Past exam of the quantum condensed matter course of the University of Cambridge / 2009 / Section A / A1

The Stokes–Einstein relation gives the translational diffusion coefficientat $T=298\mathrm{K}$. For isotropic three-dimensional Brownian motion, the mean squared displacement is $⟨|\Delta \mathbf{r}{|}^2⟩=6Dt$. Thus after one hour the root mean square displacement isThe phrase “on average” needs a convention: the mean displacement vector is zero; the mean displacement magnitude of three-dimensional Brownian motion is $⟨|\Delta \mathbf{r}|⟩=4\sqrt{Dt/\pi }=1.22\mathrm{mm}$, obtained by integrating the three-dimensional Gaussian distribution. These are displacements, not the total microscopic path length. The estimate assumes dilute unconfined water, a spherical hydrodynamic radius and constant temperature.

For a spherical micelle with $N$ tails, core radius $r$, molecular tail volume $v$ and head-group area $a$, volume and surface counting give $Nv=4\pi r^3/3$ and $Na=4\pi r^2$. Consequently $a=3v/r$. A tail cannot span a core radius larger than its maximum length $\ell$, so the amphiphile packing parameter condition isThis is a minimum area, not a maximum. The printed request reverses the geometric inequality. The limiting value corresponds to fully extended tails with $r=\ell$; larger head groups permit smaller spherical aggregates in this simple packing argument. The supplied data alone impose no finite maximum head-group area, and geometry alone does not guarantee thermodynamic micelle formation.

In the planar Debye–Hückel approximation, a surface potential decays as $\psi (x)={\psi }_s{e}^{-\kappa x}$, with $\kappa \propto \sqrt{I}$ at fixed temperature and solvent permittivity; $I$ is the ionic strength. For a monovalent salt, reducing its concentration by ten reduces $\kappa$ by $\sqrt{10}$. Calibrating the first decay from the supplied potential ratio, and assuming unchanged ${\psi }_s$, givesHere the printed meV unit is interpreted as the electrostatic potential energy of a singly charged probe; the same ratio calculation applies to potentials in mV.

This numerical answer uses an effective exponential decay calibrated from the data. The inferred initial Debye length is $5/\log (35/6.6)=3.00\mathrm{nm}$, whereas the standard dilute aqueous-electrolyte expression at room temperature gives about $0.96\mathrm{nm}$ for $0.1\mathrm{M}$ monovalent salt. The numbers therefore are not consistent with that absolute ideal-electrolyte model. Moreover a spherical colloid has an additional radius-dependent geometric factor, and its surface potential need not remain constant when salt is changed. Without a surface boundary condition and geometry, the new potential is not uniquely specified beyond the stated planar, fixed-potential approximation.