Relation between kinetic energy and temperature

[johnathon]

Where does the 3/2 come from?
$$\frac{1}{2} mv^2 = \frac{3}{2} kT$$

 

[sophiecentaur]

A particle can move in any of three directions (that's where the 3 comes from), with kt/2 being the kinetic energy carried by motion on each the x,y or z dimensions.
This link gives a short and sweet bit of book work. That Hyperphysics site is good for many things, actually.

 

[Doc Al]

Where does the 3/2 come from?
$$\frac{1}{2} mv^2 = \frac{3}{2} kT$$

There are three translational degrees of freedom, each contributing 1/2kt to the total energy. This from the equipartition theorem.

 

[Dadface]

Taking it back a step the 3kT/2 can be found by equating the two ideal gas equations,one being obtained experimentally(PV=RT) the other being obtained theoretically using kinetic theory(PV=Nmc bar squared/3)

 

[sardar]

The relation between kinetic energy and temperature can be explained by the kinetic theory of gases. According to this theory, temperature is a measure of the average kinetic energy of the particles in a substance. In a gas, the particles are in constant motion and their kinetic energy is directly proportional to their velocity.

The equation \frac{1}{2} mv^2 represents the kinetic energy of a single particle, where m is the mass of the particle and v is its velocity. When we consider a collection of particles, we can calculate the average kinetic energy by taking the sum of the kinetic energies of all the particles and dividing by the number of particles.

Now, the kinetic theory of gases also states that the average kinetic energy of gas particles is directly proportional to the temperature of the gas. This means that as the temperature increases, the average kinetic energy of the particles also increases.

When we combine these two concepts, we get the equation \frac{1}{2} mv^2 = kT, where k is a constant known as the Boltzmann constant. This equation shows that the kinetic energy of gas particles is directly proportional to the temperature of the gas.

The 3/2 in the equation comes from the fact that gas particles move in three dimensions (x, y, and z), and this affects their kinetic energy. Without going into too much detail, the kinetic energy of a particle in three dimensions can be broken down into three components, each representing the kinetic energy in one dimension. This results in an overall kinetic energy that is three times the kinetic energy in one dimension, which is where the 3/2 comes from.

In summary, the 3/2 in the equation represents the relationship between the average kinetic energy of gas particles and the temperature of the gas, taking into account the particles' movement in three dimensions.