The Quadratic Formula

by Jules Bruno
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consider the following reaction. So a reaction mixture initially contains 15 Moeller of c O B R. Two, determine the equilibrium concentration of carbon monoxide. If Casey for the reaction at this temperature is 2.15 times 10 to the minus three. All right, so in this question, they're only giving us the initial amount of this compound we're missing. Equilibrium amounts for all compounds within the equation, so we're gonna have to do a nice chart. So we're gonna set up the ice chart. And remember, in our ice chart, we ignore solids and liquids. Everything here is a gas. So it's fine. Remember, our ice chart stands for initial change and equilibrium. Since we're dealing with Casey, we know the units need to be in polarity, which they are. They're gonna plug that in. These initially are zero. Remember, we lose react INTs in order to make products, bring down everything. So we feel that our ice chart now we're gonna do our equilibrium expression K C equals products overreacted. So seo times br to over c o B r two, we're gonna plug in the values we have for each. So Casey is 2.15 times 10 to the minus three equals at equilibrium. Both of our products are equal to X, so that's X squared, divided by 15 minus X. At this point, we have to check to see Can I ignore this minus X here, the top in the bottom or not both squared so I can't take the square root. So we're gonna do next. We're gonna take the initial concentration, and you're gonna divide it by your K value. If you get a number greater than 500 you'll be able to ignore that minus X and avoid the quadratic formula. So if we plug these concentration, this concentration in 15 and divided by R. K, we get an answer of 69. So that number definitely isn't greater than 500. So we cannot ignore that minus X on. We have to keep it and do the quadratic formula. We're gonna multiply both sides now by 15 minus X, we're going to distribute, distribute. So when we distribute, we're gonna get here 3.23 times 10 to the negative four minus 2.15 times 10 to the minus three x equals X squared. This X variable has the largest power source our lead term, which means everything has to go toe it's side. So we're gonna add this to both sides here. We're gonna subtract this from both sides. Minus So now our new expression becomes X squared plus 2.15 times 10 to the minus three X minus 3 to 3 times 10 to the minus four. This represents my A might be and my seat. So the quadratic formula is negative. B plus or minus B squared minus four A. C over two A. So we're gonna start plugging these numbers in. So negative 2.15 times 10 to the minus three plus or minus B squared till 215 times 10 to the minus. Three squared minus four. Here A has a one in front of it. So that's one for C. Do not forget the negative sign. So negative. 3.23 times 10 to the minus four divided by two times one. Realize here at this point, this is plus or minus plus or minus. So I mean, we're gonna get to possibilities. All right, so I'm going to do all the math in here. Then take the square root of that number. So we're gonna get negative 2.15 times 10 to the minus three plus or minus 0. divided by two. Again, it's plus or minus. So there's two possibilities for my ex one where I am adding these two numbers together, then dividing by two or one where I'm subtracting them from each other, then dividing by two. So when I do that, I get to answers. One is 10.169 and the other one is negative. 10.19 Both of these X variables cannot be my answer. Only one of them is the correct answer. How do we check which one is the right one? Well, what you're gonna do here is you're gonna take those two X variables, and it doesn't matter where you plug it in at equilibrium. You should always get a positive number. Negative concentration. Negative amounts of equilibrium do not exist. Okay, So that would mean that the negative one I cannot use because it's not always gonna give me a positive answer back. Depending on where I put it, so this would be the total for X. Now we go back and look at what they asked me to find. Well, in the question, we're asked to figure out the equilibrium concentration of CEO at equilibrium. CEO equals X. So that is our answer. We're done. And the units are more clarity because my initial units were polarity. If my initial units were atmospheres, then my answer at the end will be atmospheres. Also, if we wanted the equilibrium concentration of, say, the reactant, you would take that X and plug it into here. So it be 15 minus that X variable to find our equilibrium concentration of our reactant. So just remember, when you find X, it doesn't necessarily mean that is the final answer. You have to go back to the ice chart, look at the equilibrium line here and see what is the equation for that particular compound. Here We were looking for carbon monoxide at equilibrium. It is just equal to X. So the X that we found is the right answer right from the beginning. But again, if I had asked for let's say, the equilibrium amount of our our reactant then I'd have to plug it into this formula to figure out my final equilibrium amount would be. So always be careful and look exactly what you need to find.