# What equation relates average kinetic energy to temperature in the high-temperature limit?

##### 2 Answers

#### Explanation:

The relationship between the kinetic energy (

Which implies that the kinetic energy is independent of the nature of the gas, it only depends on the temperature at which the gas exists.

In short, the **observable kinetic energy** (the kind we know in everyday chemistry and physics)---which is the same as the *ensemble**average of the kinetic energy*---for a **monatomic ideal gas**, is:

#\mathbf(<< barE >> = U = K = 3/2 RT)#

but the average kinetic energy for a ** single** system of monatomic ideal gases is:

#color(blue)(<< E >> = 3/2 nRT)#

So we see that the kinetic energy here **depends on the number of** **s of gas**.

But as we should know, *are* different, though that doesn't matter because this equation asks for

Furthermore, *constant*, never changes.

*Therefore, the identity of a monatomic ideal gas doesn't matter when determining its average kinetic energy. Only its temperature.*

You can read below for an interesting derivation.

**A SINGLE SYSTEM OF MONATOMIC IDEAL GASES**

We have what's called a *single* system of gases, and then we have what's called an *ensemble* of systems of gases. We're focusing on a *single* system for now.

The **average energy** of a ** single** system is defined in Statistical Mechanics as:

#\mathbf(<< E >> = -(del ln Q)/(del beta))# where, for a

monatomic ideal gas:

#Q = [((2pim)/(h^2beta))^("3/2")V]^N/(N!)# #beta = 1/(k_BT)# #k_B# is the Boltzmann constant,#1.38064852 xx 10^(-23) "J/K"# #T# is temperature in#"K"# #m# is the mass of the gas#h# is Planck's constant,#6.626xx10^(-34) "J"*"s"# #N# is the number of gas particles in the system#V# is the volume of the system

The

If we work with the first equation for a bit, we can figure out why the identity of the *monatomic ideal gas* doesn't matter.

**SIMPLIFYING THE FUNCTION THAT WILL BE DIFFERENTIATED**

First, let's figure out

#color(green)(lnQ) = ln[[((2pim)/(h^2beta))^("3/2")V]^N/(N!)]#

#= ln[((2pim)/(h^2beta))^("3/2")V]^N - lnN!#

#= Nln[((2pim)/(h^2beta))^("3/2")V] - lnN!#

#= -Nln(beta)^("3/2") + Nln[((2pim)/(h^2))^("3/2")] + NlnV - lnN!#

#= color(green)(-(3N)/2lnbeta + (3N)/2ln((2pim)/(h^2)) + NlnV - lnN!)#

**AVERAGE ENERGY OF A SINGLE SYSTEM OF MONATOMIC IDEAL GASES**

And now when we take the partial derivative with respect to

#color(green)(<< E >> = -(del lnQ)/(delbeta)) = #

#= -del/(delbeta)[-(3N)/2lnbeta + cancel((3N)/2ln((2pim)/(h^2)) + NlnV - lnN!)^("not " beta)]#

#= -(-(3N)/(2beta))#

#= color(green)(3/2 Nk_BT)#

Something cool is that

#color(blue)(<< E >> = 3/2 nRT)#

which shows that the average kinetic energy of a ** single** system of monatomic ideal gases is only dependent on the number of

**monatomic ideal gases), and the temperature (which we assumed was the same for**

*all***systems in question).**

*all***THE ENERGY WE KNOW AND LOVE??**

Okay, now we can proceed to the ** ensemble** of systems of monatomic ideal gases. The idea is, we can say that a certain number of

#<< barE >> = << E >> / n#

**This is the equivalent of us, observing kinetic energy in real life.**

In other words, this is the kinetic and potential energy we observe in physics and chemistry classes:

#\mathbf(<< barE >> = U = K = 3/2 RT)#