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The Integrated Rate Laws - Overview


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Now let's talk about the integration rate loss. So let's start with the first order Integrated rates law. And now these. You are going to be provided with these numbers. Most likely if you're sitting in the United States, they do not require you to memorize thes. However, it would be really good for you to be able to solve some complicated questions if you are at least family with ease. Um, equations. So the integrator read law for the first order is going to be Ellen A. At Time T equals two minus Katie, plus Ellen a zero. Now they can give you this a question, or they can give you the rearranged version off this equation, which is? What we can do is Ellen 80 minus Ellen. A zero equals two minus Katie. And they can give you this a question where you write Ln eight c over a zero equals two minus. Katie. Remember, if there is a substructure in in Ellen, um, equations. What you can write instead is an l N. And into the Ellen. There's a division now. How else can you write this? If you multiply this both sides by a negative, you would get a Katie. With this time inside off, this equation would flip, and this is also a way to show it. But let's stick with the very first one. What really is important about this one is if you remember, why equals two a mix plus B line equations. What you would get is why is going to be your l n a t. Your slope m is going to be native K your ex is going to be C and your B is going to be Elena zero. So if you want to draw a graph for the very first order integrated right law, the story graph first and what you want to do our why is going to be Ellen a time, see? And this is going to be t Since our X s t we are going to get a straight lines and in the straight line I wish I did not exceed in this grass in the straight line. This tangent to slow is going to be equal toe ar minus K. And this is going to correspond to our time. And this is going to correspond to our Ellen at time T and at time. Zero. What you have over here is Ellen a zero. So this is the very important graph that the first order integrates read Law provides us. But what else can we come off it when we're looking at this equation? So not the very first equation. But if you go to this last equation I wrote or here we can also come outfit the half life. So usually people tend to just memorize the half life equations. But I do not like to do that. The reason why is it's actually really easy to come off with the whole fly for yourself. So if you have to memorize your whole five t one over to this one or two cents for the Hall of Life equals to 0.693 Do I buy the case? So how we come up with we wrote Ellen, Let's make a big parentheses. A zoo. Do I buy 80? That equals two Katie. Now think about 80 at the half life. It means we had a concentration off a zero. Say it was 10. After the hall fly. Fifth time it is now five. This is halt. So that's basically what we're going to do over here. You're going to just right, Ellen, And this is going to be a zero and 80 is going to be half off a zero. That's the entire idea. So this is going to be a zero over to equals to Katie. Now, a zeros are going to cancel a shudder. And we are going to have Ln two equals to Katie. And see now this tee is a specialty. This is the T where a zero. The 80 is half off a zero. So I'm just going to write to you. Well, over to you equals to l N two. Divide by K. Now, this is the same exact same exact equation. Vit. This Just that, Ellen to it cools to 0.693 Okay, So as you can see, by using the integrated read law, we can draw a beautiful graph where the slope is showing us the M. The Y axis is showing us Ln 80 and excesses excesses showing us time as well as we can come up with the T one or two just by using the same integrated read law equation. And for our second order, integrated rates law. What we have is this equation one over 80 equals two Katie plus one over a zero. Now, let's think off how we can make a straight line graph. Mm Equals two. Excuse me. Why equals two MX plus B? Now, why is going to correspond to our 1/80 m is going to correspond to RK this time in the first order. That was a minus K, if you remember. But in this case is going to be K. And our X is going to be t again. And B is going to be one were Asia. So there's draw this graph drawing our graph are why access and our X exists that it's time now. This time, what we're going to have is a positive straight line. This is going to be our one order 80 this point over here at the time t time zero is going to be one over a zero and our slope if he draw aligned us parallel to the X axis. And if you're looking at this slope, R M is going to be equal to K. This is why that's a positive line. That's a upward line that has a slope that it's positive, if you can recall if you were graphing it for the first order Integrated read law. That graph would be going downwards. Hence the minus K. The slope would be minus K, and it will be negative. So now if you want to look at the whole five for the second order Integrated read Law, we can also drive out by ourselves easily. So what we can do is now one over 80 is going to be half off a zero a zero over to this is equal to Katie, plus one over a zero. Now let's put the easier terms in the same side. This the numerator of to the numeric trickles Sodini. No merits here, so we have to over a zero minus one over a zero equals zk etc. Therefore, we have a one over a zero and the skate goes, Ah, the denominator equals to see that it's special 1/2. So again, basically, what we did was taking the integrated right law. Instead, off this V roads, a zero or two put all the same terms on the same side and then, lastly divide both sides by the K, and then we get the whole five equation like that. And lastly, we have zero order integrated rates law. So this is basically going to be 80 equals two minus Katie, plus a zero now. Then again, we want to draw it in a graph that shows us a straight line. So there's going to be why equals two mx plus B. Our why is 80 r m is minus K, which means the slope is going to be negative again, like the first order. Uh, integration read Law and R B is going to be a zero. So if you want to draw the graph are y Axis is going to be are 80 and X axis is going to be t and what we will have is going to be and downward slopes again slope line again. And at this position where the tea is zero, we have a zero, and this slope we have is going to be cooled to minus Kate. So if you want to write the health life equation for the zeroth order reaction, what we need to do again instead off having 80 we want to write a zero over two half off a zero, and that's going to be equal to minus Katie, plus a zero. If you put the same terms in the same side, you will have a 0/2 minus a zero equals two minus Katie, and this is going to be equal to a zero over to equals to Casey and T. One or two is going to be equal to a zero over two K. Again, You do not have to memorize this or you don't want. You don't need them to provide you this formulas. You can easily come up with them once you write to integrated read law and going for 80 year A at the time T as a 0/2.

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