Undergraduate physics homework often includes an inspiring but rather unilluminating order-of-magnitude estimation of the height of the tallest possible mountain on Earth. Here, we attempt to solve the problem from first principles and enjoy our consequent edification. The key point of failure in a mountain is the elastic limit: after a certain height, plastic deformation occurs at the base. Elsewhere, we estimated elastic modulus from the hydrogen atom when computing the pitch of squealing chalk and the age of the solar system. The elastic limit is slightly different, since it describes a structural deformation that enables plastic flow; but we are not, under any circumstances, allowed to simply look up the number and avoid edification. (If you’re curious why “edification” is a common theme, ask my unofficial officemates about the running joke.)
At the molecular level, the maximum height \(h\) of a mountain is set by $$\begin{aligned} mgh = E_p, \end{aligned}$$ where \(m\) is the mass of a rock molecule and \(E_p\) is the energy required per molecule to induce plastic flow, where the lattice structure of the solid is rearranged into a disordered state. Typically, the undergraduate physics professor pulls out the value of \(E_p\) from thin air; we refuse to commit such sacrilege against desert island physics. Since silicon comes from sand which comes from mountains, we consider a mountain made of SiO\(_2\) with a molecular mass of 60 amu. To estimate the amount of energy required to disassemble a lattice — while not changing the kinetic energy (temperature) — we shall compute the excess energy required to melt a crystal once it is already at its melting temperature.
The SiO\(_2\) lattice has each Si atom bonded to four oxygens and each O bonded to two silicons; there is, for simplicity, only one microscopic configuration and thus zero entropy (\(S=0\)). As a liquid, this is replaced by a disordered phase. We can model this as allowing the molecules to move into “hole” states detached from the lattice. For \(N\) molecules, if some fraction \(f\) are allowed to move into the hole state, then the entropy is \(S = k_B \log \binom{N}{fN}\). Applying Stirling’s approximation in the large-\(N\) limit, \(S \sim -k_B N[f\log f + (1-f)\log(1-f)]\). Hence, the entropy change per molecule is $$\begin{aligned} \Delta s \sim -k[f\log f + (1-f)\log(1-f)]. \end{aligned}$$ If we take the liquid to be in the maximally disordered state, then \(\Delta s \sim k_B \log 2\).
To recover the energy to transition from solid to liquid, we also need to estimate the kinetic energy required in heating. Once again, we turn to the hydrogen atom for inspiration. From the Bohr model, we know that the Rydberg energy required to break a bond is 13.6 eV at the Bohr radius of \(0.5~\mathrm{Å}\). Linearly increasing this distance by adding two shells to reach silicon on the second row of the periodic table, we assume an atomic radius of \(1.5~\mathrm{Å}\). Since SiO\(_2\) is tightly formed crystal lattice, this is likely a good measure for the bond length as well. Fitting two atoms next to each other to form a bond, the binding energy of a single bond given by Coulomb falloff is $$\begin{aligned} E_\mathrm{bond} \sim \mathrm{Ry}\cdot\frac{0.5~\mathrm{Å}}{2\times1.5~\mathrm{Å}} \sim 2~\mathrm{eV}. \end{aligned}$$ To detach a single SiO\(_2\) from the lattice, we need to break multiple bonds. Since each oxygen is bonded to one silicon atom in the individual molecule but four silicon atoms in the lattice, removing a single molecule from the lattice requires around 3/4 of its bonds to be broken. Eliminating double-counting, an average of 3/8 of the lattice bonds must be broken to obtain individual SiO\(_2\) molecules. The remaining 5/8 fraction consists of the two bonds of an SiO\(_2\) molecule; consequently, around two bonds per molecule must be broken, giving a molecular binding energy of 4 eV. (Empirically, this is quite close: the experimental value is around 6.5 eV per molecule, so this kind of analysis is on the right track.)
Disturbingly, completely disassembling the SiO\(_2\) lattice would easily transform solid rock into a gas, but we only need to estimate the latent heat associated with transitioning from solid to liquid. Hence, we don’t truly want the binding energy. From everyday experience, we know that melting a material does not typically change its density too much — only, perhaps, 1% to 10%. Taking the geometric mean to select the middle path, we can assume that the weaker intermolecular forces can increase the bond length by about \(3\%\). Replacing the strong polar covalent bonds of the lattice with these slightly larger bonds gives $$\begin{aligned} E_\mathrm{bond} \sim \mathrm{Ry}\cdot\frac{0.5~\mathrm{Å}}{2\times1.5~\mathrm{Å}}\cdot\left(1-\frac{1}{1.03}\right) \sim 0.1~\mathrm{eV}. \end{aligned}$$ Hence, our 4 eV molecular binding energy can be replaced by 0.2 eV. Combining with our entropy estimate, this implies that the energy per molecule required for plastic flow is $$\begin{aligned} E_p \sim T\Delta s \sim \left(\frac{0.2~\mathrm{eV}}{k_B}\right)(k_B \log 2) \sim 0.1~\mathrm{eV}. \end{aligned}$$ Returning to the mountain, this implies a maximum height of \(h \sim \frac{E_p}{mg} \sim\) 10 km. For comparison, the tallest mountain from base to peak (Mauna Kea) is 10.2 km tall. Of course, our error bars are decently large, especially due to the estimate of the increased length of intermolecular bonds. If we took the full range of 1% to 10%, we would find heights that range from 5 to 50 km. Nevertheless, this is quite good for building our solution all the way up from the hydrogen atom!