# 4 compounds in equilibrium with one another - determine their equilibrium concentrations

I know that if I have two compounds in equilibrium and I know $\mathrm{\Delta G}$, then I can use $$\mathrm{\Delta G=-RTlnK}$$ to determine the concentrations of the two components at equilibrium. I have 4 compounds that are all in equilibrium with one another \begin{aligned} \mathrm{\Delta G(1,4)} ~&= ~~~3.41~ \mathrm{kcal/mol}\\ \mathrm{\Delta G(3,1)}~&=-2.21~ \mathrm{kcal/mol}\\ \mathrm{\Delta G(2,3)}~&=-0.46~ \mathrm{kcal/mol}\\ \end{aligned}

I applied the equation sequentially and found that \begin{aligned}% [1]~&=~0.34\%\\ [2]~&=~29.1\%\\ [3]~&=~13.5\%\\ [4]~&=~57\%\\ \end{aligned}

Are those the correct concentrations? Is there an "easy" way to do it?

• I am linking my question to this post, since they are very related. Aug 21, 2014 at 10:07

Using $\mathrm{\Delta G~=~-RTlnK_{eq}}$ and the 3 $\mathrm{\Delta G's}$ supplied above, we find that \begin{aligned} \mathrm{K^{1/4}_{eq}}~&=~0.003\\ \mathrm{K^{3/1}_{eq}}~&=~40.749\\ \mathrm{K^{2/3}_{eq}}~&=~2.163\\ \end{aligned} We also know that the concentrations of the 4 species must total to 1
$${[1] + [2] + [3] + [4] = 1}$$