Boltzmann Distribution

The probabilities that maximize the entropy under a constraint of constant energy obey the Boltzmann probability distribution

[@mortezakhoshbin]

How is it possible someone watch these amazing videos and dont like or comment!

[@PhysicalChemistry]

Haha, I guess not everyone enjoys PChem as much as you and I do! I definitely enjoy the feedback and interaction, though. Thanks!

[@puzzle2047]

So THAT is the equation in your channel icon! So fascinating!

[@PhysicalChemistry]

Yes! And that's the key to everything else, as you'll see if you watch more!

[@melevarck]

Thank you once again for your fantastic lectures! I have a question regarding the energy term: I sometimes see it being regarded as the potential energy of a system and sometimes as the hamiltonian of a system (encompassing the kinetic energy term). Which one is the right one? Or are both valid in different circumstances? This confuses me quite a bit, as for me it only makes sense to regard E_i as the potential energy, since the temperature term in beta already accounts for the kinetic energy of the system.

[@hindh2451]

Amazing

[@PhysicalChemistry]

Thanks

[@kuppersrocky6834]

Superb explanation!! Could we theoretically include conservation of momentum as a third constraint? And if so, would we still be able to derive the Maxwell-Boltzmann distribution from there?

[@PhysicalChemistry]

You can indeed use the Boltzmann distribution as a starting point to derive the Maxwell-Boltzmann distribution. You don't need to use conservation of momentum. Instead, the Boltzmann distribution can directly give you the distribution of velocities (kzbin.info/www/bejne/nqDRYWicr86qd7s) and then that can be converted to a distribution of speeds (kzbin.info/www/bejne/any1i4eQj7GBe68)

[@lakshaymission548]

Sir Thanks for Uploading these wonderful lectures, also is there way to communicate with you occasionally, something like a discord channel.

[@PhysicalChemistry]

You're welcome; I'm glad you appreciate them. I don't have a discord channel, but my email address is on the KZbin channel's about page.

[@luukvandenakker7838]

Hello professor, Does this derivation only hold true for the canonical ensemble due to the particle constraint? I'm really fond of your derivation using constrained optimization. In my textbooks (blundell and schroeder) both distributions are rather poorly derived, so it would be a great help if I could use a similar technique for deriving the Boltzmann distribution for the grand canonical ensemble.

[@PhysicalChemistry]

Great question! As you suspected, the Boltzmann distribution applies only for the canonical ensemble. You can follow a similar approach (still using constrained optimization) to derive the probability distribution for the grand canonical ensemble, but the details will be a little different. There are plenty of books that walk through a derivation of the grand canonical partition function / distribution function, but you'll likely want to look for books with an emphasis on statistical mechanics.

[@luukvandenakker7838]

@@PhysicalChemistry thank you sir! Do you have any recommendations? Wouldn't it also be a future video idea perhaps? Your channel has been a lifesaver and i really have come to love the subject because of it. Many thanks again!

[@PhysicalChemistry]

@@luukvandenakker7838 I'm partial to the book called Statistical Thermodynamics, by McQuarrie. It does have a good derivation of the grand canonical ensemble.

[@PhysicalChemistry]

@@luukvandenakker7838 I would enjoy making a series of videos extending these topics to include more statistical mechanics. That will probably wait until I have the excuse of teaching that course, though.

[@luukvandenakker7838]

@@PhysicalChemistry Well lets hope you will Sir. Thanks!

[@beasthunter3302]

I had a doubt . I am currently studying reaction kinetics .There ,I got to know of the Arrhenius equation .It is very closely linked with the Maxwell Boltzman distribution curve .However I am unable to understand the one thing regarding the pre exponential factor .looking at the exponential part we can say that as T approaches infinity the rate of reaction becomes equal to A (the pre exponential factor ) However we know that A itself is dependent on Temperature .It takes into account the collision frequency which itself increases with temperature .So how can we say ,based on this ,that the reaction constant reaches a finite limit ? Can you please explain .

[@PhysicalChemistry]

You're definitely right that Arrhenius kinetics are closely related to Boltzmann distribution. We often use the approximation that the Arrhenius prefactor, A, is a constant. But that's not really true, as you point out. A does have some temperature dependence. But A doesn't increase without bound as the temperature increases. It depends not only on the collision frequency, but also on geometric factors and collision cross sections. When the temperature gets too high, the probability of a collision resulting in a productive reaction actually decreases. A full description of how the Arrhenius prefactor depends on temperature can get quite complex.

[@rachealbrimberry8918]

@@PhysicalChemistry so were looking for an ideal temperature to measure collision frequency.

[@PhysicalChemistry]

@@rachealbrimberry8918 Not necessarily. The question is a little unclear. You can measure collision frequency at any temperature you like. There's not really an ideal temperature at which to measure it. The collision frequency does change with temperature, and the Boltzmann distribution can help us understand (in part) how that is true, as is discussed in some later videos on the kinetic theory of gases.

[@psychemist]

What the 'i' and 'j' indexes represt here Please help i am a newbie there but almost got it.

[@psychemist]

Please please anyone reply fast i got the exam coming ahead

[@PhysicalChemistry]

The index represents the state of the system. This could be an energy level, or confirmational state, or anything that distinguishes the particular variation that you want to calculate the probability of.

[@psychemist]

@@PhysicalChemistry what purpose is 'i' and 'j' are they the same ?

[@PhysicalChemistry]

@@psychemist They are just indices. I could have used any letter. But in this video I used j for a specific state and i inside summations where all possible states are being summed over

[@rachealbrimberry8918]

Why would you want to maximize the entropy.

[@PhysicalChemistry]

That's actually the subject of the previous ~20 videos in the sequence. You've started with the punchline! The short version is that a system will be found most often in the state that has the highest multiplicity -- or the most different ways of existing. It turns out that entropy is a way of measuring the multiplicity. So we maximize the entropy in order to predict what state we will find a system in. For the full explanation, back up to around this video: kzbin.info/www/bejne/pWrJoqpoeLh-jNE and watch in sequence from there.