Subject: thermal energy (summary)

From: [email-masked] (Computational Chemistry List)
Date: Thu, 12 May 1994 15:50:40 +0100 (MET)

Hi netters!

A long time ago (as I was remembered recently), I asked a question about
"thermal energy". I received interesting replies and now I will post them.
After that I will describe what opinion I am of now, comparing your suggestions.

Thanks for your help!
*********************************************
My original question was: 

Dear Netters! 
 
I have a question about the thermal energy, concerning rotational
barriers. How can you calculate this energy in dependence of the absolute
temperature? Some people told me it must be E = 3/2 * RT for a non-linear
molecule, but this doesn't fit to experimental data (Ethane has a
rotational barrier of approx. 3 kcal/mol and 3/2 * RT would be only 0.89
kcal/mol for T=298K). Any suggestions?
 
Kay. 
*********************************************
And now the replies. Thanks for all of them!
********************************************* 
From: Michael A. Lee <[email-masked]> 
 
I assume that you will be getting responses that explain that the
classical result for 1/2 kT per degree of freedom is modified at low
temperature due to quantum discrete states.  Low temperature is defined as
when the energy difference between states is not much smaller than the
thermal energy (kT).  
Mike 
*********************************************
From: [email-masked]
 
[ my question - deleted ] 
 
Hello Kay, 
 
It sounds like you may be confusing the rotational barrier height with the
rotational thermal energy. 
 
The rotational energy barrier is determined by a molecule's potential
energy function, and is independent of temperature. 
 
The rotational thermal energy is the average kinetic energy one would
expect to find in any given rotational degree of freedom, based on 
statistical probabilities. 
 
Hope this helps.  Let me know if you still have a question about this. 
 
Regards, 
Greg 
 
================================================================= 
Gregory King                             Internet: [email-masked] 
Eastern Reg. Res. Ctr., ARS, USDA           Voice: (1) 215 233 6675 
600 East Mermaid Lane                         Fax: (1) 215 233 6559 
Philadelphia, PA  19118-2551 
================================================================= 
********************************************* 
From: [email-masked] (Paul Soper) 
 
Kay, 
 
A distinction must be made between rotation of the molecule as whole and
internal rotation about a bond.  The first does indeed contribute 3RT/2 to
the energy of a non-linear molecule. The energy of around 3 kcal/mole
refers to internal rotation.
 
Internal rotation about a bond usually must be treated quantum
mechanically.  The energy levels can be calculated using perturbation
theory (see W.H. Flygare, "Molecular Structure and Dynamics," chapter 4).
You then need to calculate the partition function to get thermodynamic
properties (see D.A.  McQuarrie, "Statistical Mechanics," chapter 8).  You
should be able to find a treatment of internal rotation in any statistical
mechanics text.
 
- Paul Soper 
 
Paul Soper                          [email-masked] 
DuPont                                         FAX (302)-695-1717 
Tel (302)-695-1757                All the usual disclaimers apply 
********************************************* 
From: "Janet Del Bene" <[email-masked]> 
 
Kay, The 3/2 RT refers to the energy of the three degrees of rotational
freedom of a nonlinear molecule.  The rotational barrier (in ethane, for
example) is an INTERNAL rotation about a specific bond.  In order to
calculation the internal rotational energy barrier you need to optimize
the structure of the molecule to get its energy, and then optimized the
transition structure for rotation.  In ethane, for example, you would
fully optimize the staggered conformation, and then fully optimize the
eclipsed conformation.  The energy difference would be the rotational
energy barrier.

     Janet E. Del Bene 
********************************************* 
From: [email-masked] (Eugene Fleischmann) 
 
[ my question - deleted ] 
 
If I understand your question correctly, what you want is the population
split between two rotational states separated by a barrier of Er, where 
E2 > E1.  This is the population ratio:
 
P(E2)/P(E1) = e**(-Er/RT) = .006  for Er=3 kcal/mole and  
RT=.593 (i.e.T=298) 
[P(E1)/P(E2)=.994 ] 
 
therefore, the fraction of the population in state 2 with respect to the
whole is:
 
P(E2)=.006*P(E1)=.006*(1-P(E2)), P(E2)=.006/1.006=.006 or .6% 
[P(E1)=99.4%] 
 
P(E2)/P(E1)=.221 at T=1000K, therefore, P(E2)=.181 or 18.1% 
[P(E1)=81.9%] 

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
	Eugene D. Fleischmann, Ph.D. 
	Computational Chemist 
	Cray Research, Inc.             (609) 252-1250 
	121 Commons Way                 [email-masked] 
	Princeton, NJ  08540 
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
********************************************* 
From: GOVENDEM <[email-masked]> 
 
[my questions - deleted] 
 
Dear Kay, 
If my memory serves me correct, I think the above equation is only for
ideal systems of gases and not real gases, hence explaining the 
descrepancies in your results. 

Hope this helps 

MaganMagan Govender 
Dept of Chemistry 
University of Natal 
Durban 
South Africa 
********************************************* 
From: [email-masked] (Earl EVLETH p 74208) 
 
On rotational corrections to either enthalpy or  
entropy read Benson's book on Thermochemical 
Kinetics. Otherwise one goes back to Pitzer in  
the 1950s (his Quantum Chemistry text is pretty 
good also for a number of similar subjects that 
are not treated in modern texts) 
 
Evleth(Paris) 
********************************************* 
From: [email-masked] (James "Jolly" Rodgers) 
 
Dear Kay, 
 
[ my question - deleted ] 

I take it you are referring to the barrier of rotation about 
the central bond in ethane.  I prefer to refer to this as 
a "torsional barrier" to distinguish it from a barrier for molecular 
rotation. 
 
