The two atoms in a diatomic molecule exert an attractive force...

Question

The two atoms in a diatomic molecule exert an attractive force on each other at large distances and a repulsive force at short distances. The magnitude of the force between two atoms in a diatomic molecule can be approximated by the Lennard-Jones force, or F(r) = F₀ [2(σ/r)¹³ - (σ/r)⁷], where r is the separation between the two atoms, and σ and F₀ are constants. For an oxygen molecule (which is diatomic) F₀ = 9.60 x 10⁻¹¹ N and σ = 3.50 x 100⁻¹¹ m .

(a) Integrate the equation for F(r) to determine the potential energy U(r) of the oxygen molecule.

(a) Integrate the equation for F(r) to determine the potential energy U(r) of the oxygen molecule.

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use. In order to solve this problem. Two nitrogen atoms with a diatomic molecule interact through forces that can either attract or repel them depending on their separation distance. This interaction is captured by the Lenard Jones potential formula F of R equals F zero multiplied by two multiplied by sigma divided by R to the power of 13 minus sigma divided by R to the power of seven where F of R is the force between the atoms. R is their separation distance. And Sigma and F zero are constant specific to the nitrogen molecule or nitrogen F zero is 8.50 multiplied by 10 to the power negative 11 newtons. And sigma is 3.70 multiplied by 10 to the power negative 11 m. To understand the energy dynamics of this molecule derive the expression for the potential energy U of R. So that's our final answer that we're ultimately trying to solve for is we're trying to derive the expression for the potential energy U of R. And because of that, we can ignore what the values for F zero. And Sigma R means we're not trying to solve for a numerical value. So we don't really need to know what those constants represent because all we are asked to do is we're trying to derive an expression for the potential energy U of R. Awesome. With that in mind, let's look at our multiple choice answers and read them off to see what our final answer might be. A is F zero multiplied by Sigma divided by six, all multiplied by Sigma divided by R to the power of 12 minus sigma divided by R to the power of six B is F zero multiplied by Sigma divided by six, all multiplied by Sigma divided by R to the power of six. Sigma divided by R to the power of 12 C is F zero divided by six, all multiplied by sigma divided by R to the power of 12 minus sigma divided by R to the power of six. And finally D is F zero multiplied by Sigma, all divided by six, all multiplied by sigma divided by R to the power of 12 plus sigma divided by R to the power of six Awesome. So first off, let us assume that the interaction between the nitrogen atoms is purely conservative. Therefore, we could state that F of R is equal to negative du divided by Dr where capital U is denoting the potential energy. So now, at this point, we need to find the potential energy U of R by using the force F of R. And to do that, we must integrate F of R with respect to R. So we'll state that U of R is equal to negative integral of F of RDR which is equal to negative integral of when we plug in our known variables, F zero multiplied by two multiplied by Sigma divided by R to the power of 13 minus sigma divided by R to the power of seven. Don't forget your bracket. And we're derived, we're not deriving, we're taking the integration with respect to R. So when we perform this integration, we will get that U of R is equal to negative F zero, multiplied by two multiplied by Sigma to the power of 13 divided by negative 12, multiplied by R to the power of negative 12 minus Sigma to the power of seven divided by negative six, multiplied by R to the power of negative six. So we need to simplify this. And when we simplify it, we will get that U of our is equal to negative F zero divided by six, all multiplied by negative Sigma to the power of 13 divided by R to the power of 12 plus Sigma to the power of seven divided by R to the power of six. So we need to keep simplifying this equation until we can't simplify it anymore. So we need to simplify again. So when we simplify this equation again, you'll get that U of R is equal to F zero divided by six multiplied by Sigma to the power of 13, divided by R to the power of 12 minus Sigma to the power of seven divided by R to the power of six, we can simplify this again. So U of R is equal to F zero multiplied by Sigma divided by six, all multiplied by Sigma to the power of 12, divided by R to the power of 12 minus Sigma to the power of six divided by R to the power of six. And as you could see, we could simplify one last time to get our final answer. And when we do that, we will get that U of R is equal to F zero multiplied by Sigma divided by six, multiplied by sigma divided by R to the power of 12 minus sigma divided by R to the power of six. And we did it, we saw for our final answer, hooray. So looking at our multiple choice answers, the correct answer has to be the letter A. What state is that? F zero multiplied by sigma divided by six, all multiplied by sigma divided by R to the power of 12 minus sigma divided by R to the power of six. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video. Bye.

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