5.112 Principles Of Chemical Science Worksheet With Answers - Problem Set #4 Solutions - 2011 Page 14
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16Problem set #4 solutions for 5.112, Fall 2011
Page 14 of 16
The ratio of methane molecules that have the average energy E to the number of
molecules that have twice the average energy, 2 E is 3.17.
Question 17 (out of 4 points)
The following figure is a plot of the potential energy versus separation distance for two
neutral noble-gas atoms:
Lennard-Jones potential for the He-He (black) and Ar-Ar dimers.
Commonly referred to as the Lennard-Jones potential, the equation that describes these
functions is:
ε
σ
where
and
are constants that depend on the system of interest. The L-J potential is a useful
model because it captures the essential physics of interaction at long and short separation:
r
attraction between the atoms at long
and repulsion at short.
V(r)
a)
Determine which terms in
correspond to attraction and repulsion. (2 points)
⎡
⎤
12
6
⎛
σ
⎞
⎛
σ
⎞
⎢
⎥
Let us look at the equation from the question: V (r) = 4 ε
−
⎝ ⎜
⎠ ⎟
⎝ ⎜
⎠ ⎟
⎢
r
r
⎥
⎣
⎦
The attractive interaction is going to stabilize the formation of the molecule, so it has the
6
⎛
σ
⎞
negative sign in front of it. The attractive interaction term is therefore:
⎝ ⎜
⎠ ⎟
r
The repulsive interaction is going to destabilize the dimer, so it will have the positive sign
12
⎛
σ
⎞
in front of it. The repulsive interaction is therefore:
.
⎝ ⎜
⎠ ⎟
r
b)
What does the depth of the potential energy well correspond to (indicated by the black
arrow)? (1 point)
ε )
The depth of the potential energy well,(
is the binding energy between two neutral
noble atoms (or the dissociation energy of the dimer)
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