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.

 

[charng]

I can provide some insight into this topic. The temperatures at which molecular energy storage modes can be defined depend on the type of molecule and its internal structure. Generally, for simple molecules such as diatomic molecules, translational energy modes are active at room temperature, while rotational modes become active at higher temperatures, and vibrational modes at even higher temperatures.

The activation of these modes is a result of the increase in thermal energy at higher temperatures, which allows the molecules to overcome the energy barriers and move in different ways. This is known as the Boltzmann distribution, which describes the distribution of energies among molecules in a gas.

The analysis of these energy storage modes is typically done assuming an ideal gas, where the molecules are assumed to be point particles with no interactions between them. This is a simplification that allows for easier calculations and provides a good approximation for many gases at low pressures and high temperatures.

However, in reality, gases do have interactions between molecules, and these can affect the behavior of the gas and the activation of energy modes. Additionally, the relationships between specific heat at constant volume (Cv) and constant pressure (Cp) also play a role in the analysis of energy modes. These relationships depend on the type of molecule and its internal structure and can vary for different gases.

In summary, the definition of temperatures for molecular energy storage modes depends on the type of molecule and its internal structure. The analysis of these modes often assumes an ideal gas and considers the relationships between Cv and Cp. However, these assumptions should be carefully considered and may not always accurately represent real-world conditions. I hope this helps in your understanding of gas dynamics and combustion.