# Vibrational Excitation and Heat Capacity

## Homework Statement

(a) Given V, α, κT, μJT, and CP, calculate CV at 90.0 bar and 308 K for carbon dioxide gas.

(b) If carbon dioxide's vibrations were fully excited, then CV would be 4R. What's the percent vibrational excitation at 90 bar and 308K?

## Homework Equations

Both constant pressure and constant volume heat capacities are molar heat capacities when written. V is molar volume, μ is Joule-Thompson coefficient, α is thermal expansion coefficient, and κT is isothermal compressibility coefficient. I'm given most of the values, so this becomes a simple plug-in, but there's something that's not right.

$$C_P - C_V = α/κ_T (V + C_Pμ_{JT})$$

V = 0.06647 L/mol
α = 0.03296 K-1
κT = 0.01086 bar-1
μJT = 0.2427 K/bar
CP = 250.7 J/K*mol

## The Attempt at a Solution

$$C_P - C_V = α/κ_T (V + C_Pμ_{JT})$$

$$C_P - C_V = \frac{0.0329 bar}{0.01086 K} (0.06647 L/mol + (250.7 J/K*mol)(0.2427 K/bar))$$

Looking at the rightmost product in the RHS,

$$(250.7 J/mol)(0.2427 bar^{-1})\frac{0.01L*bar}{1J} = 0.608 L/mol$$

Since units match with V, I can add,

$$C_P - C_V = \frac{0.0329 bar}{0.01086 K} (0.06647 L/mol + 0.608 L/mol)$$

$$C_P - C_V = \frac{0.0329 bar}{0.01086 K} (0.675 L/mol)\frac{1J}{0.01L*bar}$$

$$C_P - C_V = 204.8 J/mol*K$$

Since heat capacity at constant pressure is given, I can find heat capacity at constant volume

$$C_V = 45.9 J/K*mol$$

The problem comes with (b)

$$Percent Vibrations = \frac{45.9 J/K*mol}{(4)(8.314 J/K*mol)}*100$$

This comes out to ~138%

It doesn't make any sense for the heat capacity to be higher at a lower temperature since vibrational excitations aren't fully excited. I have absolutely no idea what could be wrong though, I've double checked units and am stumped.

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BvU
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Can you explain how you come from 0.06647 + 0.0608 to 0.675 ?

Can you explain how you come from 0.06647 + 0.0608 to 0.675 ?
Sorry, that was a typo while writing the post. Should've been 0.608, not 0.0608. The numbers calculated after that were done using 0.608 though.

haruspex
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I don't know the physics, so I'm just trying to follow the algebra. Where did you use the given temperature and pressure?

From how I see it, pressure and temperature aren't really meant to be directly plugged in.

Most (if not all, I do know heat capacities and compressibility factors are affected) of the values of the partials listed are dependent on either temperature or pressure (if not both) so I guess the pressure/temperature is being implictly used or else you couldn't have a value for α, κT μJT, and CP.

Or I could be totally wrong.

haruspex
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From how I see it, pressure and temperature aren't really meant to be directly plugged in.

Most (if not all, I do know heat capacities and compressibility factors are affected) of the values of the partials listed are dependent on either temperature or pressure (if not both) so I guess the pressure/temperature is being implictly used or else you couldn't have a value for α, κT μJT, and CP.

Or I could be totally wrong.
OK, so the temperature and pressure numbers are redundant.
The only other suggestion I have is that it may be wrong to equate the percentage of vibrational excitation to the percentage that Cv is of its maximum.

I had considered that, but I quickly fell away from it since it wouldn't take away the fact that the calculated Cv has no reason to be higher than the Cv with fully excited vibrational levels anyways which leads me to believe there's a problem somewhere in the calculation I made for (a), since calculating 4R is as trivial as can be.

If I had to think of another way of calculating (b), I figure that at room temperature rotational and translational energy levels are pretty much fully excited.

So at 90 bar and 308K, there's (3/2)R coming from translation and (2/2)R (I'm assuming CO2 as linear) coming from rotation... summing both approximately leads to 20.785 J/mol that's contributing to heat capacity in both cases (fully excited vibrations and not fully excited vibration).

So for (a), this would mean there's 25.115 J/mol being contributed to heat capacity by vibrational energy levels... On the other hand, subtracting 20.785 from 4R to find the vibrational energy level contribution when vibrations are fully excited lead to 12.471 J/mol. Comparing this to the result from (a) doesn't make any sense.

EDIT:

Could the 4R be wrong?

CO2 has 3 atoms, so there's 3(3) - 5 modes (3 translational and 2 rotational are taken). So there's 4 modes of vibration, and I know that energy is partitioned as both kinetic and potential energy for vibrational modes. So the contributon from vibrational modes would be 2(4/2)R = 4R? So the heat capacity with everything fully excited would be (3/2)R + (2/2)R + 4R?

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haruspex
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I had considered that, but I quickly fell away from it since it wouldn't take away the fact that the calculated Cv has no reason to be higher than the Cv with fully excited vibrational levels anyways which leads me to believe there's a problem somewhere in the calculation I made for (a), since calculating 4R is as trivial as can be.

If I had to think of another way of calculating (b), I figure that at room temperature rotational and translational energy levels are pretty much fully excited.

So at 90 bar and 308K, there's (3/2)R coming from translation and (2/2)R (I'm assuming CO2 as linear) coming from rotation... summing both approximately leads to 20.785 J/mol that's contributing to heat capacity in both cases (fully excited vibrations and not fully excited vibration).

