9780030839931, Chapter 25, Problem 2P

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Chapter 25, Problem 2P Problem Equation of State of the Harmonic Crystal Derive the form (25.7) for the pressure in the harmonic approximation, by substituting the harmonic form (25.6) of the internal energy U into the general thermodynamic relation (25.5). (Hint: Change the integration variable from T to X = and integrate by parts with respect to X, taking due care of the integrated terms.) Step-by-step solution Step of 4 Solution: The Helmholtz free energy F is defined as, Here U is the internal energy, T is the temperature and S is the entropy. Since the entropy S and the internal energy U related by, Therefore pressure P will be, P Here F is the Helmholtz free energy, V is the volume and T is the temperature. Step of 4 The pressure entirely in terms of the internal energy will be, P = (1) If the small oscillation assumption is valid, then the internal energy of an insulating crystal should be accurately given by the result of the harmonic approximation: (2) Step of 4 To obtain the pressure P in the harmonic approximation, substitute equation (2) in equation (1), we get, P av = (3) P Substitute, T' = (k) (4) So, dT' = (5) Putting the value of equation (4) and equation (5) in equation in the third term of equation (3), we get, P (6) Step of 4 The third term of the above equation is in the form of standard integration and the value of that integral will be equal to zero. So, the pressure P in the harmonic approximation will be, P (7) Therefore, P = + + - ks