# Heat Capacity and Specific Heat ```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
```