The following question concerns the constant pressure heatcapacity, C p . a) Der

Question

The following question concerns the constant pressure heatcapacity, Cp.

               a) Derive the following relationship

               Cp = - T(¶2G/¶T2)p                                                                                          (1.1)

               b) The temperature dependence of the molar Gibbs free energy of apure chemical substance is often fit to an expression of theform

               G = G° + aT + bT2 + cT lnT +d/T                                                                         (1.2)

where G° is the molar Gibbs free energy for some referencestate, and G°, a, b, c, and d are constants. Use eq 1.1 tofind the constant pressure molar heat capacity for a substancewhose Gibbs free energy is given by eq 1.2.

Explanation / Answer

Cp = dH/dT P dG = -SdT + VdP dG/dTP = -S dS/dTP =Cp/T            Recall that dS = Q/T      SodS/dTP = (1/T)Q/TP = Cp/T Cp/T = dS/dTP = -d[dG/dTP]/dTP =-d2G/dT2P Cp = -T*d2G/dT2P b) dG/dT = a + 2bT + c + c*lnT - d/T2 d2G/dT2= 2b + c/T - 2d/T3 Cp = -T*d2G/dT2 = -2bT - c +2d/T2

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