS.V, vib$. However, it is small and can also be negative as well as positive.

The second post is crucial too $ΔS.V$. It results from the large number of possibilities to distribute the point defects to the various lattice sites. If one looks at a crystal with $NMe$ Atoms of the variety $\text{Me}$in which is still $NV$ There are spaces $NMe+NV$ Total grid places. All spaces and all $\text{Me}$- Atoms are individually indistinguishable. In this case, the number of arrangements is equal to the number of permutations. One denotes with $Ωconfig$ the number of spaces available $NMe+NV$ To distribute grid positions, the following applies to the resulting contribution to entropy:

$ΔΔS.V, config=k⋅lnΩconfig=k⋅lnNMe+NV!NMe!⋅NV!$

$ΔS.V, config$ is always positive and is referred to as configuration entropy or as a contribution to entropy in terms of position statistics. The mole fraction of the vacancies in the crystal at the given values ​​of pressure and temperature can be calculated using the Gibbs-Helmholtz equation $xV$ be calculated:

$ΔGV=NVΔGV−TΔS.V, config$

The size $ΔGV$ denotes the change of $G$ per vacancy, which results from the change in the enthalpy of binding and the entropy of oscillation. She is so long from $NV$ regardless of how the defects do not interact. This applies in general. for small defect concentrations. The term $−TΔS.V, config$ contains the contribution of the configuration entropy. The addition of the contributions leads to a minimum of the Gibbs energy at the equilibrium number $NV, GGW$ the defects.

## Video: Basic Concepts about Sensors and Transducers (July 2022).

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