If 0.500 mol each of phosphorus trichloride and chlorine are injected into a 1.00-L container at 60oC, find the equilibrium concentrations of all three species in the equilibrium mixture.

PCl5(g) <===> PCl3(g) + Cl2(g) K - 12.5 at 60oC

This is my ICE Table...

---PCl5(g) <===> PCL3(g) + Cl2(g)
I--(0.00)-------(0.500)----(0.500)

C--(x)----------(-x)-------(-x)

E--(x)----------(0.500-x)--(0.500-x)

K=([PCl3(g)][Cl2(g)])/[PCl5(g)]

Subbing in the values I get
12.5 = (0.500-x)^2 / x
12.5x = 0.250 + x^2
-x^2 + 12.5x - 0.250 = 0

Then I sub this into the quadratic formula...

x=(-12.5(+-)root((-12.5)^2-4(-1)(-0.250)))/2(-1)

I get
(using + root)x=0.02003210281
(using - root)x=12.4799679

My textbook says the correct answer should be 0.0185.

Side Issue:I heard that using the negative root in the quadratic formula is useless in Chemistry, is that true?

Yes, this is because your 'x' value refers to a certain amount of concentration of a substance. And you cannot have a negative amount of substance.

Similarly, in mathematics and physics, when you solve quadratics related to something like length, you omit the negative root since you can't have negative lengths.

As for your answer, you aren't that inaccurate. I think your slight deviance from the answer is because you omitted the '2ab' term from '(a-b)^2' when forming your quadratic.

I solved another question similar to this and used the negative root in the quadratic equation and got the correct answer.

I saw your new post above in which you talk about the negative root and I've addressed that there. I hope it is to your satisfaction. If you have further question about it you can follow up at that post. In this particular "rule", that business that you've heard, is right some of the time and wrong some of the time. It all depends upon what you let x stand for. IF x stands for a concentration, length, mass, etc then the rule is correct because you cn't have a negative mass, length, or concentration etc.. However, if x stands for something else and you must add or subtract something from the x to get the final answer, then that rule is wrong. Again, see more discussion at the other post you made.Hope this helps.

Let's go through the calculation to find the equilibrium concentrations of all three species in the mixture.

First, we set up the initial, change, and equilibrium (ICE) table:
---PCl5(g) <===> PCL3(g) + Cl2(g)
I-- (0.00) -------- (0.500) ------ (0.500)
C-- (x) ----------- (-x) ----------- (-x)
E-- (x) ----------- (0.500-x) ----- (0.500-x)

The equilibrium constant expression for this reaction is K = ([PCl3(g)][Cl2(g)])/[PCl5(g)]. Substituting the given values, we have 12.5 = (0.500-x)^2 / x.

Now, let's solve for x using quadratic equation. The equation is: -x^2 + 12.5x - 0.250 = 0.

Plugging in the values into the quadratic formula, x = (-b ± √(b^2 - 4ac)) / (2a).

Using the positive root, x = (-12.5 + √(12.5^2 - 4(-1)(-0.250))) / (2(-1)) = 0.02003210281. Using the negative root, x = (-12.5 - √(12.5^2 - 4(-1)(-0.250))) / (2(-1)) = 12.4799679.

The positive root, x = 0.02003210281, corresponds to the concentration of PCl5 that reacts with PCl3 and Cl2. However, this cannot be the equilibrium concentration because it is larger than the initial concentration of PCl5 (0.00 mol).

The negative root, x = 12.4799679, is not physically meaningful in this context because it would imply a negative concentration. In chemistry, concentrations cannot be negative. Therefore, we discard this root.

To find the correct answer, we need to use an alternative method. We can solve the quadratic equation by factoring it or by using the quadratic formula with the positive root only. By factoring the equation 0 = -x^2 + 12.5x - 0.250, we obtain (x - 0.02003210281)(x - 12.4799679) = 0. Setting each factor equal to zero, we find two possible values for x: x = 0.02003210281 and x = 12.4799679.

Now, we need to determine which root is physically meaningful. Since the equilibrium concentration cannot exceed the initial concentration of PCl5 (0.500 mol), we choose x = 0.02003210281 as the equilibrium concentration.

Therefore, the equilibrium concentrations of all three species in the equilibrium mixture are:
[PCl5(g)] = 0.500 - x = 0.500 - 0.02003210281 = 0.4799678972 mol/L
[PCl3(g)] = x = 0.02003210281 mol/L
[Cl2(g)] = x = 0.02003210281 mol/L

As for your side issue, using the negative root in the quadratic formula is not useless in chemistry. The negative root may be meaningful in different contexts or scenarios. It is important to consider the physical meaning of the solution and to select the appropriate root based on the given conditions in the problem.