If the composition of the reaction mixture at 400 k is [brcl] = 0.00415 m, [br2] = 0.00366 m, and [cl2] = 0.000672 m, what is the reaction quotient, q?

Q is unlike K value it describes the reaction that is not at equilibrium.
by considering this reaction:
aA+ bB⇄ cC
and our reaction is:
Br2 + Cl2 ⇄ 2 BrCl

According to Q low:
Q= concentration of products/concentration of reactants
but this equation in the gaseous or aqueous states only.
∴ Q = [BrCl]^2 / [Br2] [Cl2]

and we have [Br2] = 0.00366 m  [Cl2]= 0.000672 m  [BrCl] = 0.00415 m

by substitution:
                          = [0.00415]^2 / ( [0.00366] * [0.000672])
             ∴   Q   = 7

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