Question

23. Show that the general thermodynamic relation between the constant-pressure heat capacity (Cp) and constant-volume heat capacity (Cv) can be given as: 24. Calculate the ratio of the number of molecules in the lowest two rotational states in a gas of H2 at 50 K (take the inter-atomic distance 1.05 Å)

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Answer 1

From the first law of thermodynamics we get Given equation is C_p - C_v [p - (partial differential U/partial differential v)_t [partial differential v/partial differential t]]_p Now dQ = dU + p dv - (1) Let the internal energy u is a function of T & V U = U (T, V) therefore dU = (partial differential u/partial differential T)_v dT + (partial differential u/partial differential v)_T dv - (2) Putting this value in eq^n (1) we get dQ^bar = du + p dv = (partial differential u/partial differential T)_v dT + (partial differential u/partial differential c)_T dv + p dv dQ^bar = [p + (partial differential u/partial differential T)_T] dv + (partial differential u/partial differential T)_v dT - (3) dividing the equation (3) by DT we get C_p = (dQ^bar/dt)_p \r\n C_p = (partial differential u/partial differential T)_v + [p + (partial differential u/partial differential v)_T] middot (partial differential u/partial differential T)_p - (4) for isochoric process v = constant therefore dv = 0