Above the critical temperature \u03b1 0 you should find G 11 G 22 which is a simple

Above the critical temperature α 0 you should find g

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. Above the critical temperature ( α > 0), you should find G 11 = G 22 , which is a simple consequence of rotational invariance. Below the critical temperature, ( α < 0), choose the state with h m 1 ( r ) i = q | α | and h m 2 ( r ) i = 0 in zero field. You should find G 11 6 = G 22 , and determine both functions in 3 dimensions. 3. We compute physical properties of the van der Waals gas at different points in the phase diagram in the P , T , plane. For all of the questions below, you may use the “Landau” approximation to the Gibbs free energy derived in class, and obtain answers to leading non-vanishing order in x T - T c and/or y P - P c . In this approximation, we can write the Gibbs free energy, G ( P, T ) G ( P, T ) N = G c N + y - x 2 b m + x + 4 by 24 b 2 ! m 2 + a 1944 b 5 m 4 (4) 2
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where the value of m has to be chosen at each P and T to minimize G . Further, this optimum value of m is related to the particle density, ρ by 1 ρ = 1 ρ c + m + 1 6 b m 2 (5) (a) Sketch the phase diagram in the P , T plane. Find the equation of the line representing the liquid-gas transition. (b) Compute the discontinuity in the density, ρ L - ρ G , along this line. How does this discontinuity vanish upon approaching the critical end-point at P = P c and T = T c ?
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