Sketching this function for potential energy of two atoms in a molecule

[Nano-Passion]

Homework Statement

The potential energy of two atoms in a molecule can sometimes be approximated by the Morse function, where r is the distance between the two atoms and A, R, And positive constants with S<<R. Sketch this function for 0<r<∞...

##U(r) = A ( (e^{(R-r)/(S)}-1)^2 - 1) ##

This is just one part of the problem but it is the part bothering me so I'll leave out the rest of the problem. This should be an easy problem, as indicated by the (*).

The Attempt at a Solution

##U(r) = A ( (e^{(R-r)/(S)}-1)^2 - 1) ##
I rewrite ##e^{(R-r)S}## as ##e^{R/S-r/S}##

The only way I can think about graphing this is to ignore all other things except for ##e^{R/S-r/S}## and manipulating R/S

Since S<<R, ##R/S→∞##

But then that just gives me nonsense! What else can I do?

Expanding ##(e^{R/S-r/S})^2## does not help either. Ignoring all constants and values except for r does not help either because ##e^{R/S-r/S}## = e^-r which will give me the wrong sketch.

 

[Chestermiller]

Plot U/A as a function of r/R, with R/S as a parameter for each curve. Try a value of R/S equal to 10 to start with. Play with different values of R/S.

 

[Nano-Passion]

Plot U/A as a function of r/R, with R/S as a parameter for each curve. Try a value of R/S equal to 10 to start with. Play with different values of R/S.

As a function of r/R? I'm not following and I don't understand the motivation behind this. =/

 

[Chestermiller]

As a function of r/R? I'm not following and I don't understand the motivation behind this. =/

You'd like to be able to display all the information about this function on a single graph. You can do this if you use the dimensionless variables U/A, r/R, and R/S. When r/R = 1, U/A = -1; thus, this point should be the same on all the curves, irrespective of R/S. Rewrite the equation in the form:

(U/A) = (exp(-R/S (r/R -1)) -1)² -1

You can explore the behavior in the region r close to R by expanding in a Taylor series in (1-r/R):

(U/A) ~ ((1 - (R/S)(1-r/R)) -1)² -1 = -1 + (R/S)²(1-r/R)²

At very small r/R, you approach

(U/A) ~ exp (2R/S)

At very large r/R, you approach

(U/A) ~ -2 exp(-R/S (r/R))

 

[Nano-Passion]

You'd like to be able to display all the information about this function on a single graph. You can do this if you use the dimensionless variables U/A, r/R, and R/S. When r/R = 1, U/A = -1; thus, this point should be the same on all the curves, irrespective of R/S. Rewrite the equation in the form:

(U/A) = (exp(-R/S (r/R -1)) -1)² -1

You can explore the behavior in the region r close to R by expanding in a Taylor series in (1-r/R):

(U/A) ~ ((1 - (R/S)(1-r/R)) -1)² -1 = -1 + (R/S)²(1-r/R)²

At very small r/R, you approach

(U/A) ~ exp (2R/S)

At very large r/R, you approach

(U/A) ~ -2 exp(-R/S (r/R))

Thanks.