Hello, this video is the first in a series
of videos that talk about ideal gases. I think we all know or think we know what an ideal gas is,
we deal with it all of the time. However, it turns out that we can treat
some other things as ideal gases as well including electron gases and photons,
but we will get to that in due course. We start by reminding ourselves
of the ideal gas law PV = NRT. This was first presented
by Jacques Charles in the long time ago. He lived from 1746 to 1823 and he studied
the behavior of gases among other things. More formally, ideal gases exist in the domain where
the Maxwell-Boltzmann limit holds. And this is a limit where the particles
are on average are far enough apart that they don't really interact with each
other except briefly during collisions. The collisions are necessary to
establish thermodynamic equilibrium, but on average,
they don't feel the effect of each other. So just a reminder that this is
the Maxwell-Boltzmann partition function. And if you factor out the gamma term,
then you get this sum called Q, the third equation on the slide,
which is a sum over k, which is a quantum state index e to the minus beta
epsilon k where epsilon is the energy. This is called the molecular
partition function and it's a function only of molecular
properties and temperature through beta which is, remind you is 1 over kt. So first we assume that we could
separate the translational and internal energy modes of the molecule. And this is the substance
making up the gas and this is a reasonable assumption if there are no
external fields operating on the gas. So we write that epsilon, the energy is the translational
energy plus the internal energy. Now if we just note in
general that e to the sum of two numbers a plus b is equal to the
product of e to the a times e to the b. And if a and
b are independent of each other, that is changing a doesn't
necessarily affect b, then we can write the sums
as shown in the last slide, the sum over a and b of e to
the minus a times e to the minus b. And that can split into two sums the sums
a e to the minus a and e to the minus b. So that allows us to write the partition
function at the log of the partition function more formally as e to the minus
alpha times the sum of the e to the minus beta epsilon for translation
times the sum for internal motion. So that the length of
the Maxwell-Boltzmann partition function, lane of the partition function is e to
the minus gamma times the translational molecular partition function nv times
the internal molecular partition function. And it turns out that the same
argument applies to internal modes so that we can write the internal partition
function as the product of the rotation vibration and electronic modes. And so therefore the problem of
finding the fundamental relation for ideal gases devolves to finding the cues. Remembering that
the the fundamental regression is equal to minus k in
the partition function for whatever particular
representation that you were in. Thanks very much and have a great day.