at an equilibrium mixture of PCl5, PCl3, and Cl2 has partial pressures of 217.0 Torr, 13.2 Torr and 13.2 Torr respectively. a quantity of Cl2 is injected into the mixture, and the total pressure jumps to 263.0 Torr (at the moment of mixing). the system then re-equilibrates. calculate the new partial pressures after equilibrium is reestablished.
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To determine the new partial pressures after the system re-equilibrates, we need to use the principles of Le Chatelier's principle.
1. Write the balanced chemical equation for the reaction between PCl5 and Cl2:
PCl5(g) ⇌ PCl3(g) + Cl2(g)
2. Based on the given information, we have the initial partial pressures:
PCl5 = 217.0 Torr
PCl3 = 13.2 Torr
Cl2 = 13.2 Torr
3. Since Cl2 is injected into the mixture and the total pressure after injection is 263.0 Torr, we can determine the change in pressure:
ΔP = 263.0 Torr - (217.0 Torr + 13.2 Torr + 13.2 Torr)
= 263.0 Torr - 243.4 Torr
= 19.6 Torr
4. According to Le Chatelier's principle, an increase in pressure will shift the equilibrium to the side with fewer moles of gas molecules.
5. The reaction shows that for every 1 mole of PCl5 that reacts, 1 mole of Cl2 is consumed, and 1 mole of PCl3 is produced. Therefore, the number of moles of gas remains the same during the reaction.
6. Since the number of moles of gas does not change, the increase in pressure is solely due to the reduction in volume caused by the injected Cl2.
7. Based on the molar ratios, we can determine the partial pressure changes for PCl5, PCl3, and Cl2:
ΔPCl5 = -19.6 Torr
ΔPCl3 = +19.6 Torr
ΔCl2 = 0 Torr (no change)
8. Calculate the new partial pressures by adding the pressure changes to the initial pressures:
New PCl5 = 217.0 Torr - 19.6 Torr = 197.4 Torr
New PCl3 = 13.2 Torr + 19.6 Torr = 32.8 Torr
New Cl2 = 13.2 Torr
Therefore, the new partial pressures after the system re-equilibrates are:
PCl5 = 197.4 Torr
PCl3 = 32.8 Torr
Cl2 = 13.2 Torr
To calculate the new partial pressures after equilibrium is reestablished, we need to use the concept of Le Chatelier's principle. According to this principle, if a stress is applied to a system at equilibrium, the system will adjust in such a way as to counteract the stress and restore equilibrium.
In this case, the stress is the injection of additional Cl2 into the mixture. Since Cl2 is being added, the system will shift towards consuming Cl2 and producing more PCl5 and PCl3 in order to restore equilibrium.
We can start by setting up an expression for the equilibrium constant of the reaction:
PCl5 ⇌ PCl3 + Cl2
The equilibrium constant expression is given by:
Kp = (PCl3)(Cl2) / (PCl5)
Now, let's consider the initial partial pressures of the substances given in the question:
PCl5 = 217.0 Torr
PCl3 = 13.2 Torr
Cl2 = 13.2 Torr
Since the total pressure after injecting Cl2 is 263.0 Torr, we can calculate the initial partial pressure of Cl2 as follows:
Total pressure = PCl5 + PCl3 + Cl2
263.0 Torr = 217.0 Torr + 13.2 Torr + Cl2
Cl2 = 263.0 Torr - 217.0 Torr - 13.2 Torr
Cl2 = 32.8 Torr
Now, we can substitute these initial partial pressures into the equilibrium constant expression and solve for the new partial pressures:
Kp = (PCl3)(Cl2) / (PCl5)
264.8 = (13.2 + x)(32.8 + x) / (217.0 - x)
Simplifying the equation gives us a quadratic equation:
265.1568 - 13.2x + x^2 = 7171.6 - 249.6x
Rearranging the terms in the equation:
x^2 - 13.2x - 223.6x + 7171.6 - 265.1568 = 0
Combining like terms:
x^2 - 236.8x + 6906.4432 = 0
Using the quadratic formula, we can solve for x:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values a = 1, b = -236.8, and c = 6906.4432 into the quadratic formula, we get two solutions for x: x1 = 37.3 and x2 = 199.5.
Since the partial pressure of Cl2 cannot exceed the total pressure, we can discard the x2 value. Therefore, the new partial pressure of Cl2 after equilibrium is reestablished is:
Cl2 = 32.8 Torr + x1 = 32.8 Torr + 37.3 Torr = 70.1 Torr
Finally, we can calculate the new partial pressures of PCl5 and PCl3:
PCl5 = 217.0 Torr - x1 = 217.0 Torr - 37.3 Torr = 179.7 Torr
PCl3 = 13.2 Torr + x1 = 13.2 Torr + 37.3 Torr = 50.5 Torr
Therefore, the new partial pressures after equilibrium is reestablished are:
PCl5 = 179.7 Torr
PCl3 = 50.5 Torr
Cl2 = 70.1 Torr