advertisement

1.6.1 Heat Capacity and Specific Heat Many books (e.g. Physics by Resnick and Halliday, p. 468) define the heat capacity C of an object as the amount of heat required to heat it 1 C. Heat is a form of energy. Suppose we heat up the object. Let Q = amount of energy supplied to the object T = change in temperature of the object Then the heat capacity is Q C = T The specific heat of a substance is the heat capacity of a unit mass of the substance. In order to relate heat capacity to a derivative, let T = temperature E(T) = amount of energy needed to raise the temperature of the object from temperature 0 to temperature T If we heat an object up, then Q = E(T + T) - E(T) So C = E(T + T) - E(T) T If we let T approach 0 and take the limit in the previous formula then we would have dE C = dT Suppose that for 1 kg of water one has E(T) = 4200 T - 0.4 T2 + 0.004 T3 + 7 10-5 T4 - 4 10-7 T5 where E is measured in joules and T is the Celsius temperature. (See the file Heat Capacity of Water where I obtained this formula by curve fitting.) What is the heat capacity at 30 C? dE C = dT = 4200 - 0.8 T + 0.012 T2 + 2.8 10-4 T4 - 2 10-6 T4 If one substitutes T = 30 one obtains C = 4192 joules/C. 1.6.2 - 1