Here is the derivation of the kinetic energy as it relates to temperature.

This video can be found on YouTube at https://youtu.be/sMnMDqnUgbQ

The kinetic energy of a single molecules is

If you have "N" number of molecules then the formula becomes

If you are counting your groups of molecules by the mole, then the formula for a collection of molecules can be written as

Where,

A "dozen" is a collection of 12 items. Simularly, a "mole" is a collection 6.022 x 10²³ items.

In thermodynamics, the kinetic energy of a gas is more commonly referred to as the gas's internal energy. The variable "U" is used for the internal energy. This means the formula now becomes.

This is also expressed as

What's important about this formula is the relationship between temperature and internal energy. Often the problems will want to know if the internal energy changes. e.g. Does the internal energy increase or decrease? The change in temperature is used to indicate a change in the internal energy. If the temperature goes up, then so does the internal energy.

Note: When talking about temperature you have to use "Kelvins" and not "degrees Kelvin." Kelvin is the only temperature scale that relates to energy levels. If you double the temperature in Kelvins, then you also double the gas's internal energy. Going from 50K to 100 K means the internal energy of the gas is doubled. Going from 50° C to 100° C is means the energy changes by a factor of 1.86 instead of 2. This is because the Celsius scale is NOT based on absolute thermal energy like the Kelvin.

In thermodynamics the gas in a container is "the system." Mechanical work can by done on the system or by the system or on the surroundings or by the surroundings. Making the volume small is what happens when work is done on the gas by the surroundings. The prepositions on and by are highlighted because they will come to mean something mathematically.

There is another way to do "work" on the gas. And that is by adding thermal energy to removing thermal energy from a gas.

With mechanical energy work can be done to increase a system's energy. With unorganized collections of molecules, the work that is done can now be mechanical and/or thermal.

Terminology: Sometimes the term "sink" is used instead of "reservoir." Using this terminology, the two reservoirs above would be described as a, "hot sink," and a, "cold sink."

When less dense air rises it is because more dense air is heavier and falls. As it falls the less dense air is pushed out of the way -upwards. Warmer air is less dense and rises. When it rises the molecules take the thermal energy with them.

The mechanism called conduction occurs when thermal energy is transfered from one location to another by a third object.

The rate the energy is transfered is called the power and is measured in Watts in the SI system. The unit of a Watt is named after James Watt and therefore abbreviated with a capital "W."

When two bodies are connected by a third, solid body, the equation describing the relationship between the two bodies is

"k" is the thermal conductivity constant in W/(m•K). This can be found in a table like the one on wikipedia. This table describes substances and how well they conduct thermal energy.
"A" is the cross sectional area of the beam connecting the two bodies in m².
"L" is the length of the beam connecting the two bodies in meters.
"ΔT" is the difference in temperature in either Celsius or Kelvin.
"Q" is the thermal energy in Joules.
"t" is the time in seconds.

"Q/t" describes how quickly the energy is transfered. This makes "Q/t" the "power."

Thermal energy from the Sun reaches the Earth without any molecules to carry or transfer the energy. This is radiation. Instead energy is carried by electromagnet waves.

Emissivity defines how reflective on object is. If a body reflects all of the light, and thermal energy, that radiates to it, then it will have an emissivity of 0. If a body absorbs all of the light and thermal energy that radiates to it, then it has an emissivity of 1. This object will be black and if it truly absorbs all of the energy radiated onto it, then it is called a "black body." Every thing else lies on the spectrum between 0 and 1. Dark objects are closer to 1 and light objects are closer to 0.

Good absorbers of thermal energy are also good emitters of thermal energy. As an example black clothes get warm more quickly than white clothes. Black clothes also cool down more quickly than white clothes when moved to the shade or indoors.