# Is $Q$ dependent on the $\epsilon$ which is choose in the expression Gibb's free energy of a chemical reaction?

In this lecture by mit ocw(*), Moungi Bawendi states that we can have some change in our chemical system containing reactants and products by scaled by some $$\epsilon$$ amount. Using this set up and the equation which gives chemical potential as a function of pressure (**) he derives the $$\Delta G$$ as a function of $$\epsilon$$ as shown below:

$$\mu_{i} (T , P_{tot} ) = u_{i}^{o} (T) + RT \ln \frac{P}{P_o}$$

and, with the expression for $$\Delta G$$ in terms of chemical potentials:

$$\Delta G = \sum_{i=prods} \mu_i \nu_i - \sum_{i= reacts} \mu_i \nu_i$$

He gets,

$$\Delta G(\epsilon) = \epsilon [ \Delta G^{o} + RT \ln \frac{ (P_d)^{v_d} (P_c)^{v_c} }{ (P_b)^{v_b} (P_a)^{v_a} }]$$

Now, this expression I have a few questions about:

• Is the pressure ratio term on the right hand side dependent on $$\epsilon$$?

• How does one correctly interpret the final equation ? Is it related to calculus of variations ? I ask this because it looks very similar ot the derivation of fundamental theorem of calculus of variations (found here)

**: page-4 of these notes from mit ocw