# ReactionEquilibrium:Background:Reaction Equilibrium Standard Properties

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The thermodynamic chemical potential ${\displaystyle \mu }$ of a component i in a mixture can be separated into several contributions

${\displaystyle \mu _{i}=\mu _{2}^{0}(T)+kTln\rho +kTlnx_{i}+kTln\phi _{i}}$

where ${\displaystyle \rho }$ is the number (or molar) density, ${\displaystyle x_{i}}$ is the mole fraction, and ${\displaystyle \phi .}$ is the fugacity coefficient. ${\displaystyle \mu _{i}^{0}}$involves contributions to the chemical potential that (as indicated) depend only on temperature ${\displaystyle T}$; ${\displaystyle k}$ is Boltzmann's constant. If we define a "standard state" as one where the substance behaves as an ideal gas ${\displaystyle (\phi _{i}=1)}$, is a pure component ${\displaystyle (x_{i}=1)}$, and is at unit density ${\displaystyle (\rho =1)}$, then all the other terms drop out, and the chemical potential is just ${\displaystyle \mu _{i}^{0}}$. Thus a fancy way to refer to ${\displaystyle \mu _{i}^{0}}$is as "the standard-state chemical potential for the pure substance in the ideal-gas state at unit density." Note that we're not saying that such a state exists physically; we're just saying that we're ignoring all those other contributions to the chemical potential and looking just at ${\displaystyle \mu _{i}^{0}}$. Because for a pure substance the chemical potential equals the molar Gibbs free energy, ${\displaystyle \mu _{i}^{0}}$ is more commonly referred to as the standard-state free energy.

The other contributions originate from the intermolecular interactions (causing ${\displaystyle \phi _{i}}$ to differ from unity), and the entropic contributions that involve counting distinguishable permutations of the positions of atoms of different species, and the center-of-mass movement of the molecules through the system's volume. If these are neglected, then the remaining contributions to ${\displaystyle \mu _{i}}$ come from the properties of a single molecule of species i considered in isolation. Thus we can develop expressions for ${\displaystyle \mu _{i}^{0}}$ by examining a single molecule of that species.

Statistical mechanics says that ${\displaystyle \mu _{i}^{0}}$ can be evaluated by summing all the distinguishable states of a single molecule holding its center of mass fixed, with each state weighted by the Boltzmann factor, ${\displaystyle \exp(-u/kT)}$, where u is the energy. In particular

${\displaystyle \exp(-\mu _{i}^{0}/kT)=\sum _{states}\exp(-u/kT)}$

For the monatomic species, there is only one state, which we take to have zero energy. So for these species the standard-state free energy is zero

${\displaystyle \mu _{R}^{0}=\mu _{B}^{0}=0}$

For the dimeric species, the sum over states is an integral in which the atoms are integrated over all positions for which their separation r is within the attractive region ${\displaystyle \mathrm {K} \sigma and the center of mass is fixed. The energy is the same for all such states, equal to negative ${\displaystyle \epsilon }$. The value of this integral is equal to the volume of the spherical shell where the radius is between the limiting values.

${\displaystyle \exp(-\mu _{dimer}^{0}/kT)={\frac {1}{\sum }}\pi \sigma ^{2}(1-\kappa ^{2})\exp(+\varepsilon /kT)}$

or

${\displaystyle \mu _{dimer}^{0}=\varepsilon -kT\ln \left[\pi \sigma ^{2}(1-\kappa ^{2})/\sum \right]}$

Two points to note.  First, this is written for a two-dimensional system. Second is the presence of the factor ${\displaystyle \Sigma }$, which is a symmetry number. It is 1 for the heteronuclear dimer RB, and 2 for the homonuclear dimers RR and BB. It is needed to account for the fact that in homonuclear dimers, the configuration obtained when the two atoms exchange positions is is indistinguishable from the original; the sum over states counts both and so must be reduced by 2 to compensate for the double counting.

The standard-state enthalpy and entropy can be evaluated using the thermodynamic relations

${\displaystyle h_{i}^{0}=\left({\frac {d({\frac {\mu _{i}^{0}}{kT}})}{d({\frac {1}{kT}})}}\right),s_{i}^{0}=\left({\frac {d\mu _{i}^{0}}{dT}}\right)}$

For the monomeric species these are zero. For the dimer

${\displaystyle h_{dimer}^{0}=\varepsilon }$

${\displaystyle s_{dimer}^{0}=k\ln \left[\pi \sigma ^{2}(1-\kappa ^{2})/\sum \right]}$

Thus, through variation of dimer potential parameters ${\displaystyle \varepsilon }$ and ${\displaystyle \kappa }$ we can adjust independently the standard enthalpy and entropy of the reactions in this system.