# Vibrational Excitation and Heat Capacity

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

(a) Given V, α, κ_{T}, μ_{JT}, and C_{P}, calculate C_{V} at 90.0 bar and 308 K for carbon dioxide gas.

(b) If carbon dioxide’s vibrations were fully excited, then C_{V} 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 κ_{T} 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}

κ_{T} = 0.01086 bar^{-1}

μ_{JT} = 0.2427 K/bar

C_{P} = 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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