This introductory physical chemistry course examines the connections between molecular properties and the behavior of macroscopic chemical systems.

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This introductory physical chemistry course examines the connections between molecular properties and the behavior of macroscopic chemical systems.

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Module 4

This module connects specific molecular properties to associated molecular partition functions. In particular, we will derive partition functions for atomic, diatomic, and polyatomic ideal gases, exploring how their quantized energy levels, which depend on their masses, moments of inertia, vibrational frequencies, and electronic states, affect the partition function's value for given choices of temperature, volume, and number of gas particles. We will examine specific examples in order to see how individual molecular properties influence associated partition functions and, through that influence, thermodynamic properties. Homework problems will provide you the opportunity to demonstrate mastery in the application of the above concepts.

- Dr. Christopher J. CramerDistinguished McKnight and University Teaching Professor of Chemistry and Chemical Physics

Chemistry

Let's review the key concepts that we saw this week.

You've had some chance to do some self assessments.

There'll be a homework where you have a chance to put some things into practice

with actual maybe numerical examples, but I'll try to emphasize the conceptual

things that are going to be useful as we move forward.

So first, a partition function for an atom involves only translational and

electronic partition functions. That's different from a diatomic system

where other things contribute, but for an atom, that translational partition

function, which, and this is true for all ideal gases.

Depends only on the mass of the atom, or the molecule if it's another ideal gas

that isn't atomic, and a convention that's chosen for a volume of a box that

allows the solution of the particle in a box Shrodinger equation.

For an atom, we looked at an electronic partition function as involving

Typically, at most, the ground state and maybe the first excited state, and their

respective degeneracies. In the absence of a populated excited

electronic state, a monotonic ideal gas has a molar internal energy of

three-halves RT. So one-half rt for each direction of

translation and the heat capacity is three halves r because heat capacity is

the derivative of internal energy with respect to temperature.

The monatomic ideal gas partition function is consistent with the ideal gas

equation of state. As all ideal gases depend on volume the

same way. And as pressure is determined from a

differentiation of the partition function with respect to volume, indeed all ideal

gases, monatomic, diatomic, polyatomic are consistent with the ideal gas

equation of state. When we go to the diatomic gas, instead

of only translation and electronic contributions to the total energy.

There's also a rotational contribution and a vibrational contribution.

A difference between the atomic case and they diatomic case is in the atom we took

the ground state electronic energy to be zero.

In fact, in the diatomic case, we take zero to be the separated atoms and the

ground state energy is the energy that's recovered as they're allowed to bond.

So that is minus The dissociation energy, which is the opposite, the energy

required to split them apart. And we also made the simplifying

assumption that we just wouldn't worry about excited states, so there is no

population of states above the ground state.

The vibrational partition function for a diatomic can actually be determined as a

convergent sum, so it's a sum that has a solution as it goes from one to infinity.

And the ground state vibrational energy is taken to be equal to the zero-point

vibrational energy, one half plunks constant times the vibrational frequency.

The vibrational temperature, which is a convenient construct that allows us to

think about What will contributions be relative to an external temperature to

the heat capacity that arise from vibration for instance.

BUt in any case it's the vibrational frequency divided by Boltzmann's

constant. At temperatures below that vibrational

temperature, the contribution to the molar internal energy is just the zero

point energy. And to the heat capacity it's nothing.

And then as we warm up towards and ultimately through the vibrational

temperature, we contribute an additional RT to the internal energy and an

additional R to the heat capacity. The rotational modes also can be thought

of as having associated with them a temperature So the rotational

temperatures defined differently and it's shown on this slide, H bar squared

divided by twice the moment of inertia times Boltzmann's constant.

And the reason we might want to inspect the rotational temperature is, that we

can determine a useful rotational partition function by switching a sum to

an integral. As long as that rotational temperature is

much lower than the temperature that we're exploring for our gas.

That gives rise to very dense energy levels, and hence, we can integrate.

And when we do that for a diatomic, you get a contribution of RT to the molar

internal energy, you get a contribution of R to the molar heat capacity.

Increasing degeneracy but and the dense levels associated with rotation mean that

the population of the rotational levels rise beginning from the ground state and

then ultimately begin drop off as the energies go higher and higher.

In the absence of populating an electronic excited state, which we've

said we really won't consider. 0r vibrational excited states.

Then the diatomic has a total molar internal energy of 5 halves RT: 3 halves

from translations and 1, or two halves, from rotations.

And the molar heat capacity, the temperature derivative of that, 5 halves

R. Rotation for polyatomic ideal gases has

equivalent looking partition functions to the diatomic if the polyatomic is linear,

so it just has a different moment of inertia.

And when it's a nonlinear then there are spherical, symmetric, or asymmetric top

partition functions that one can look up. They have sort of a common t to the 3 as

temperature dependence we talked about. And the rotation that arises in those

polyatomics that aren't linear, we contribute rt and r to the internal

energy and the molar heat capacity when it is linear, so 2 halves in each case.

When it's non-linear, you get three-halves, because there are three

different unique rotations. So three-halves rt and three-halves r,

for the non-linears. And finally the vibrations in the

polyatomic. There are many instead of one, like in a

diatomic. They each contribute.

And so you get a product of partition functions or a sum of energies.

And one uses the same expressions as in the diatomic.

But summed or products of. Alright, well, that was a lot of bullets.

It might seem like there are a lot of critical concepts.

Hopefully many of them are tied together, because a lot of them have to do with

distinguishing between one atom, two atoms, three atoms.

Are they linear? Are they nonlinear?

But if you feel comfortable with it, we're well equipped then to move on.

And we will be moving on, we're going to continue to make motion on the tracks.

And we're actually going to dive into classical thermodynamics to some extent.

We're still going to have a molecular picture which is going to inform our

treatment of thermodyamics, because because we're chemists.

but we're going to take a look at the first law of thermodynamics with a

molecular perspective. See you then.

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