We are going to make a very rough estimate of how much pressure must be applied to a typical solid to compress it to the point where the potential energy reference level $u_0$ of the individual atoms becomes positive. Ordinarily, for a typical solid, $u_0$ is around -0.2 eV .
(a) If the interatomic spacings are typically 0.2 nm , how many atoms are there per cubic meter?
(b) Roughly, how much work (in joules) must be done on one cubic meter of this solid to raise $u_0$ to zero?
(c) Work is equal to force times distance parallel to the force ( $F \mathrm{~d} x$ ) but, by multiplying and dividing by the perpendicular surface area, this can be changed into pressure times volume ( $-p \mathrm{~d} V$ ). Because solids are elastic, the change in volume is proportional to the change in applied pressure, $\mathrm{d} V=-C \mathrm{~d} p$, and the constant $C$ is typically $10^{-17} \mathrm{~m}^5 / \mathrm{N}$. With this background, calculate the work done on a solid as the external pressure is increased from 0 to some final value $p_f$.
(d) With your answer to part (c) above, estimate the pressure that must be exerted on a typical solid to compress it to the point where $u_0$ becomes positive (Figure 4.4, top right).
(e) What is a typical value for the variation of $u_0$ with pressure, $\partial u_0 / \partial p$, at constant temperature and atmospheric pressure in a solid?