[skp524]

a) Based on equipartition of energy, determine the heat capacity,Cp, of CO2 gas assuming ideal gas behavior.  Consider one mole of CO2.

My attempt:
1.5 RT for translational motions, 1 RT for rotational motions, 1 RT for vibrational motions
 U= (1.5+1+1)RT=3.5 RT
  Cv,m= 3.5R , Cp,m - Cv,m= R  
Thus , Cp for one mole CO2 gas = 4.5R =37.4 JK-1mol-1

b) Carbon dioxide consists of 4 vibrational modes, one symmetric stretching, one asymmetric stretching and two bendings and the vibrational frequencies are 1388cm-1,2349 cm-1 and 667 cm-1 , respectively. Planck suggested only discrete energy level are  allowed for vibrational motions. Based on this assumption, he derived a forumla for the average energy of an oscillator at temperature T,
<E>= hv/[exp(hv/kT)-1]. Determine the heat capacity , Cp of CO2 in light of this. Compare the value with that obatined from part (a), which one would be more accurate and why?

My attempt:  
I tried to split the calculation into 2 parts, one for translational+rotational, another for vibrational
U= 3.5RT=(1.5+1+1) (k*Na)T
=2.5(k*Na)T +1(k*Na)T
=2.5RT+ Na(kT) * 1
=2.5RT+ Na<E>*(k/k)
=2.5RT+ (k*Na/k)*<E>
=2.5RT+ (R/k)<E>

Replace <E>=kT with <E>=hv/[exp(hv/kT)-1]

Differentiate <U> with respect to T
Cv=2.5R+(R) (1/k *d<E>/dT)
Cv=2.5R+R {[(hv/kT * exp(hv/2kT)]/[exp(hv/kT)-1]  }^2
Cv=2.5R+R*x^2*{[exp(x/2)]/[exp(x)-1]}^2
Cv=2.5R+ Rf
 , where x =  hv/kT and f =x^2*{[exp(x/2)]/[exp(x)-1]}^2

As there are 4 vibrational modes, there would be f(v1), f(v2) and two f(v3)

Cp-Cv=[2.5R+R(f1+f2+2f3)]+R=3.5R+R(0.055+0.002+0.449*2)=3.5R+0.955R
=4.455R=37.0 JK-mol-1

As a result, the value from part (b) is more accurate in accordance with the standard value at 298K , 36.94 JK-mol-1. For the possible reasons behind, I suggest that energies are  not equally shared among different frequencies, therefore we need to take the average energy of "different" oscillators ( i.e. different frequencies) into consideration. In addition, from part (a), the Cp was found to be actually a constant, i.e. 4.5R , that is temperature-independent. In fact, temperature can affect heat capacity, so part (b) 's approach may be more suitable.

Can anyone give comment to my deduction to let me know whether it is correct?

[Enthalpy]

Agree with about everything here.

CO₂ is THE small molecule whose Cp is difficult to evaluate quickly, because its has many vibration modes that get excited around room temperature. Stiff molecules like H₂ and N₂ behave just like perfect gases around RT - provided the gas' density is far from the liquid's one.

If your gas is hotter, any molecule gets as complicated as CO₂ is at RT. Especially in flames, heat capacity varies a lot, notably from H₂O and CO₂. And then you have equilibrium shift (with CO). At higher temperature, like during the atmospheric entry of a spacecraft, excited electronic states get populated, O₂ dissociates, and even N₂ would at higher speed.

Your CO₂ estimation by hand was complicated enough to tell that this is a job for a software.