### Integration Matcher

Match the charts of these functions to the charts of their integrals.

### Brimful

Can you find the volumes of the mathematical vessels?

### Graphic Biology

Several graphs of the sort occurring commonly in biology are given. How many processes can you map to each graph?

# Lennard Jones Potential

##### Stage: 5 Challenge Level:

The Lennard Jones potential has several features which might make it a suitable model for reality. This is best realised by looking at the plot of the function:

(a) As the separation of the two atoms increases, the attraction between them increases and tends to zero at infinite distance. This is sensible, since as two atoms approach each other from a large separation, their potential energy slowly drops as they are attracted together. Mathematically this is seen by the fact that both terms in the potential energy expression tend to zero as $r$ tends to infinity.

(b) There is a potential energy minima, which is the stable atomic separation. We know that the atoms ARE attracted to each other by van der Waals attractions, and so it makes sense that there will be some fixed distance apart that they will remain. Mathematically, this is as the turning point of the function, where the gradient is equal to zero.

(c) As the separation of the atoms decreases further, the potential rises sharply, which indicates that it is highly unfavourable for the atoms to be squashed together further. This is seen in reality, where two neutral atoms do not increasingly approach each other indefinitely! Mathematically, this is the $\left(\frac{\sigma}{r}\right)^{12}$ dominating the other term, which leads to a very positive potential as $r$ decreases.

The $W(r)$ potential curve differs from the Lennard-Jones potential as it has a term to the power of $9$ as opposed to $12$. Consequently, the curve still tends to zero at infinity, still has a potential energy minima, and increases sharply with small $r$. Therefore it could well yield a good match with reality with appropriate values of the constants.