Question

Heat capacity of solids at low temperatures follows Debye's law, c~T^3, where c is the heat capacity, and T is the temperature in kelvin. Assume that for some particular sample 10J of heat is required to increase the sample's temperature twice from 10K to 20 K. How much heat would be needed to heat that sample from 20K to 40K?

Please show steps/assumptions.

Thanks,

Answer 1

The amount of heat required to change the temperature of a substance of heat capacity C is

The heat capacity of solids at low temperature is directly proportional to the cube of the temperature.

where k is the constant of proportionality.

The total heat required to change the temperature of a solid from temperature T_{​​​​​​i} to T_{​​​​​​f}​​​ can be determined by integrating the above differential

Given that a sample require 10J energy to change temperature from 10K to 20K

The heat required to change the temperature of the sample from 20K to 40K is