Thermodynamics - Series Reaction - Estimating mols of reactants based on Free Energy of Rxn Question


  Three reactions occur simulteneously (assume ideal) at $\pu{T = 0^\circ C}$.
  
  $$\ce{A&lt;=&gt;B&lt;=&gt;C&lt;=&gt;D}$$
  
  a) With $\Delta_r G^\circ_\text{AB} = \pu{400 J/mol}$, $\Delta_r G^\circ_\text{BC} = \pu{-100 J/mol}$, and $\Delta_r G^\circ_\text{CD} = \pu{-200 J/mol}$. If you start with 10 moles of A, calculate how many moles of B, C, and D you have when the system reaches equilibrium. Show all your work. (Hint: mass can neither be created nor destoyed)
  
  b) Order the equilibrium concentrations from largest to smallest. Explain how you might be able to intuitively guess this ranking.


I know that given deltaG_RxN, one can calculate Keq, and use Keq to find the moles at equilibrium of reactants and products. Since the products at the end of one reaction became the reactants of the second, I fed in the mol amount of the first equation into the second equation.



Confused because in discussion - we went over how to do part B. and I don't think my math lines up.


positive delta G_RXN = more reactants than products, and vice versa
thus in the first rxn there should be more [A] than [B] @ equilibrium


in the second rxn there should be more [C] than [B]

and the third rxn there should be more [D] than [C]. 


what did i do wrong math-wise?
What you do is write $[D]=[C]K_{CD}$, $[C]=[B]K_{BC}$, and $[B]=[A]K_{AB}$. So, $$[C]=K_{BC}K_{AB}[A]$$ and $$[D]=K_{CD}K_{BC}K_{AB}[A]$$So, $$[A]+[B]+[C]+[D]=(1+K_{AB}+K_{BC}K_{AB}+K_{CD}K_{BC}K_{AB})[A]=10$$