Heat (Energy) Transfer and Thermal Equilibrium

"I know not with what weapons World War III will be fought, but World War IV will be fought with sticks and stones."

Albert Einstein

As we have seen in the zeroth law of thermodynamics, when two objects are placed in contact heat (energy) is transferred from one to the other until they reach the same temperature (are in thermal equilibrium). When the objects are at the same temperature there is no heat transfer.

Since heat is a form of energy it can be measured in Joules. Historically, since heat was believed to be a separate form of energy, thermodynamics was developed using a unit of energy unique to heat - the calorie. The relationship between Joules and Calories is known as the Mechanical Equivalence of Heat,

1 Calorie = 4.184 Joules

Definition of the (kilo)-Calorie:

1 kcal = Heat required to raise 1 kg of water through 1 ⁰C

Note that the calorie which descibes the energy content on food labels is actually 1 kcal.

Definition of the Btu:

The British Thermal Unit (Btu), still in use in the USA, is defined in a similar way to the calorie, but using British units,

1 Btu = Heat required to raise 1 lb of water through 1 ⁰F

Note that,

1 kcal = 3.968 Btu = 4184 Joules = 3086.7 ft.lbs

Given these definitions, the amount of heat required to change the temperature of a mass, m by is given by,

thermeq_eqn1

where c is the Specific Heat (capacity) of the substance of which the mass is composed. The units of c are J/kg.⁰C. Notice that for water c = 1 kcal/kg.⁰C (= 4184 J/kg.⁰C), so that the value of c for other substances is always relative to that of water.
Also, due to the similar way in which calories and Btus are defined if the specific heat of a substance is 'x' kcal/kg.⁰C it is also 'x' Btu/lb.⁰F.

The Heat Capacity of an object, C, is defined by,

thermeq_eqn2

the heat needed to raise an object by one degree.

Knowledge of specific heats and/or heat capacities and the fact that energy must be conserved allows us to determine the equilibrium temperature of two objects initially at different temperatures by demanding that,

Heat lost by hot object = Heat gained by cold object

where we ignore heat gained or lost from/to the surroundings. The study of heat gained and lost in this manner is often called Calorimetry.

fire divider

Change of Phase and Latent Heat

For most substances there are three "phases" - solid, liquid and gasesous. During a phase change the structure (not the composition) of the substance changes.

Typically in a solid the composite atoms or molecules are arranged in a rigid lattice structure - each atom or molecule is largely fixed in space.
When the substance melts turning into a liquid, the lattice structure breaks down and each atom/molecule is free to move through the space occupied by the liquid, but the substance still maintains a definite surface.
On vaporizing the substance no longer has a definite surface, the atoms/molecules behave like a gas, having very little interaction with each other.

These phase changes occur at definite temperatures, e.g. water freezes at 0 ⁰C and boils at 100 ⁰C.

To undergo a change of phase heat must be added or extracted from the substance. While the heat is added or subtracted there is no change in temperature, therefore the heat required to complete the phase change is the called Latent (or hidden) Heat.

The heat required to melt (or freeze) a substance is the Latent Heat of Fusion, L_F

Latent heat of fusion of water/ice = 80 kcal/kg = 333 kJ/kg

The heat required to vaporise (condense) a substance is the Latent Heat of Vaporization, L_V

Latent heat of vaporisation of water/water vapor = 540 kcal/kg = 2256 kJ/kg

thermeq_diag

When considering the process reaching thermal equilibrium using Heat lost by hot object = Heat gained by cold object if either object is likely to undergo a phase change we must include the amount of heat gained or lost in the phase change,

thermeq_eqn3

Note that for a system in thermal equilibrium if ice and water are present the temperature must be 0 ⁰C. If heat is added, ALL the ice must melt before the temperature will rise above 0 ⁰C.

thermoanim2

There is this farmer who is having problems with his chickens. All of the sudden, they are all getting very sick and he doesn't know what is wrong with them. After trying all conventional means, he calls a biologist, a chemist, and a physicist to see if they can figure out what is wrong. So the biologist looks at the chickens, examines them a bit, and says he has no clue what could be wrong with them. Then the chemist takes some tests and makes some measurements, but he can't come to any conclusions either. So the physicist tries. He stands there and looks at the chickens for a long time without touching them or anything. Then all of the sudden he starts scribbling away in a notebook. Finally, after several gruesome calculations, he exclaims, 'I've got it! But it only works for spherical chickens in a vacuum.'

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu