A-Values - Online Tutor, Practice Problems & Exam Prep

Calculating Chair Equilibrium

All right, guys. So in these videos, I'm going to teach you how to use free energy in kilojoules per mole to calculate the exact percentages of each conformation that you would get of a cyclohexane. Again, these videos might be beyond the scope of what your professor wants you to know for your test. So I'm going to leave it up to you to if you need to know this or not. You've been warned, so it's time to get into this. It turns out that we can use that delta G value that we get from our A values to calculate those exact percentages at any given temperature. Now the way we do this is through the Gibbs free energy equilibrium constant equation. Just so you guys know, if this equation looks familiar, it's not unique to this type of problem. In fact, pretty much any process that you can describe a free energy difference in can, you can determine an equilibrium through this equation. So this is a very important equation for all of chemistry, not just for cyclohexanes. Okay? Now as you can see, what it says is that let's just go through one term at a time. It says that the delta G for the change in free energy is equal to the negative R. Now remember that R is the gas constant that we used to use in general chemistry and there were 2 different values of R that we used to use. Just note that the one we're using is in joules per mole, so that's 8.134. That's going to be important in a second. Then temperature, temperatures in Kelvin. Remember that I mean it's been a while since we dealt with temperature, but remember that 0 degrees Celsius is equal to 273.15 degrees Kelvin. That's going to be a conversion that we have to use in a little bit. Then you're going to multiply that by the natural log of the equilibrium constant. Well, in order to solve any of these problems for percentages, we need to know the value of the equilibrium constant. Because equilibrium constant by definition tells you what's your products over your reactants. I need to know that fraction. If we go ahead and we solve for K_E, I did the math for you. Don't worry. What we get is that the K_E is equal to the negative delta G over the R times the T, all to the e. If we can just plug in these variables, we're going to get the equilibrium constant. Now we know what R is. We know what T is. Your calculator tells you what E is. All we need is negative delta G. Do we have a way to find that? Yes, guys. That's through our A values. Our A values tell us what the free energy changes as we go axial. Awesome. Now I do want to make one note of the delta G. Notice that this is negative delta G. But everything that we solved when we're doing A values, we were actually solving for positive delta G because we were actually looking at the less stable one. We're looking at how much energy do we have to put into the system to go to axial. When we use this equation, we're actually going to be inputting the positive delta G here and that's fine. What we're going to be getting is a number that describes basically how we're going to that less stable value. So then over here, what we have is that then we get that K_E and now we can solve for the percentages using the definition, products over reactants. Once we get that positive K_E number, that positive K_E number means that we're actually going towards the favored direction. I'm not sure if you guys remember but if you have a K_E over 1, that means you're going to the more favored direction. I'm just telling you guys right now, if we use a positive number for delta G, we're going to get also a positive number. I'm sorry. We're going to get a number that's above 1 for K_E. We're going to get this greater than 1. What that means is that our definition of K_E has to be the products over the reactants, meaning the more favored conformation over the less favored. Just letting us know that the way that we've arranged this equation, the way that your textbook does it, is that it always does the more favored over the less favored. Meaning that when you get this positive value, it's going to tell you what percentage you're going to have of the equatorial. And then that minus 100 will be your axial. Now here it says that K_E is equal to X over one over X. That just has to do with the definition of equilibrium constant. How K_E is what your X is what you're making. That's your product. So then one minus whatever you made would be your reactants. Then we don't really want K_E here. We want X because we're really trying to figure out how much of this product we're going to get. So if we solve for X, I did that for you. What you're finally going to get is that X is equal to K_E over K_E+1. If you want to put it in a percentage term, it's times 100. Now that was a ton of words that I just said, a lot of numbers and symbols. I do not need you guys to perfectly understand this as much as I just need you to memorize it and know how to use it. If your professor wants you to solve this on your exam, then these equations should be in your mind. You should have memorized this equation. You should have memorized the definition of K_E or how to solve for X. Now we're going to focus on the actual working part, on the actual part that we determine percentages which is so cool. I'm a huge Orgo nerd as you guys know. This is fun. Getting to determine the exact percentage of each cyclohexane. This first one will be a worked example and we'll go ahead and start off with this first one. I'm just going to pause the video and then we'll come back and we'll solve this one together.

Estimate the equilibrium composition of the chair conformers of the following cyclohexanes at room temp:

