ฝัง
- เผยแพร่เมื่อ 9 ก.พ. 2025
- Short physical chemistry lecture on average ensemble energy.
The average energy of a system is a weighted average of the energy of each level weighted by its probability. This expression is equivalent to calculating the energy by taking the negative partial derivative of the natural logarithm of the partition function with respect to the inverse temperature beta.
Notes Slide: i.imgur.com/taQ...
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Short question: It isn't covered in this video, but Helmholtz FE is related to Q by: ΔA = -kTln(Q_f/Q_i). Using conformational change of a macromolecule as an example, how is it possible that the partition function changes upon simple change in shape along a given reaction coordinate, i.e. torsional angle. And, strangely enough, ensemble averages, not partition functions, are usually employed as Q_f and Q_i when using this formula. Any discourse on this would be great.
Really helpful. Thankyou so much!
Thanks Shivam.
Maybe I missed the reasoning behind taking the partial of ln(q)? This was the beginning of the derivation of the average energy in terms of the partition function, correct? If this is true, why did it start with ln(Q)? Thanks for your help!!
Hi John. This was a case of starting with knowing the answer. Starting with that expression, we can show that it is equivalent to the average energy. So this is more of a proof than a derivation. If you wanted to do it in the traditional way, just reverse the steps and the logic at each step. The final result ends up having 1/Q dQ/dB, which by the chain rule equals d(lnQ)/dB, which I suppose chemists / physicists believe is an easier expression to remember.
TMP Chem Thank you for your fast response to my question!! That makes sense now thank you!
since you're taking the "partial derivative" of ln(Q) with respect to beta, there must be a factor you consider to be constant, what would that be?
by the way, this was an excellent video!
Great question. If you study statistical mechanics at a higher level, you'll learn that there are multiple different kinds of "ensembles" (microcanonical, canonical, grand canonical, etc.) and the variables which each depend on are slightly different. In this chapter, I'm always assuming we're working with the canonical ensemble, whose partition function is a function of the temperature (or beta), volume, and number of particles.
TMP Chem wow, it amazes me how vast chemistry is... thanks for the reply!
The more in depth you study a topic, the more you realize there is to know that you don't yet know. The process of becoming an expert in something is to develop a greater and greater appreciation for the depth of your own ignorance. This is why the wise are so hesitant to make claims outside their expertise, and the foolish claim they know it all.
I'm still not sure what ensemble energy is. is ensemble E the same with ensemble? what's the relationship b.w ensemble(or ensemble E?) and ?
Ensemble energy is just the expectation value of energy for the system, or what we call "internal energy" in the macroscopic classical thermodynamics taught in the rest of this course. So ensemble energy is , which is U in the rest of the course.
Thank you. your answer is simple and clear.