I was doing an experiment whereby I had to measure the specific heat ratios of certain gases. Carbon dioxide came out to be around 1.3, and checking with the accepted values this is close. My question is, the only way I can justify this answer is if $\ce{CO2}$ had 7 active degrees of freedom at room temperature. This implies that the molecule is vibrating. I am not a chemist and I thought that molecules only store energy in this degree of freedom at high temperatures.

  • $\begingroup$ Your results implies just rotational degree of freedom. Why you see it as to require activated vibrations? 7/5 is 1.3 $\endgroup$ – Alchimista Mar 14 '19 at 8:54
  • $\begingroup$ Its 1.4, the tables in my textbook have the specific heat of C02 listed has 1.28, which is closer to 9/7. $\endgroup$ – Vishal Jain Mar 14 '19 at 10:02
  • $\begingroup$ I see. I was astray by you saying " the only way I can justify this answer is if C02 had 7 active degrees of freedom at room temp". $\endgroup$ – Alchimista Mar 14 '19 at 10:07

Yes, the vibrational modes are present at any temperature, including absolute zero where the lowest vibrational energy levels only are populated; the zero-point levels. The molecule is linear ( OCO ) so it has $3N-5=4$ vibrational modes for $N=3$ atoms. The symmetric stretch corresponds to both CO bonds stretching in phase, and asymmetric stretch to one CO stretching while the other compresses (this has highest frequency), and there are two degenerate bends where the OCO angle changes (lowest frequency). If the z-axis is along the OCO bonds, bends are in the zx and in the zy planes. Wikipedia has lots of diagrams showing these vibrational normal modes.

I have calculated the heat capacity as shown in the figure. The translational value is 3R/2 at all but the lowest temperatures (a few K) and similarly for rotational levels, which have a value of R, again above about 50 Kelvin. (The rotational constant is 0.3902 wavenumbers). The vibrational heat capacity from the modes (1388, 667,667, and 2349 wavenumbers) gives the rising curve on the figure. At 300 K the value is about 3.4 R or 28 J/mol/K. It is clear that only a few vibrational quanta are populated at room temperature. The high temperature value of the heat capacity adds another 4R (each vibration counts R) and is not reached until about 4000 K. Clearly never obtained in practice.

CO2 heat capacity

Total heat capacity for CO2 vs temperature. The blue line is the total heat capacity, the lower horizontal line has a value R, the upper one R+3R/2. The top horizontal line has no particular importance and is the value at 300 K and is approximately 3.4R as mentioned in the comments.

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  • $\begingroup$ Wait, are you saying that there are 4 vibrational modes present at room temperature? $\endgroup$ – Vishal Jain Mar 14 '19 at 10:03
  • $\begingroup$ The point is not how many modes are there or how many are vibrational. The point is that the four modes (that of course exist) may or not be excited by heat. At room temperature the answer is no, and we are let with 5. Discrepancies might come from experiment, from the fact that the gas is real, from the fact that a small percentage might be populated statically. I would use for gamma the ratio 7/5, but perhaps I am wrong.@Vishal Jain $\endgroup$ – Alchimista Mar 14 '19 at 10:19
  • $\begingroup$ As would I, its just slightly weird that N2, another linear molecule, which also has 5 degrees of freedom, has a specific heat ratio of almost exactly 1.4 at room temperature, whereas C02 has a lower value of gamma with the exact same number of degrees of freedom available to it. $\endgroup$ – Vishal Jain Mar 14 '19 at 10:40
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    $\begingroup$ Yes 4 vibrational modes at any temperature. As @Alchimista notes if you are measuring heat capacity the population of these vibrations really does matter, also the number of (whole body) rotational levels excited and the translational energy both need to be calculated. All these can be obtained via statistical mechanics using the Partition Function. You need only the pressure, mass, vibrational frequencies and bond lengths and the temperature of course. $\endgroup$ – porphyrin Mar 14 '19 at 12:07
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    $\begingroup$ @Alchimista I don't think "almost 7 for accessible DOG" is the right wording. There are 5 total translation + rotation degrees of freedom. The additional R worth of contribution to Cv at RT is not strictly 2 additional DOG, but a sum of contributions from the four vibrational modes (mostly the two bending ones), each of which has a small fraction of excited states such that the contribution is not the expected 1R for each one, but just a fraction of that. Coincidentally, the sum of the four fractions turns out to be about R at room temp, a misleadingly round number. $\endgroup$ – Andrew Mar 14 '19 at 15:41

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