Michael S. answered • 13d
B.S. in Chemistry, Indiana University; Organic Chem Teaching Intern
Kc = 3.77
You are given one equilibrium and asked about a different one, so this is a K-manipulation problem. There are only two rules, and you apply them in order.
Rule 1 — reverse the reaction → take the reciprocal of K
Rule 2 — multiply the coefficients by n → raise K to the power n
Start: 2 HBr(g) ⇌ H2(g) + Br2(g), Kc = 7.04 × 10−2
Target: ½ H2(g) + ½ Br2(g) ⇌ HBr(g)
Step 1 — reverse it
H2 and Br2 need to be on the left and HBr on the right, so flip the given equation:
H2 + Br2 ⇌ 2 HBr, K1 = 1 / (7.04 × 10−2) = 14.20
Step 2 — halve the coefficients
Every coefficient is multiplied by ½, so K is raised to the ½ power — a square root:
Kc = (14.20)1/2 = 3.77
Order does not matter, but doing both does. Halving first then reversing gives 1/√(0.0704) = 1/0.2653 = 3.77, the same answer. What matters is applying both operations. Stopping after the reciprocal leaves you with 14.2 — the value for a reaction producing 2 HBr, not 1.
Why the exponent rule works
It falls out of the expression itself. For H2 + Br2 ⇌ 2 HBr:
K1 = [HBr]2 / ([H2][Br2])
For the halved version:
Kc = [HBr] / ([H2]1/2[Br2]1/2)
That second expression is exactly the square root of the first. The rule is not something to memorise blindly — it is what happens to the exponents when you scale a balanced equation.
Sanity checks
• Reversing a reaction with K < 1 must give K > 1, and 14.2 duly exceeds 1.
• Taking the square root of a number greater than 1 makes it smaller but keeps it above 1: 3.77 sits between 1 and 14.2, as it must.
• K is never negative and never zero, so if either shows up, something went wrong structurally.
One thing to be careful about: these rules apply to the same reaction rewritten, not to adding two different reactions. When you add equilibria, you multiply their K values — that is a third rule, and it is easy to reach for the wrong one under time pressure.