cis-1,3-diethylcyclohexane

Hey everyone, so in this practice question a, it says, "Estimate the equilibrium composition of the chair conformers of the following cyclohexanes at room temperature." So here we're looking at cis-1,3-diethylcyclohexane. Now, here there are 2 ways that we can draw this structure. One way that we can draw it I'm doing the bow tie version. So, here, one way we can draw this structure where it's cis-1,3 is, let's say that this is carbon 1 here and I'll make this one carbon 3. One way I can draw this is having both of them be equatorial up. Remember, cis means pointing the same direction, Here ethyl (Et) and here ethyl (Et). Now, besides this way we could do a ring flip, and doing this ring flip, we're going to switch from equatorial to axial, but remember we have to maintain direction. Now, I'm gonna say, I'm gonna make this carbon 1, and this is 2, and then this is 3 from this ring flip. Or to be axial up, ethyl would have to be here, so it's pointing axial up. And for it to be up again, ethyl would have to be here. So, these are the 2 ways we can draw cis-1,3-diethylcyclohexane. In this one, we have 2 equatorials that are up, and here we have 2 axials that are up. Now, we know in terms of stability that the equatorial position is more stable, but we need to calculate the composition, the percentages of these 2. So, here we're going to say to do that, we first have to determine what our equilibrium constant will be. The formula that we use here is your equilibrium constant \( K_e \) equals \( e \) to the negative \( \Delta G \) divided by \( RT \). And, what we're going to say here is we have to talk about the flip energy that's involved in this. So, if we're talking about flip energy in terms of my ethyls going from these equatorial positions to axial positions, each one would involve 8 kilojoules of energy. And since there are 2 of them that's a total of 16 kilojoules per mole of energy. This value is in terms of \( \Delta G \). We're going to say here, \( R \) is equal to \( 8.314 \, \text{joules}/\text{mole} \times \text{K} \). Because it's in joules, that means I need to convert the kilojoules into joules. So, here, that will come out to be 16,000 joules. We're gonna say \( e \) to the negative \( 16,000 \, \text{joules}/\text{mole} \) divided by \( 8.314 \, \text{joules}/\text{mole} \times \text{K} \). We're told that this is happening at room temperature. Room temperature is 25 degrees Celsius. To change it into Kelvin we add 273.15 to it. So that's 298.15 Kelvin which we're gonna put here, \( 298.15 \, \text{K} \). And it's important to write these units so we can see what cancels out. Joules cancel out, moles cancel out, kelvins cancel out. So this will be \( K \) equals \( e \), and we're gonna say plug in negative \( \Delta G \) divided by \( RT \). When we do that, we're gonna get our value of 6.454686. And when we punch that into our calculators, we're gonna get 635.674. And we're gonna say here that this is a very large equilibrium constant case, so it's very product favored. What can we do with it? Since we have our equilibrium constant \( K \) now, we can plug it into our formula. We're gonna say, \( K_e = \frac{K_e}{K_e+1} \). So that's \( \frac{635.674}{635.674+1} \). When we do that we're gonna get 0.9984. We have to find the percentages because we're looking for composition we multiply this by a 100 that gives us our percentage of \( 99.84\% \). What this number tells me is that the more stable one of these 2 Cyclohexane Chairs would be \( 99.84\% \) of the time in that form. So again, equatorial positions are more stable, so this would be \( 99.84\% \) of the time. We'd find this form, and then we do \( 100\% \) minus \( 99.84\% \) that's \( 0.16\% \). So, \( 0.16\% \) of it would be in this form the less stable form. In terms of our equilibrium arrows here or our arrows, since this left side is way more stable we'd have a bigger arrow pointing towards it. And what's on the right is less stable since it's in the axial position, we'd have a smaller arrow towards it. So this is the way we depict our 2 chair conformations of cis-1,3-diethylcyclohexane. And by the utilization of our formulas, we find out that the more stable one, it represents \( 99.84\% \) of the composition, and the less stable one only represents \( 0.16\% \). So, you can see how much more skewed the more stable form is than the less stable form is. So, that's how we approach this particular question.

Estimate the equilibrium composition of the chair conformers of trans-1-methyl-3-phenylcyclohexane at room temperature.

What percentages did you get? Did you get more axial this time or less? Let's check it out. The first thing we have to do is we have to draw our cyclohexane to figure out are we adding these values, subtracting them. Trans 1,3 means that I'm actually always going to have one axial and one equatorial because I have to face different directions. One's going up, one's going down. That means that we're going to be subtracting values because you never have both of them axial at the same time. One of them is taking energy and one of them is giving energy back by going to the equatorial. What are those values? Well, we learned before that we had the values 12.6 for phenyl and the value for methyl was 4.2. Subtract it from each other, that would give us a total energy cost of 4.2 kilojoules per mole. All right. So that was easy enough. So, as you can notice, this one is a lot easier to flip than the 2 ethyl groups because again, you don't have them both in the axial position at the same time. Now that we know that, let's go ahead and move to the actual equation part. I'm just going to skip the drawing the equation all out. I'm just going to draw the numbers in. So, what the numbers would be is 8400 because I'm using 1000 as a multiplier, so 84008.134×298.15e. What does that give us? Well, let's check it out. 84008.134×298.15 gives me 3.46. Now I'm going to take the exponent of that and what I'm getting from my kinetic energy is now a number of 31.93. What do you guys think about that kinetic energy? It's a lot smaller than the one I was dealing with at the top. So, to get the X, to solve for X, I'm going to say that X is equal to 31.9332.93. Now, to get the exact number, instead of typing this into my calculator, I'm just going to type this number, 31 divided by (the previous answer plus 1). No. Oh my gosh. I messed it up again. There we go. Equals, yes, I didn't mess it up. Equals 0.9696 which, if I multiply by 100, gives me 96.96%. Isn't that awesome? That's going to be the value of which conformer? Obviously equatorial because of equatorial preference. Let's draw the other one. The other conformation looks like this, where I now have my phenyl faced up and my methyl faced down. What I find is that 96.96% of my conformer is going to be that one. And then I'm going to get only 3.04% as my axial. Now that sounds like a really small amount to have as axial. But think about how much more I have than this one. Because of just the difference in ΔG, it made it a lot more likely to have this and a lot less likely to have that one. Isn't that interesting? Awesome, guys. So anyway, that's the way that you calculate exact conformation percentages. Hopefully, you don't have to do this on your exam. But now that you know it, you'll be prepared for anything that comes up. All right, guys. Let's go ahead and move on to the next video.