Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / Section G / 31 / c

The hard-core chain does not undergo an energetic low-temperature collapse. For allowed configurations the only finite energy is bond stretching. If $k{\sigma }^2\gg k_{B}T$, bond lengths remain near $\sigma$ and temperature mostly changes the small bond-length fluctuations, not the distribution of angular conformations. The end-to-end distance therefore fluctuates over an expanded self-avoiding-coil distribution even at low temperature, with an approximate scale $\sigma (N-1{)}^{\nu }$ and $\nu \simeq 0.588$ for sufficiently long three-dimensional chains.

For $N=10$, the scaling is qualitative and finite-chain effects matter. An equilibrated distance-versus-sweep trace is stationary, with correlated excursions; a histogram of that trace is a distinct plot, not a curve of probability density against simulation step. Neither the temperature in kelvin nor the spring constant is specified, so an exact histogram cannot be calculated. The illustrative figure compares like-for-like coil traces rather than asserting a unique numerical simulation result.

At room temperature, the same hard-core exclusion remains temperature independent. If the spring remains stiff, the conformational distribution remains broadly similar to the low-temperature expanded-coil distribution. Finite bond-length fluctuations increase with $\sqrt{k_{B}T/k}$ and can slightly enlarge the length scale and distribution width. The radial bond measure also favours lengths slightly above the harmonic minimum.

The comparison depends on the dimensionless stiffness $k{\sigma }^2/(k_{B}T)$, not on the word “room” alone. There is no attractive energy in this model to produce a low-temperature globule and high-temperature coil transition. Monte Carlo decorrelation speed can change with temperature and proposal scale, but the premise already stipulates adequate equilibrium sampling in both cases.