Vibrational Excitation and Heat Capacity

1. The problem statement, all variables and given/known data

(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?

2. Relevant 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.

[tex] C_P – C_V = α/κ_T (V + C_Pμ_{JT}) [/tex]

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

3. The attempt at a solution

[tex] C_P – C_V = α/κ_T (V + C_Pμ_{JT}) [/tex]

[tex] C_P – C_V = \frac{0.0329 bar}{0.01086 K} (0.06647 L/mol + (250.7 J/K*mol)(0.2427 K/bar)) [/tex]

Looking at the rightmost product in the RHS,

[tex] (250.7 J/mol)(0.2427 bar^{-1})\frac{0.01L*bar}{1J} = 0.0608 L/mol [/tex]

Since units match with V, I can add,

[tex] C_P – C_V = \frac{0.0329 bar}{0.01086 K} (0.06647 L/mol + 0.0608 L/mol) [/tex]

[tex] C_P – C_V = \frac{0.0329 bar}{0.01086 K} (0.675 L/mol)\frac{1J}{0.01L*bar} [/tex]

[tex] C_P – C_V = 204.8 J/mol*K [/tex]

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

[tex] C_V = 45.9 J/K*mol [/tex]

The problem comes with (b)

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

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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