At very low temperatures, debye showed that the heat capacity of solids is approximated by: , where the parameter θd, called the debye temperature, is proportional to the typical vibrational frequencies in that solid. since diamond is very tightly bonded, its vibrational frequencies are large and θd is large (1890k, compared to 92k for solid ar). this debye relation holds pretty well when t < 0.2 θd.
a. calculate δh for heating diamond at constant pressure from 200 to 300k. hint: (for solids and liquids, you can generally assume that cv, m = cp, m, whereas they differ by r for ideal gases.)
b. ten grams of diamond at 200k is put into an adiabatic vessel of fixed volume containing ar gas at 315 k and 1 atm. pressure. when this system (ar gas plus diamond) comes to thermal equilibrium, the final temperature drops to 300k. what is the volume of this adiabatic vessel? (hint: neglect the small volume required for the diamond itself, so that the volume can be directly calculated from the amount of ar gas. since ar is an ideal monatomic gas at these conditions, cv, m = 3/2 r.)