Science:Math Exam Resources/Courses/MATH 180/December 2017/Question 06

MATH 180 December 2017

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[hide] Question 06

The Buckingham potential $$V(r)={\frac {A}{e^{Br}}}-{\frac {C}{r^{6}}},\quad r>0,$$ where ${\textstyle A}$, ${\textstyle B}$ and ${\textstyle C}$ are positive constants, is a refinement of the Lennard-Jones potential that describes the potential energy ${\textstyle V(r)}$ of a diatomic molecule where the atoms are a distance ${\textstyle r}$ apart. Find and justify all vertical and horizontal asymptotes of ${\textstyle V(r)}$.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

[show] Hint 1
Near the vertical asymptotes, the function tends to (plus or minus) infinity.

[show] Hint 2
The horizontal asymptotes are approached by the function for very positive or negative values of ${\textstyle r}$.

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[show] Solution

Observe that if $V(r)={\dfrac {A}{e^{Br}}}-{\dfrac {C}{r^{6}}}$ approaches positive/negative infinity, then either ${\dfrac {A}{e^{Br}}}$ or $-{\dfrac {C}{r^{6}}}$ approaches positive/negative infinity, as $r\to a$, for some $a$. In other words, if $r=a$ is a vertical asymptote of $V$, then $r=a$ is a vertical asymptote of ${\dfrac {A}{e^{Br}}}$ or of $-{\dfrac {C}{r^{6}}}$. Recall that the exponential function can never be zero, so ${\dfrac {A}{e^{Br}}}$ is finite for any $r$. On the other hand, the rational function $-{\dfrac {C}{r^{6}}}$ is not defined at $r=0$, which makes $r=0$ a candidate for a vertical asymptote of $-{\dfrac {C}{r^{6}}}$ and hence of $V$. In fact, $$\lim \limits _{r\to 0^{+}}V(r)=\lim \limits _{r\to 0^{+}}\left({\dfrac {A}{e^{Br}}}-{\dfrac {C}{r^{6}}}\right)=A-\lim \limits _{r\to 0^{+}}{\dfrac {C}{r^{6}}}=-\infty .$$ So ${\textstyle r=0}$ is the only vertical asymptote. For horizontal asymptotes, since ${\textstyle V(r)}$ is only defined for ${\textstyle r>0}$, we find that $$\lim \limits _{r\to +\infty }V(r)=\lim \limits _{r\to +\infty }\left({\dfrac {A}{e^{Br}}}-{\dfrac {C}{r^{6}}}\right)=0-0=0.$$ So the only horizontal asymptote is ${\textstyle V=0}$. Answer: the vertical asymptote is ${\textstyle {\color {blue}r=0}}$ and the horizontal asymptote is ${\textstyle {\color {blue}V=0}}$.