(III) The potential energy of the two atoms in a diatomic (two...

Question

(III) The potential energy of the two atoms in a diatomic (two-atom) molecule can be approximated as (Lennard-Jones potential)

U(r) = -(a/r⁶) + (b/r¹²) ,

where r is the distance between the two atoms and a and b are positive constants.

(e) Let F be the force one atom exerts on the other. For what values of r is F > 0 , F < 0 , F = 0?

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem, the potential energy between two oxygen atoms in an 02 molecule due to their interaction can be expressed as U of R is equal to negative A divided by R to the power of eight plus B divided by R to the power of 16. With R being the distance between the atoms and A and B positive constants representing various forces. For what values of R does the force F between the atoms become greater than zero? Less than zero and equal to zero. So that's our end goal. Ultimately, we're trying to figure out for what values of R does the force F between the atoms become greater than zero? Our first answer, less than zero, our second answer and equal to zero, our third answer. And that's what our ultimately, what we're trying to solve for is those three separate conditions. Awesome. We're also given some multiple choice answers. Let's read them off to see what our final answer pair or answer set, I should say might be. So A is R is less than two multiplied by B divided by A, all to the power of 1/8. And F is greater than zero. R is equal to two multiplied by B divided by A to the power of 1/8 and F equals zero and R is greater than two multiplied by B divided by A to the power of 1/8. And F is less than zero and B is R is less than B divided by A to the power of 1/16. F is greater than zero. R is equal to B divided by A to the power of 1/16. And F is equal to zero and R is greater than B divided by A to the power of 1/16. And F is less than zero and C is R is less than a divided by B to the power of 1/8. F is greater than zero. R is equal to a divided by B to power of 1/8 and F equals zero and R is greater than a divided by B to the power of 1818. And F is less than zero. And finally, D is R is less than a divided by two multiplied by B all to the power of 1/8. F is greater than zero. R is equal to a divided by two multiplied by B to the power of 1/8 and F is equal to zero and R is greater than A divided by two, multiplied by B to the power of 1/8 and F is less than zero. OK. So firstly, or first off, let us make the following assumptions. Firstly, let us assume that the negative A divided by R to the power of eight term suggests an attractive force between the atoms is present. While the positive B divided by R to the power of 16 term suggests that a repulsive force is present and that it dominates at a very short distances. And secondly, let us assume that and let's write this down in blue, let us assume that F is equal to negative DU by Dr Awesome. So moving right along, let us recall and note that the force F is given by F equals negative DDR of and using our equation that was provided to us in the prom itself negative A divided by R to the power of eight plus B divided by R to the power of 16. When we perform the derivative with respect to R, we will find that it's equal to eight A divided by R. The power of nine minus 16 multiplied by B divided by R to the power of 17. OK. So recap here, it's eight multiplied by A divided by R to the power of nine minus 16, multiplied by B divided by R to the power of 17. And we got this, when we perform the derivative to our initial equation that was provided to us in the prom itself. And we derived it with respect to R. So now at this point to find the equilibrium position where F equals zero, so let's make a note here. So this is F equals zero condition. We will need to set the expression equal to zero and isolate and solve for R like. So, so let me do that. We will get that eight multiplied by A divided by R to the power of nine minus 16, multiplied by B to the power. So B divided by R to the power of 17 and set this equal to zero. So now using algebra will start rearranging this equation to isolate and solve for R by itself. So when we start to do that, we'll get that eight multiplied by A divided by R to the power of nine is equal to 16. We will move 16/16, multiplied by B divided by R to the power of 17 rearranging. Again, will give us a multiplied by R to the power of eight is equal to two multiplied by B getting R by itself. Finally, we will get that R is equal to two multiplied by B divided by A, all to the power of 1/8. Fantastic. We did it. So now we need to note that the repulsive force will be and let's write this in red. So this is the repulsive force. So R is less than two multiplied by B divided by A, all to the power of 1/8. And this is in respect to F is greater than zero. And this is the repulsive force. So we'll write repulsive, so we know repulsive Awesome. So now let's write this in blue or write the attractive force. So let's say attractive, OK, attracted. Awesome. And the attractive force condition will be R is greater than two multiplied by B divided by A all to the power of 1/8. And this is for the condition where F is less than zero. And the equilibrium position, let's write this in black and to keep things simple, I'll just write E for equilibrium. So the equilibrium position will be R equals two multiplied by B divided by A all to the power of 1/8. And this is for the condition F equals zero. And that's it we solve for this problem. Hooray, we did it. So we solve for our three separate conditions. Fantastic. So looking at our multiple choice answers, the correct answer has to be the letter A where R is less than two multiplied by B divided by A to the power of 1/8 for F is greater than zero. And then R equals two multiplied by B divided by A to the power of 1/8 is equal to the condition F equals zero and R is greater than two multiplied by B divided by A to the power of 1/8 and Vanessa's corresponds to the F is less than zero condition. Thank you so much for watching. Hopefully that helped and I can't wait to see in the next video. Bye.

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