Answer
$$K' =1.1$$
Work Step by Step
1. Invert the first reaction and multiply the coefficients by 2:
$$2 N_2O (g) \leftrightharpoons 2 N_2(g) + O_2(g)$$
$$K_1' = (\frac{1}{K_1})^2 = (\frac{1}{2.7 \times 10^{-8}})^2 = 1.37 \times 10^{15}$$
2. Inver the second reaction and multiply the coefficients by 2:
$$4 NO_2(g) \leftrightharpoons 2 N_2O_4(g)$$
$$K_2' = (\frac{1}{K_2})^2 = (\frac{1}{4.6 \times 10^{-3}})^2 = 4.7 \times 10^{4}$$
3. Multiply the third reaction by 4:
$$2 N_2(g) + 4 O_2(g) \leftrightharpoons 4 NO_2(g)$$
$$K_3' = (K_3)^2 = (4.1 \times 10^{-9})^2 = 1.7 \times 10^{-17}$$
4. Sum all the new expressions, multiplying their constants
$$2N_2O + 4NO_2 + 2N_2 + 4O_2 \leftrightharpoons 2N_2 + O_2 + 2 N_2O_4 + 4 NO_2$$
- Remove the repeated ones.
$$2N_2O (g) + 3O_2(g) \leftrightharpoons 2 N_2O_4(g) $$
$$K' = (1.37 \times 10^{15})(4.7 \times 10^4)(1.7 \times 10^{-17}) = 1.1$$
