In chemical engineering, we frequently need to evaluate the amount of heat required to raise the temperature of a given material from a value T1 to T2. If C(T) is the specific heat of the material, the amount of energy needed is ≡T T mC T T 1 2 ( )d , where m is the mass of the material, since the specific heat is the energy required to raise the temperature of unit mass of the material by unit temperature. The average specific heat Cav is given by [ ] ∫ ( ) ( ) − T T C T T T T 1 2 d 2 1 . The specific heat of a material is given, in J/kg K, as where T0 is the reference temperature of 100 K. For 1 kg of the material, compute the total energy, in joules, needed to raise the temperature from 100 to 1000 K. Also determine the average specific heat over this temperature range. Use the trapezoidal rule, and reduce the segment size, starting with 100 K, until the results remain essentially unchanged with further reduction.