The mixture reaches equilibrium at 325 °C according to the following balanced equation: $$N_2O_4(g) \rightleftharpoons 2NO_2(g)$$ colourless brown When equilibrium is reached, it is observed that the colour of the gas in the syringe is brown - NSC Physical Sciences - Question 6 - 2018 - Paper 2

The forward reaction is ENDOTHERMIC because a decrease in temperature (as in cooling the system by dipping in ice water) favours the exothermic reverse reaction. According to Le Chatelier's principle, the system shifts in the direction that counteracts the temperature decrease, leading to a higher concentration of $N_2O_4$.

DECREASES. Reducing the volume of the syringe increases the concentration of gases, thus favouring the side with fewer moles, which is the $N_2O_4$.

REMAINS THE SAME. The equilibrium constant is dependent on temperature and not on changes in pressure or volume.

INCREASES. Decreasing the volume increases concentrations, which in turn increases the frequency of collisions and thus the rates of both the forward and reverse reactions.

Let the initial moles of $N_2O_4$ be $X$.

At equilibrium, 20% of $N_2O_4$ has decomposed:

Moles of $N_2O_4$ at equilibrium = $0.8X$, Moles of $NO_2$ formed = $2 * 0.2X$ = $0.4X$.

Using the formula for equilibrium constant:

$K_c = \frac{[NO_2]^2}{[N_2O_4]}$

Substituting in the values gives:

$0.16 = \frac{(0.4X / 2)^2}{0.8X / 2}$

Solving for $X$: $0.16 = \frac{(0.2X)^2}{0.4X}\Longrightarrow 0.16 = \frac{0.04X^2}{0.4X}\Longrightarrow 0.16 = 0.1X\Longrightarrow X = 1.6.$