At $500 \ K$,for a reversible reaction $A_{2(g)} + B_{2(g)} \rightleftharpoons 2 AB_{(g)}$ in a closed container,$K_C = 2 \times 10^{-5}$. In the presence of a catalyst,the equilibrium is attained $10$ times faster. The equilibrium constant $K_C$ in the presence of a catalyst at the same temperature is:

Using the data provided,find the value of the equilibrium constant for the following reaction at $298 \ K$ and $1 \ atm$ pressure: $NO_{(g)} + \frac{1}{2} O_{2(g)} \rightleftharpoons NO_{2(g)}$
$\Delta_{f} H^0(NO_{(g)}) = 90.4 \ kJ \cdot mol^{-1}$
$\Delta_{f} H^0(NO_{2(g)}) = 32.48 \ kJ \cdot mol^{-1}$
$\Delta S^{\circ} = -70.8 \ J \cdot K^{-1} \cdot mol^{-1}$
$\text{antilog}(6.4) = 2.51 \times 10^6$ (Note: Calculation based on standard thermodynamic relations)

At $298 \ K$,
$N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g), K_1 = 4 \times 10^5$
$N_2(g) + O_2(g) \rightleftharpoons 2NO(g), K_2 = 1.6 \times 10^{12}$
$H_2(g) + \frac{1}{2}O_2(g) \rightleftharpoons H_2O(g), K_3 = 1.0 \times 10^{-13}$
Based on the above equilibria,the equilibrium constant of the reaction,
$2NH_3(g) + \frac{5}{2}O_2(g) \rightleftharpoons 2NO(g) + 3H_2O(g)$
is $.......... \times 10^{-33}$ (Nearest integer).

At $600 \ K$,$2 \ mol$ of $NO$ are mixed with $1 \ mol$ of $O_2$.
$2 \ NO_{(g)} + O_{2(g)} \rightleftarrows 2 \ NO_{2(g)}$
The reaction occurring as above comes to equilibrium under a total pressure of $1 \ atm$. Analysis of the system shows that $0.6 \ mol$ of oxygen are present at equilibrium. The equilibrium constant for the reaction is $.........$ (Nearest integer).

$2NOBr(g) \rightleftharpoons 2NO(g) + Br_2(g)$. If nitrosyl bromide $(NOBr)$ is $40\%$ dissociated at a certain temperature and a total pressure of $0.3 \ atm$,what is the $K_P$ for the reaction $2NO(g) + Br_2(g) \rightleftharpoons 2NOBr(g)$?

Fill in the blanks:
$a$. If a reaction is in equilibrium,then the rate of the forward reaction and the rate of the backward reaction will be ....... .
$b$. If $\Delta G$ is negative,then thermodynamically the reaction takes place in the ...... direction.
$c$. The equilibrium constant gives ........ information for a reaction.