Calculate $ K_C $:

The equilibrium constant in terms of concentrations ($ K_C $) is given by the ratio of the forward reaction rate constant ($ k_f $) to the backward reaction rate constant ($ k_b $):

$ K_C = \frac{k_f}{k_b} = \frac{1}{2500} $

Convert $ K_C $ to $ K_P $:

To convert the concentration equilibrium constant ($ K_C $) to the pressure equilibrium constant ($ K_P $), use the following relation:

$ K_P = K_C (RT)^{\Delta n_g} $

where:

$ R = 0.0831 \, \mathrm{L \, atm \, mol^{-1} \, K^{-1}} $

$ T = 1000 \, \mathrm{K} $

$\Delta n_g = 2 - 1 = 1$ (change in the number of moles of gas, $ \Delta n_g $, from reactants to products)

Substituting these values in, we get:

$ K_P = \frac{1}{2500} \times 0.0831 \times 1000 $

Calculate $ K_P $:

Solving the above expression gives:

$ K_P = 0.033 $

Therefore, the equilibrium constant $ K_P $ for the reaction at 1000 K is 0.033.