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Figure. 6.10 shows heat capacity of a crystal vs temperatue in terms of the Debye theory. Here C_(cl) is classical heat capacity Theta is the Debye temperature. Using this plot, find: (a0 The Debye temperature for silver if at a temperature T= 65K its molar heat capacity is equal to 15J (mol.K), (b) the molar heat capacity of aluminum at T= 80K if at T= 250K it is equal to 22.4J(mol.K) (c )the maximum vibration frequency for copper whose heat capacity at T= 125K differs form the classical value by 25%. |
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Answer» Solution :(a) By Dulong and Petit's LAW, the classical heat capacity is `3R= 24.94J//K`- mol e. THUS `(C )/(C_(cl))= 0.6014` From the graph we see that this value of `(C )/(C_(cl))` corresponds to `(T)/(Theta)= 0.29` Hence `Theta=(65)/(0.29)~~ 224K` (b) `22.4J` mole - K corresponds to `(22.4)/(3xx8.314)=0.898`. From the graph this corresponds to `(T)/(Theta)~~ 0.65`. This gives `Theta= (250)/(0.65)~~ 385K` Then `80K` corresponds to `(T)/(Theta)= 0.208` The corresponding value of `(c )/(C_(cl))` is `0.42`. Hence `C= 10.5J//mol e-K` (c ) We calculate `Theta` from the datum that `(C )/(C_(cl))= 0.75 at T= 125K`. The `x`-coordinate corresponding to `0.75` is `0.40`. Hence `Theta=(125)/(0.4)= 3125K` Now `kTheta= ħ omega_(max)` So `omega_(max)= 4.09xx10^(13) rad//sec` |
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