# 15.1: Heat Capacities

When we supply heat energy from a bunsen burner or an electrical heating coil to an object, a rise in temperature usually occurs. Provided that no chemical changes or phase changes take place, the rise in temperature is proportional to the quantity of heat energy supplied. If q is the quantity of heat supplied and the temperature rises from T1 to T2 then

$q = C * (T_{2} – T_{1})$

OR

$q = C * (\Delta T)$
where the constant of proportionality C is called the heat capacity of the sample. The sign of q in this case is + because the sample has absorbed heat (the change was endothermic), and (ΔT) is defined in the conventional way.

If we add heat to any homogenous sample of matter of variable mass, such as a pure substance or a solution, the quantity of heat needed to raise its temperature is proportional to the mass as well as to the rise in temperature. That is,

$q = C * m * (T_2 – T_1)$

OR

$q = C * m * (\Delta T)$

The new proportionality constant C is the heat capacity per unit mass. It is called the specific heat capacity (or sometimes the specific heat), where the word specific means “per unit mass.”

Specific heat capacities provide a convenient way of determining the heat added to, or removed from, material by measuring its mass and temperature change. As mentioned [|previously], James Joule established the connection between heat energy and the intensive property temperature, by measuring the temperature change in water caused by the energy released by a falling mass. In an ideal experiment, a 1.00 kg mass falling 10.0 m would release 98.0 J of energy. If the mass drove a propeller immersed in 0.100 liter (100 g) of water in an insulated container, its temperature would rise by 0.234oC. This allows us to calculate the specific heat capacity of water:

98 J = C × 100 g × 0.234 oC
C = 4.184 J/goC

At 15°C, the precise value for the specific heat of water is 4.184 J K–1 g–1, and at other temperatures it varies from 4.178 to 4.219 J K–1 g–1. Note that the specific heat has units of g (not the base unit kg), and that since the Centigrade and kelvin scales have identical graduations, either oC or K may be used.

Example $$\PageIndex{1}$$: Specific Heat of Water

How much heat is required to raise the temperature of 500 mL of water (D = 1.0) from 25.0 oC to 75.0 oC, given that the specific heat capacity of water is 4.184 J K–1 g–1?

Solution:

q = 4.18 J/goC × 500 g × (75.0 - 25.0)
q = 104 500 J or 104 kJ.

Table $$\PageIndex{1}$$ Specific heat capacities (25 °C unless otherwise noted)

Substance phase Cp(see below)
J/(g·K)
air, (Sea level, dry, 0 °C) gas 1.0035
argon gas 0.5203
carbon dioxide gas 0.839
helium gas 5.19
hydrogen gas 14.30
methane gas 2.191
neon gas 1.0301
oxygen gas 0.918
water at 100 °C (steam) gas 2.080
water at T=[1] liquid 0.01°C 4.210
15°C 4.184
25°C 4.181
35°C 4.178
45°C 4.181
55°C 4.183
65°C 4.188
75°C 4.194
85°C 4.283
100°C 4.219
water (ice) at T= [2] solid 0°C 2.050
-10°C 2.0
-20°C 1.943
-40°C 1.818
ethanol liquid 2.44
copper solid 0.385
gold solid 0.129
iron solid 0.450