Partition Function

The sum of all the Boltzmann factors for a system is called the partition function. This is an important quantity not only because it keeps the Boltzmann probability distribution normalized, but also because it can provide some information about the effective number of accessible states.
There is an unfortunate couple of algebra errors toward the end of the video, I apologize. The last result should be
exp(+E_min / kT) Q
See the comments below by @AbRijkeboer and @terminator3598, and thanks to both of them for spotting the mistake.

[@MusafirHoonYaro]

Prof. Stuart - you videos are absolutely a class of their own! You are a very effective teacher.

[@PhysicalChemistry]

I'm very happy you liked them

[@atanunath]

11:52 you forgot the kT in Exp(Emin/kT) and continued writing just Exp(Emin). Extremely beautiful lecture though, so clear and elegant.

[@PhysicalChemistry]

You're absolutely right, that is a mistake. I need to fix this lecture to repair that mistake. In the meantime, perhaps your comment can help prevent any confusion for others, so thanks for the correction.

[@JFBond-zs8xf]

Reviewing a course in statistical thermodynamics taken at McGill 39 years ago, I was having trouble with the partition function. Your lecture made it crystal clear, thank you! I will return to your site.

[@PhysicalChemistry]

Great! I'm glad the video helped. See you next time you need a refresher

[@mortezakhoshbin]

Beauty of all science is hidden in PChem. Great as other lectures

[@PhysicalChemistry]

Hopefully not too hidden! My goals is to make it plain to see.

[@venuslee2085]

These lectures saved my grade :) Thank you so much

[@PhysicalChemistry]

You put in the work to save your own grade, but I'm happy to have provided an assist.

[@itsawonderfullife4802]

very insightful. thank you prof.

[@AbRijkeboer]

Your answer to @terminator3598 still leaves me with exactly the same question. The lecture shows an apparent undue sign change (as a second thought correction) effectively leading to the number of accessible states being the summation of Exp(-(Ei + Emin)/kT) rather than Exp(-(Ei - Emin)/kT), taken into account @atanunath below as well. Furthermore, supposedly in the numerator of Pj the exponential term changes to Exp(-(Ej - Emin)/kT)? NB: Excellent video's with very clear examples and explanantions. Many thanks for making these available.

[@PhysicalChemistry]

Yes, I see. You're absolutely right. There are a couple of different algebra errors. My math, and my explanation, both got pretty sloppy at the end of this video. I apologize. Since Q = Σᵢ exp(−Eᵢ / kT) but we want to calculate the number of accessible states as (call it Q') Q' = Σᵢ exp( −(Eᵢ−Emin)/kT ) = exp(+Emin/kT) Σᵢ exp(−Eᵢ/kT) then Q' = exp(+Emin/kT) Q so if you want to gauge the number of accessible states using the properly calculated partition function when the ground state is not zero, just multiply it by exp(+Emin/kT) And yes, if you're changing the zero of energy in one place, you need to do so everywhere you use energies, including in the equation for probability, as you point out

[@AbRijkeboer]

Thank you, also for the vast number of videos you made on high level topics. Very educational. I am a 74 year old chemical engineer (MSc) trying to fresh up and catch up on physical chemistry matters.

[@PhysicalChemistry]

@@AbRijkeboer You're welcome. I'm impressed with your passion for learning!

[@terminator3598]

Why did you take the reciprocal of exp(Emin/kT) when you took it out of the summation? Shouldn’t the adjusted partition function be exp(Emin/kT)Q?

[@PhysicalChemistry]

You are absolutely right. The adjusted partition function should definitely be exp(+Emin/kT) Q. Thanks for spotting that, and sorry for my sloppy algebra (This is an edit to a previous comment. In my prior response, I hadn't watched through my clumsy and erroneous "fixing" of the equation.)

[@Afreen891]

Sir please make videos of fermi dirac and bose einsten equations

[@PhysicalChemistry]

Thanks for the suggestion. Those topics were more advanced than I wanted to include for this course, but they are on the list of videos to record when I go more in depth into stat mech topics.

[@davelochen]

The lectures are excellent and for someone who is just starting PChem, it would be helpful if the order of the playlist could be specified. Thanks.

[@PhysicalChemistry]

Thanks, glad you like them. If you want to start from the very beginning, the very first video in the sequence is the one on Mutually Exclusive Events (kzbin.info/www/bejne/rJW2iIB_nZeilac) Each individual playlist should already be in order (don't shuffle!). If you're asking about what order the playlists themselves should be in, I'll add that to the channel's "About" page. Good question / suggestion, thanks!

[@Ghostrider-ul7xn]

Could you make a Playlist separately for statistical mechanics if there isn't one already? Your lectures are excellent but it would really save my time if you could organize them into one.

[@PhysicalChemistry]

The videos are for a physical chemistry course, but it's entirely structured around statistical mechanics (see here for an explanation: kzbin.info/www/bejne/hpLKe6SOiamMkMk ). So if you start with the first video (kzbin.info/www/bejne/hYe5lJiGntSoaKs ) and go through in order, clicking on the image that pops up at the end of each video, it will walk you through the full course. There are also 31 different playlists for different topics / chapters. See the Playlists page for the channel.

[@mortezakhoshbin]

I think at 13:00 a KT in the denominator is missed in the last equation. Emin/KT

[@PhysicalChemistry]

You're absolutely right. Those Boltzmann factors should definitely be exp(-E_min / kT)

[@akif1741]

9:32: I think this is only the case if you know that the partition is 10^10 at T→∞. If the partition function is 10^10 at 298K, then I guess there would be a LOT more accessible energy levels

[@PhysicalChemistry]

You're right that the partition function changes (and increases) with temperature. If it is 10^10 at 298 K, it will be larger (perhaps MUCH larger) as T → ∞. The basic point being made, though, is that Q tells you how many states can be accessed, qualitatively, at that temperature. That's true regardless of whether Q = 1 or Q = 3 or Q = 10^10 or larger.

[@lookthesky9632]

Hi Professor Steven, may I ask what is difference between partition function and Boltzmann Distribution? I am just confused those terms. Thank you very much.

[@PhysicalChemistry]

The partition function is the one that is usually labeled Q or q (or sometimes Z). It is the *sum* of a bunch of exponentials. It is kind of a count of how many states are important. The Boltzmann distribution (short for Boltzmann probability distribution) tells you the *probability* of being in a particular state. It is usually labeled p_i or something similar. The partition function appears in the denominator of the Boltzmann probability distribution.

[@lookthesky9632]

@@PhysicalChemistry I really appreciate it!

[@strangeWaters]

I know the footage could be mirrored but I choose to believe that you learned to write backwards to make these videos

[@scuitinos]

Good afternoon Professor, could you explain how I arrived at the Freundlich's isotherm equation? On the other hand, can you comment on which book you recommend to study adsorption? Greetings and thank you very much for your channel, do you have an email to write to you?

[@luthomampunye497]

2. Consider a system of non-interacting gas particles in a grand canonical ensemble at temperature T = 300 K and chemical potential = 10 kJ/mol. Calculate the partition function (E) for this system.