The potential energy of two atoms in a diatomic molecule is...

The potential energy of two atoms in a diatomic molecule is approximated by $U(r) = (a/r^{12}) - (b/r^6)$, where $r$ is the spacing between atoms and $a$ and $b$ are positive constants. (a) Find the force $F(r)$ on one atom as a function of $r$. Draw two graphs: one of $U(r)$ versus $r$ and one of $F(r)$ versus $r$. (b) Find the equilibrium distance between the two atoms. Is this equilibrium stable? (c) Suppose the distance between the two atoms is equal to the equilibrium distance found in part (b). What minimum energy must be added to the molecule to $dissociate$ it$-$that is, to separate the two atoms to an infinite distance apart? This is called the $dissociation$ $energy$ of the molecule. (d) For the molecule CO, the equilibrium distance between the carbon and oxygen atoms is 1.13 $\times$ 10$^{-10}$ m and the dissociation energy is 1.54 $\times$ 10$^{-18}$ J per molecule. Find the values of the constants $a$ and $b$.

The potential energy of two atoms in a diatomic molecule is approximated by $U(r) = (a/r^{12}) - (b/r^6)$, where $r$ is the spacing between atoms and $a$ and $b$ are positive constants. (a) Find the force $F(r)$ on one atom as a function of $r$. Draw two graphs: one of $U(r)$ versus $r$ and one of $F(r)$ versus $r$. (b) Find the equilibrium distance between the two atoms. Is this equilibrium stable? (c) Suppose the distance between the two atoms is equal to the equilibrium distance found in part (b). What minimum energy must be added to the molecule to $dissociate$ it$-$that is, to separate the two atoms to an infinite distance apart? This is called the $dissociation$ $energy$ of the molecule. (d) For the molecule CO,
the equilibrium distance between the carbon and oxygen atoms is 1.13 $\times$ 10$^{-10}$ m and the dissociation energy is 1.54 $\times$ 10$^{-18}$ J per molecule. Find the values of the constants $a$ and $b$.

Transcript

00:01 Problem 7 .35 is about a model for the potential energy function between two atoms in a diatomic molecule.

00:12 This is called the leonard jones potential.

00:15 It's pretty commonly used for modeling how diatomic molecules work.

00:23 And we want to find a bunch of things out about this.

00:26 So what is the force as a function of the distance, first of all? and when we want to sketch the graphs of the force and the potential energy, we want to find the equilibrium disk separation r.

00:54 I'm sorry, this is part of part a.

01:11 Just ignore the b here.

01:13 Anyway, the equilibrium distance and just determine whether it's a stable or unstable equilibrium.

01:20 Assuming that the molecules are at the separation of our equilibrium, we want to know what the dissociation.

01:27 Energy is, which is the energy that must be added to the system in order to get them to be freed from each other.

01:36 And then we're told some information about carbon dioxide, and we want to find what are the values of a and b for that.

01:47 So, the force is going to be entirely in the radial direction.

02:03 It's going to be the negative derivative of the potential energy.

02:14 So we see here this is r to the negative 12th.

02:19 So we'll have a negative and then another negative.

02:25 And so this will be 12a over r to the 13th.

02:34 And then this will have a negative sign here...

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