Anyway, I believe the correct formula for the ensemble average 
for available thermal energy is actually 3/2 * nRT where n is 
the number of atoms.  For  ethane, it is thus 8 * 0.89 = 7.1 kcal/mole. 
This is the average total energy that is available to the molecule 
and includes contributions due to stretching and bending as well as 
molecular translation and rotation.  Thus, the energy is divided over 
a number of degrees of freedom, with a range of atomic speeds given by the 
Boltzmann distribution.  At any particular instant, some degrees of 
freedom will have very little kinetic energy while others may have 
considerable kinetic energy. 
 
What you are perhaps more interested in is the rate at which structures 
of a particular energy are observed (such as the transition state in a 
conformational interconversion). 
 
This is:  Rate = dN/dt = A * exp (- delta E/ RT) 
Or the integrated form: delta N  =  A * delta t * exp (- delta E/ RT) 
 
where A is a preexponential factor which is dependent on entropic factors; 
t is time; and delta E is the difference in energy between the 
structure of interest and that of a stable state, often presumed to be 
the global minimum. 
 
The preexponential factor is basically the probability of obtaining 
a particular structure, given sufficient kinetic energy is available. 
It can have contributions due to a number of factors (for example 
orientational factors) and I believe that the preexponential factor 
is usually derived experimentally, although it can in principle be 
derived by theoretical means, such as through a molecular dynamics 
simulation. 
 
Anyway, the essence of the above equation is that the number of times that 
a particular structure is observed decreases exponentially as the energy 
of the structure increases, increases in an exponential manner with regard 
to absolute temperature, and also increases linearly as one observes for 
longer periods of time. 
 
I don't know offhand what the preexponential factor is for torsional 
interconversion in ethane, I can say though that on the time scale of 
molecular dynamics simulations (i.e. picoseconds) it is common to observe 
fluctuations in conformational energy of 3 to 5 kcal/mole, so for ethane 
on an experimental time scale (i.e. usually milliseconds or longer), 
one would expect to see rapid conformational interconversions. 
 
Hope this helps! 

				Sincerely, 

				James "Jolly" Rodgers 
				Dept. Of Medicinal Chemistry 
				University Of Minnesota 
********************************************* 
From: "Carlos F. S. Castro" <[email-masked]> 
 
[my question deleted] 
 
Dear Mr. K. Kreidler, 
 
Concerning the calculations of rotational barriers, maybe i can have the
answer. Using ab initio molecular calculation you can estimate them
throught a vibracinal frequency job. In the Gaussian program, which i use,
one of the out- puts is the thermal energies (hartree/particle). Doing it
in the two forms, the lower one and the highest one in energy and getting
the difference should give you the answer desired: for Ethane, the forms
should be staggerd and eclipsed, i guess. 
If i'm right, please call back, because i would like to know for sure.  
Thanks, 
Carlos F. S. Castro Comp.
Chem. Lab.  
University of Brasilia, Brazil.
********************************************* 
From: "Joao O.M.A. Lins" <[email-masked]>
 
     Dear Dr. Kreidler, 
 
     I would like to receive the summary of this question. 
     Thanks in advance, 

					  Joao/// 
********************************************* 
From: [email-masked] 
 
dear Kay 
 
did you received any answer to your interesting inquire? 
 
[ my question - deleted ] 

thank you 
	     Davide 

------------------------------------------------------------------- 
Davide M. Proserpio - Istituto di Chimica Strutturistica Inorganica 
Universita' di Milano,   Via Venezian, 21 -  20133  Milano,   Italy 
phone +39-2-70635120 fax 70635288 - [email-masked] 
------------------------------------------------------------------- 
 
From: "RUBEN E. VENEGAS" <[email-masked]> 
 
Hi Kay, 
 
I saw your posting regarding thermal energy a while ago.  I was thinking 
I was going to see some replies, but nothing so far.  Have you got any 
replies , if so could you share them with me ?? 
I really appreciate it.  
my e-mail is 
[email-masked] 
 
Thank you very much. 
 
Ruben Venegas

********************************************* 

And now I will a summary of what I think is right, after I have compared all the
results.

There is a difference between 1. the rotation of the whole molecule and
2. the rotational barrier along one specific bond. First depends on the
degrees of freedom of the molecule, which is actually 1.5 nRT in the case
of a non-linear molecule (with n being the number of atoms). So this
energy depends on the temperature.

The rotational barrier height along one bond is described by the potential
function of the molecule, and this function is temperature independent.

If you have a difference in energy between to states, you must use
Boltzmann's equation to calculate the distribution of the species on these
states.

So you cannot attach a sharp energy value to a chosen temperature, but
always have a distribution of energy. 

What I wanted to describe, was the second case. Now, after your advice and
a bit of thought it's seems all so simple.

Again, thanks for your help and my apologies for letting you wait for so
long! 

Bye bye.

Kay Kreidler.