# Why do we use Ḡ (“G bar”) as a notation for partial molar free energy?

For partial molar entropy there is a bar that is positioned above the $$G$$ with a subscript I that is sometimes added to the $$G~(\bar{G}).$$ The bar I read sometimes denote the "partial nature of the $$G$$".

However, is it not rhetorical. I am looking for the explanation as to what is so partial about this quantity with regard to it's occurrence in the equation.

• The bar denotes we took the derivative with respect to composition. Molar free energy would just be the free energy divided by "n" the composition. For a pure component these are the same, but not for a mixture. Not everyone uses a bar also...Like all notation, you have to pay attention with each new article you read, what they mean. – Charlie Crown Jul 15 at 4:11

## 1 Answer

Thermodynamic properties are functions of composition, which can be represented as $$(\chi _1,...,\chi _N)$$ where $$\chi _i$$ is the mole fraction of component $$i$$ present in the system. Now imagine a change in the composition such that the amount of component $$i$$ is increased by a very small quantity $$dn _i$$ (alternately, imagine the system to be infinitely large), and a thermodynamic property $$X$$ is changed by $$dX$$. Then (using the notation you describe) we define the partial molar property as $$\bar{X}=\left(\frac{\partial X}{\partial n_i}\right)_{p,T,n_j}$$ This notation means that we make a limiting, infinitely small change in $$n_i$$, such that the change in $$X$$ is for all practical purposes nil, ie $$\Delta X=0$$. The partial molar quantity describes the slope of the property $$X$$ (which can be represented as a surface in a multidimensional manifold) along the composition coordinate $$n_i$$, at a specific composition $$(\chi _1,...,\chi _N)$$.