Can you get this USNCO question?

EquilibriumNational 2020

A gas X reacts reversibly and independently to form two different gases, Y and Z: $\ce{X(g) <=> Y(g)}$ $K_Y$ $\ce{X(g) <=> Z(g)}$ $K_Z$ At a certain temperature, the time evolution of the pressures of the three gases are shown. What is the equilibrium constant $K_Y$ at this temperature? Time 0 1 2 3 4 5 Partial Pressure (atm) $P_X$ $P_Y$ $P_Z$

8% of national qualifiers got this right

Explanation

draft

At equilibrium the partial pressures level off (flat parts of the curves at large $t$ ). Reading the graph at $t=5$ :

$P_X \approx 0.35\text{ atm}$
$P_Y \approx 2.8\text{ atm}$
$P_Z \approx 1.8\text{ atm}$

For the equilibrium $\ce{X <=> Y}$ :

$K_Y = \dfrac{P_Y}{P_X} = \dfrac{2.8}{0.35} \approx 8.0$

So $K_Y \approx 8.0$ , matching (B).

This value (≈3.0) is just the equilibrium partial pressure of $Y$ itself, mistaking $P_Y$ for the equilibrium constant instead of dividing by $P_X$ .

Correct: $K_Y = P_Y/P_X$ using the leveled-off (equilibrium) pressures, $2.8/0.35 \approx 8.0$ .

This comes from using the transient overshoot peak of $P_Y$ (≈3.2) divided by an early, not-yet-equilibrium $P_X$ value, giving a ratio close to 9.3 instead of the true equilibrium values.

Incorrect—the curves clearly flatten out by the later times shown, indicating the system has reached equilibrium, so $K_Y$ can be calculated from the plateau pressures.

Concept · Equilibrium constant from graphical partial pressure data