Heat as Energy
When describing heat as energy it's supposed to be used to describe when energy is transferred from one place to another. So, this thing called heat flow is an energy transfer only happens because a difference of temperature is at hand. Like, if you had a piece of ice in your hand. The ice is melting because you are losing heat energy that is being put into the ice. That's right, perhaps you thought it was the other way around, but it's not. The two objects (your hand, and the piece of ice) are trying to reach an equilibrium with each other.
Specific Heat

The amount of energy needed to change the temperature of a certain object is different depending on the object. Every substance has a certain value for the amount of heat required to change its temperature. This is called the specific heat, known c.

 

Q is the heat energy. M is the mass and T the temperature.
This is the more familiar form of the equation.
Conservation of Energy

This is just like anything else. Except that it's been expanded to fit in with heat. If something in a system loses heat, then something must gain heat right? Yeah . . that would be about it. So if you look at the equation you'll see that before equals after.

 

Problem:

You drop a 0.05 kg piece of hot coal at a temperature of 200oC into a bucket of water that has 0.4 kg at 20oC. The final temperature 22.4oC, what is the specific heat of the metal?

You just use that formula up there!

(0.05 kg)(cx)(200oC - 22.4oC) = (0.4 kg)(4186 J/kgxCo)(22.4oC - 20oC)

cx = 453 J/kgxCo
Latent Heat

Usually when you add heat to an object the temperature increases. But sometimes when heat is added the temperature does not rise! Like when a phase change occurs. Say you have a piece of ice and you were trying to melt it. The ice would reach a temperature of 0oC and not raise anymore in temperature. It would stay at that temperature until all the ice had melted. This is a special hidden heat, called Latent Heat. That's the L in the equation.

Q = mL

There are several things, the Latent Heat of Fusion and the Latent Heat of vaporization. The first meaning used when the phase change occurs from a solid to a liquid. The latter occurs when a liquid turns to a gas.

Heat Transfer by Conduction
Pretend you're sitting in front of a fire, and you stick a rod in the fire. You feel the rod get warmer and warmer. You think the energy is flowing through the rod. But actually, the atoms that make up the rod are vibrating faster and faster. The faster they vibrate the hotter it is. Slowly the atoms collide with each other until it reaches your hand. The rate at which conduction occurs depends on the type of material.

You can find the rate of flow of heat with this equation:
Convection
Convection is heat transferred by the movement of a heated surface. Like when you stick your hands over a fire. You feel the expanded air move past your hands. Radiators work with convection. Hot air raises to the top of the room, pushing the cool air down, which the radiator takes in and spits out as warm air pushing the cool air down again. It sets up a continuous air current, so warm air is constantly being circulated.

You can find the rate at which heat is transferred by a fluid to a surface area with this equation: A being the surface area. h is the convection coefficient. h depends on several factors, like the shape of the surface and whether or not the surface is horizontal or vertical.

Radiation
When you place yours hands to the side of a fire. You feel the heat even though you're not touching the fire, or feeling the air currents. All objects radiate energy through electromagnetic waves. The type of radiation that occurs with heat is infrared radiation.
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