Step-by-step explanation:
The balanced equation for the reaction is:
2 NO2(g) + O2(g) ⇌ 2 N2O5(g)
The pressure equilibrium constant, Kp, for this reaction is given by:
Kp = (P(N2O5))^2 / (P(NO2))^2 * P(O2)
where P is the partial pressure of each gas at equilibrium.
We are given the initial partial pressures of NO2 and O2, as well as the mole fraction of N2O5 at equilibrium. We can use this information to calculate the partial pressures of each gas at equilibrium.
Let x be the change in partial pressure of NO2 and O2 due to the reaction, and let y be the partial pressure of N2O5 at equilibrium. Then we have:
2 NO2(g) + O2(g) ⇌ 2 N2O5(g)
Initial: 3.8 atm 7.3 atm 0
Change: -2x -x 2y
Equilibrium: 3.8-2x 7.3-x y
From the mole fraction of N2O5, we know that:
y / (3.8-2x + 7.3-x + y) = 0.13
Simplifying this gives us:
y / (11.1 - 3x + y) = 0.13
Multiplying both sides by (11.1 - 3x + y) gives us:
y = 0.13 (11.1 - 3x + y)
Expanding this out gives us:
y = 1.443 - 0.39x + 0.13y
Solving for y in terms of x gives us:
y = (1.443 - 0.39x) / (1 - 0.13)
y = (1.443 - 0.39x) / 0.87
Now, we can plug this expression for y into the expression for Kp:
Kp = ((1.443 - 0.39x) / 0.87)^2 / ((3.8-2x + y) / 11)^2 * ((7.3-x) / 11)
Simplifying this expression gives us:
Kp = ((1.443 - 0.39x) / 0.87)^2 / ((3.8-2x + (1.443 - 0.39x) / 0.87) / 11)^2 * ((7.3-x) / 11)
We can solve for x numerically using a solver or by graphing the function and finding its root. The resulting value of x can then be used to calculate Kp.
Assuming that the temperature and volume remain constant throughout the reaction, the pressure equilibrium constant Kp for the reaction at the final temperature of the mixture is approximately 2.12 x 10^-4 (in units of atm^2).