Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 1.10-1

THE CONDITIONS OF EQUILIBRIUM

Physics — Statistical Mechanics · fundamental-equations-postulates

Problem

The following ten equations are purported to be fundamental equations of various thermodynamic systems. However, five are inconsistent with one or more of postulates II, III, and IV and consequently are not physically acceptable. In each case qualitatively sketch the fundamental relationship between SS and UU (with NN and VV constant). Find the five equations that are not physically permissible and indicate the postulates violated by each. The quantities v0v_0, θ\theta, and RR are positive constants, and in all cases in which fractional exponents appear only the real positive root is to be taken. a)  S=(R2v0θ)1/3(NVU)1/3b)  S=(Rθ2)1/3(NUV)2/3c)  S=(Rθ)1/2(NU+RθV2v02)1/2d)  S=(R2θv03)V3/NUe)  S=(R3v0θ2)1/5[N2VU2]1/5f)  S=NRln⁡(UV/N2Rθv0)g)  S=(Rθ)1/2[NU]1/2exp⁡(−V2/2N2v02)h)  S=(Rθ)1/2(NU)1/2exp⁡(−UVNRθv0)i)  U=(v0θR)S2Vexp⁡(S/NR)j)  U=(Rθv0)NV(1+SNR)exp⁡(−S/NR)\begin{aligned} &a)\; S = \left(\frac{R^2}{v_0\theta}\right)^{1/3}(NVU)^{1/3}\\ &b)\; S = \left(\frac{R}{\theta^2}\right)^{1/3}\left(\frac{NU}{V}\right)^{2/3}\\ &c)\; S = \left(\frac{R}{\theta}\right)^{1/2}\left(NU + \frac{R\theta V^2}{v_0^2}\right)^{1/2}\\ &d)\; S = \left(\frac{R^2\theta}{v_0^3}\right)V^3/NU\\ &e)\; S = \left(\frac{R^3}{v_0\theta^2}\right)^{1/5}[N^2VU^2]^{1/5}\\ &f)\; S = NR\ln(UV/N^2R\theta v_0)\\ &g)\; S = \left(\frac{R}{\theta}\right)^{1/2}[NU]^{1/2}\exp(-V^2/2N^2v_0^2)\\ &h)\; S = \left(\frac{R}{\theta}\right)^{1/2}(NU)^{1/2}\exp\left(-\frac{UV}{NR\theta v_0}\right)\\ &i)\; U = \left(\frac{v_0\theta}{R}\right)\frac{S^2}{V}\exp(S/NR)\\ &j)\; U = \left(\frac{R\theta}{v_0}\right)NV\left(1 + \frac{S}{NR}\right)\exp(-S/NR) \end{aligned}

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