So for (a), this would mean there's 25.115 J/mol being contributed to heat capacity by vibrational energy levels... On the other hand, subtracting 20.785 from 4R to find the vibrational energy level contribution when vibrations are fully excited lead to 12.471 J/mol. Comparing this to the result from (a) doesn't make any sense.

EDIT:

Could the 4R be wrong?

CO2 has 3 atoms, so there's 3(3) - 5 modes (3 translational and 2 rotational are taken). So there's 4 modes of vibration, and I know that energy is partitioned as both kinetic and potential energy for vibrational modes. So the contribution from vibrational modes would be 2(4/2)R = 4R? So the heat capacity with everything fully excited would be (3/2)R + (2/2)R + 4R?
I tried to get up to speed on the subject by scanning the web, but it's just made me more confused than ever.
http://chemwiki.ucdavis.edu/Physica...py/Vibrational_Spectroscopy/Vibrational_Modes agrees there are 4 modes for CO2.
At http://www.chemicalforums.com/index.php?topic=50913.0 I see
Cp for one mole CO2 gas = 4.5R =37.4 JK-1mol-1
(and that Cp is pretty much constant). That's way less than you have for Cp.
Using the summation at http://www.chemeddl.org/alfresco/se...b-9416-4761-b280-3763385046c9/stat_thermo.pdf, for your temperature of 308K, I get Cv = 1.01R. But note that when I plug in T=293 I do not get the result they show there - I get 0.93R - so maybe I'm misreading the formula.
http://userpages.umbc.edu/~lkelly/HeatCap.PDF says
Cv,m = 3/2 R + 2(1/2R) + (3N – 5) (1/2R) for a linear molecule
which gives 4.5R, no?
http://www.peacesoftware.de/einigewerte/calc_co2.php5, for your temp and pressure, gives Cv = 46.7 JK-1mol-1 = 5.6R, Cp = 361 JK-1mol-1 = 43.4R (using your R=8.314 J/K.mol).

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I tried to get up to speed on the subject by scanning the web, but it's just made me more confused than ever.
http://chemwiki.ucdavis.edu/Physica...py/Vibrational_Spectroscopy/Vibrational_Modes agrees there are 4 modes for CO2.
At http://www.chemicalforums.com/index.php?topic=50913.0 I see

Cp for one mole CO2 gas = 4.5R = 37.4 JK-1mol-1

(and that Cp is pretty much constant). That's way less than you have for Cp.
From my understanding, 1/2R is distributed to each kinetic degree of freedom for translational and rotational motion. However, since vibrational motion allows for potential energy, 1/2R is distributed to both the kinetic and the potential energy per vibrational degree of freedom.

If I'm wrong, only 1/2R is distributed per vibrational degree of freedom, then indeed, it's Cv = (3/2 + 1 + 4/2)R = 4.5R. If I'm right, Cv = (3/2 + 1 + 4)R = 6.5R.

...In either case, neither of those are equal to 4R. Although if I'm wrong and Cv = 4.5R, the percent vibrational excitation would still be >100%. :(

Anyways, I've been led to believe that CP (or Cv) is definitely not constant in general. Perhaps CO2 is sort of an oddball because of how fairly excited its vibrational modes get at room temperature like the commenter in the link you posted said?

Using the summation at http://www.chemeddl.org/alfresco/se...b-9416-4761-b280-3763385046c9/stat_thermo.pdf, for your temperature of 308K, I get Cv = 1.01R. But note that when I plug in T=293 I do not get the result they show there - I get 0.93R - so maybe I'm misreading the formula.
Can't access the link, but that Cv seems unlikely to me. That's less than (3/2)R which is given by translational degrees of freedom which is pretty much continuous at any temperature.

http://userpages.umbc.edu/~lkelly/HeatCap.PDF says

which gives 4.5R, no?
http://www.peacesoftware.de/einigewerte/calc_co2.php5, for your temp and pressure, gives Cv = 46.7 JK-1mol-1 = 5.6R, Cp = 361 JK-1mol-1 = 43.4R (using your R=8.314 J/K.mol).
I hadn't thought of looking up values of CP, that's a pretty good idea. I can't access the last link you gave me, but I'm assuming it's a calculator for constant volume heat capacity for CO2 given pressure and temperature. I got curious and tried using the CP value you gave me instead of the one I was given in the equation in the OP, but the corresponding CV value that came out didn't come close to the one you gave me.

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Yeah, that worked. The Cv value you were telling me was the contribution to Cv from vibrations, I misunderstood and thought you meant total Cv.

I plugged in their numbers and got the result they got, so I went ahead and put in my temperature and got what you got, 1.01R. Taking that as the contribution to heat capacity from vibrations and summing it with translation + rotation gives me a total heat capacity of 29.2 J/K*mol.

That gives a vibrational percentage of 87% which iirc is what is supposed to be the answer. I have no idea why the equation in the OP isn't giving something even remotely close, but I'm clueless as to what...

Guess I'll have to wave the white flag at this point.

EDIT:

Out of curiosity, I went ahead and plugged in T = 40000K and got a contribution from vibrations of 3.999. So I'm pretty confident that Cv for CO2 is 6.5R and not 4R.

EDIT:

Well, I figured out what was wrong. It was lack of reading comprehension. The 4R was meant to be the contribution from vibrational excitation and I was mistakenly using it as total Cv. The total Cv would be (1.5 + 1 + 4)R = 6.5R. The way to calculate the % would be (45.9 - 2.5R)/(4R) * 100%.

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