Consider the following reactions in diluted aqueous phase in a batch reactor:

$A+BR$

$R+BS$

The feed contains only A and B$.$ Prove the following concentration relations after a finite time into the reactions.

$\frac{c_S}{c_{A0}}=1-\frac{c_A}{c_{A0}}-\frac{c_R}{c_{A0}}$

$\frac{c_R}{c_{A0}}=\frac{2(c_{A0}-c_A)}{c_{A0}}+\frac{c_B-c_{B0}}{c_{A0}}$

Are these relations still valid if the reactions are gas phase reactions in a batch reactor?

Hint: Since more than one reaction is involved, when constructing the stoichiometric table, use the molar extent of each reaction instead of the conversion of a single species.