Kinetics: Defining Temperatures for Molecular Energy Storage Modes

[engineer23]

When can you define temperatures for the molecular energy storage modes (i.e. translational, rotational, vibrational)? I have seen the graphs of this where translational modes are at room temperature and then as temperature increases, the rotational and later vibrational modes are "activated." What assumptions are inherent in this sort of analysis (ideal gas? relationships between Cv and Cp?)...

Thanks for any insight into this! I am starting a course on combustion and am trying to familiarize myself with some background on gas dynamics, etc.

 

[pkleinod]

When can you define temperatures for the molecular energy storage modes (i.e. translational, rotational, vibrational)?

Roughly speaking, you can do this when the populations of the individual states can be characterised by a Boltzmann distribution, since the parameter temperature determines the whole distribution of populations over the states. Also, if you speak, for example, of a rotational temperature, then it is necessary, strictly speaking, that the rotational mode be decoupled from the other modes. This is rarely the case but you could still speak of the
rotational temperature of all the molecules in a particular vibrational mode. Confusing? Let me give a real example:

Consider the reaction H + Cl2 producing HCl(v,J) + Cl. The heat of reaction ends up in the translation of the products (i.e. kinetic energy) and in the internal energy of the the rovibrational states of HCl characterised by the vibrational quantum number v and the rotational quantum number J. The internal energy of a particular state (v,J) CANNOT
be written as E(v) + E(J) because of the coupling between the rotational and vibrational modes. It is, however, still useful to talk of a vibrational temperature by considering the distribution of the populations HCl(v,*), where I mean that for a given v, you sum over the populations of all the rotational states, assigning an average energy to HCl(v,*); it could be that the distribution of populations of HCl(v.*) over all v might be able to be simulated by a Boltzmann distribution characterised by some temperature Tv, which would then by called the vibrational temperature. Similarly, you can find the populations over J for all states belonging to a particular v, and this might be able to be simulated by a Boltzmann distribution characterised by a rotational temperature. Note that there could be a different rotational temperature for each v. Also, if you do this experiment in the laboratory, the results will depend strongly on how long after the collision process the measurements are taken: rotational distributions "relax" toward their equilibrium distributions much more quickly than the vibrational distributions. This is because vibrational energy transfer on collision is much less efficient than that of rotation and translation.

This is all a bit vague and fraught with "ifs" and "buts" but I hope it is of some help.
If you wish to read more about such experiments, just google "J.C. Polanyi" for his work on the above reaction and others.