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It says in my course notes that a particle has so-called "kinetic energy"

$E=\frac{3}{2}kT=\frac{1}{2}mv^²$

Where does this formula come from? What is k?

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3 Answers

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The above equation solves for the average kinetic energy of a gaseous particle at a given temperature. k is known as Boltzman's constant, $k_B = 1.3806503 × 10^{-23}\frac{m^2kg}{s^2K} $ and is equal to the ideal gas constant divided by Avagadro's number, $\frac{R}{N_A}$.

So where does the equation come from?

The short answer: The equation above is derived from the ideal gas law as well as the experimentally verified fact that 1 mole of any gas at STP occupies a constant volume (measured to be 22.4L). We can use this relation with the mass of the given particle to prove that average kinetic energy is proportional only to temperature of the gas.

The long answer: This page provides an in-depth derivation of the formulas above.

Hope this helps!

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This follows from the equipartition theorem. The equipartition theorem states that in thermal equilibrium, the average energy of each degree of freedom (each independent way the system can move) is $k_B T/2$, where $T$ is the temperature and $k_B$ (or just $k$) is called the Boltzmann constant. There are three independent directions in which a gas particle can move (three independent components of velocity), so the total kinetic energy is $3\times k_B T/2$. It is important to realise that this is a statistical formula that is only true on average for a large number of particles in equilibrium: each individual particle may in fact have a kinetic energy different from this. The Boltzmann constant provides a link between the microscopic and macroscopic worlds, by relating the typical average energies of microscopic particles to the energies required to change the temperature of a macroscopic mass by a measurable amount.

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I've looked at the notes. There are several things going on here, some of them not stated. And we have to make some assumptions. One is that the average kinetic energy of a particle moving in a direction $v$ is $mv^2/2$, where $m$ is the mass of the particle. This isn't too hard to prove, but doing so would take us far afield.

Another assumption is that there is an average energy per particle in a gas and that this average depends on temperature. By a rather complicated argument one can show that the average energy per particle is $kT/2$, where $T$ is the absolute temperature in kelvins and $k$ is a constant known as Boltzmann's constant.

The argument then goes as follows: The particle can move in three independent directions. That is, its motion will change its $x$, $y$, and $z$ components independently. So we have to multiply our formulas by three, which almost give us the formula you have above.

The difference is the $v^2$ term. That's defined as the sum the average energies in the three independent directions, $v^2 = v_x^2 + v_y^2 + v_z^2$. Since all directions are equivalent and independent, we assume that each contributes equally to the sum. Thus $v^2 = v_x^2/3$, where I've picked $v_x$ for convenience, I could have used any of the $v$'s.

Putting all the bits together gives us the formula above.

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