A 12.0-g block of aluminium is heated from 80 K up to 180 K, under constant volume conditions. We have to find the amount of heat required according to (a) the classical theory of heat capacity and (b) Einstein’s quantum theory of heat capacity. The Einstein temperature for aluminium may be taken to be 290 K.

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Molar mass of aluminium is 26.98 g. Therefore, the number of moles of aluminium atoms in 12.0 g of aluminium will be

(a)

According to the classical kinetic theory the internal energy per mole at temperature T is given by . Therefore, change in internal energy of 12.0 g of aluminium when it is heated from 80 K to 180 K at constant volume will be

(b)

According to Einstein’s theory of heat capacity the molar internal energy is given by the function

Therefore, change in internal energy of 12.0 g of aluminium when it is heated from 80 K to 180 K at constant volume will be