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For interacting systems, it may not be possible to solve the statistical mechanics exactly or even in any well-controlled approximation scheme. Sometimes, however, it is possible to make useful progress just using dimensional analysis. (a) Consider a gas of classical particles of mass m in a volume V interacting via a pair potential U(r). Sketch a typical form of U(r). Suppose that for a particular class of substances, U(r) has the form U(r) = ε u(r/σ). The meaning of this is that the energy scale is set by ε and the length scale by σ. For example, different gases might have different hard core radii σ and binding energies ε. Working in the canonical ensemble, show that all substances in this class have the same equation of state when expressed in suitably scaled variables. i.e. p* = Π(v*, T*), where starred quantities are scaled pressure, volume per particle and temperature, and Π is a function that dimensional considerations alone cannot determine. (b) Show that if there is a critical point for this class of fluids, then pcvc/Tc is a constant independent of the particular fluid. (c) The above theory works very well for gases like Neon, but significant departures are observed at low temperatures for gases like He and H2. These are due to quantum effects. By dimensional analysis, show that the scaled equation of state should have the form p* = Π(v*, T*, η) where Π is some function that we cannot determine by these dimensional considerations alone, and η = h /[σ mε ]. (d) Classically, the critical temperature kBTc/ε is a constant, but quantum mechanically, it depends on η. By plotting an appropriate graph (you may use a computer if you wish) estimate the critical temperature of the isotope 3He. You will need the following data. 4He H2 kBTc/ε 0.529 0.896 σ (Å  2.56 2.93ε/kB (K) 10.2 37 You should justify what assumptions you find it necessary to make. The mass of the proton is 1.67 × 10−27 kg, and kB = 1.38 × 10−23 J K−1. (e) In part (c), why did we use σ for the length scale and not v1/3 (v ≡ V/N)? |
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