Discussion
theoretical definition
One has to be careful when defining temperature not to confuse it with heat. Heat is a form of energy. Temperature is something different. We could begin with a technical definition, but I would prefer to start with a question. How hot is it? The answer to this question (or a question like this) is a measurement of temperature. The hotter something is, the higher its temperature. Therefore I would like to propose the following informal definition …
Temperature is a measure of hotness.
Don't go looking for "hotness" in any dictionary (except maybe a slang dictionary). There is no such word. I made it up. Despite this fact, I believe that most speakers of English will understand this neologism. Unfortunately this won't do for scientific purposes. Quantities in science are typically defined operationally (through the process by which they are measured) or theoretically (in terms of the theories of a specific discipline). We will begin with a theoretical definition of temperature and end with an operational definition.
Let's review what you should already know.
1. A system possesses energy if it has the ability to do work.
2. Energy comes in two basic forms: the kinetic energy of motion and the potential energy of position.
3. Energy is conserved; which is to say, it cannot be created or destroyed. When one form of energy decreases another form must increase.
The archetypal example of this is the rock at the top of a hill. Due to its height above the bottom of the hill it possesses gravitational potential energy. Give it a push and it will start to roll. If we assume the ideal situation of a closed system where no energy is lost on the way down, then the rock's initial potential energy will equal its final kinetic energy.
Now take the archetypal example one step further. Assume the rock crashes into a wall. Neither the rock nor the wall are made of rubber, so the rock comes to a halt. Now it appears as if we have violated the law of conservation of energy. The kinetic energy is lost and nothing has come along to replace it. Where has the energy gone?
The answer to this question is inside the rock (and inside the wall). The energy has been transformed from the external energy visible as the motion of the rock as a whole to the internal energy of the motion of the invisible parts that make up the rock (and the wall). The two energies are identical in size, but different in appearance. External energy is visible because it is organized. The translational kinetic energy of a rock is due to coordinated motion. All the parts move forward together. The rotational energy is also coordinated. The parts all rotate together around the center of mass. In contrast, the internal kinetic energy of a rock is invisible since the pieces are so small and numerous and their motion is completely uncoordinated. Their motions are statistically random with a mean value of zero making the energy invisible to macroscopic beings like us. Potential energy can also exist in external and internal forms. I won't provide you with an example here but I will say that external potential energy is relatively obvious. Internal potential energy is somewhat obscure.
If you believe that objects can have internal energy, then it shouldn't be much of a stretch to believe that they can exchange this energy. This is known as thermal contact. The irreducible bits and pieces of objects responsible for carrying the internal energy are known as atoms — from the Greek "α τομή" [a tomi] meaning "can't be cut" — but belief in atoms is not a necessity. It just makes life easier. (Surprisingly, much of thermal physics and thermodynamics was worked out before atoms were generally considered real.) Since we are dealing with large numbers of atoms in uncoordinated motion, there will be times and places where the transfer of internal energy will run in one direction and different times and places where the transfer of internal energy will be in the opposite direction. Since the numbers are so unimaginably large, we really don't care about what happens to any one atom. All that we can observe in such cases is the net or overall transfer of internal energy. This is known as heat. If the net exchange of internal energy is zero; that is, if no heat flows from one region to another; then the whole system is said to be in thermal equilibrium.
Since heat is defined as internal energy in transit from one place to another, nothing can be said to have heat or store heat. Instead we say that heat flows from one place to another. The direction is indicated by the sign in front of the number. Use "+" when heat flows into a place and "−" when it flows out. Heat can travel left, right, up, down, forward, or backward, but that's not usually the way it's described. Heat is a form of energy and energy is scalar, therefore specific directions and angles and all the rest of that vector stuff should be ignored (initially, anyway).
Heat is a form of energy and the unit of energy is the joule, therefore heat should be measured in joules. Before this fact was known, however, heat had its own special units; like the calorie and the british thermal unit (Btu), for example. These units still pop up from time to time. But enough about heat. Let's get back to temperature. What is it?
Two regions have the same temperature when there is no net exchange of internal energy between them.
In it's most primitive sense, temperature is what determines the direction of heat flow. The net transfer of internal energy between regions in thermal contact is out from the region with the higher temperature and into the region with lower temperature. In more concise terms, heat flows from hot to cold. That's the theoretical definition of temperature.
operational definition
