We are going to make a very rough estimate of how much pressure must be applied

Question

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? Work is equal to force times distance parallel to the force (Fdx) but, by multiplying and dividing by the perpendicular surface area, this can be changed into pressure times volume (-pdV). Because solids are elastic, the change in volume is proportional to the change in applied pressure, dV = -Cdp, and the constant C is typically 10^-17 m^5/N. With this back- ground, calculate the work done on a solid as the external pressure is increased from 0 to some final value p_f. 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). What is a typical value for the variation of u_0 with pressure, partial differential u_0/partial differential p, at constant temperature and atmospheric pressure in a solid?

Explanation / Answer

6. given, potential energy reference level for a typical solid, uo = -2 eV

a. given atomic spacing, a = 0.2 nm

so, in 1 metre length, number of atoms = 1/0.2*10^-9 = 5*10^9 atoms

so toal number of atoms in 1 m^3 = N = (5*10^9)^3 = 1.25*10^29 atoms

b. Work done = Potential energy raise per atom * number of atoms

W = uo*N = 2.5*10^29 eV

1 eV = 1.6*10^-19 J

so, W = 4*10^10 J

c. given, dV = -CdP

here C = 10^-17 m^5/N

now, work done = -pdV

dW = -pdV = C*P*dP

so integrating from 0 to W and 0 to Pf

W = C[Pf]^2/2

d. for Uo to become +ve

W = 4*10^10 J = C*[Pf]^2/2 = 10^-17[Pf]^2/2

Pf = 8.944*10^13 Pa

e. W = uoN

dW = N*dUo = C*P*dP

so, dUo/dP = CP/N

as temperature is constant and P = Patm = 1.01*10^5 Pa

dUo/dP = 10^-17*1.01*10^5/1.25*10^29 = 8.08*10^-42 J/Pa

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