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
But as we should know,
Furthermore,
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
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)#