Help With calculating molar volume?

your question..

Help With calculating molar volume? Calculate the molar volume of carbon dioxide at 400 K and 30 atm, given that the second virial coefficient (B) of CO2 is -0.0605 L mol-1. Compare your result with that obtained using the ideal-gas equation.

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virial equation of state

.. PV / (nRT) = 1 + B x (n/V) + C x (n/V)² + D x (n/V)³ + … ∞

we want molar volume.. V/n… so let’s call that Vm and rewrite this as

.. PVm / (RT) = 1 + B / Vm + C / Vm² + D / Vm³ + … ∞

now in this case, we’re going to assume V/m is high enough (meaning the density is low) so that the C and D and the rest of the terms to ∞ drop out. leaving

.. PVm / (RT) = 1 + B / Vm

rearranging

.. PVm² / (RT) = Vm + B

.. PVm² = RTVm + RTB

.. PVm² – RTVm = RTB

.. Vm² – (RT/P) Vm = RTB/P

this is a quadratic. let’s complete the square

.. Vm² – (RT/P) Vm + (RT / 2P)² = RTB/P + (RT / 2P)²

then

.. (Vm – RT / 2P)² = RTB/P + (RT / 2P)²

then taking sqrt and rearranging

.. Vm = (RT / 2P) ±√(RTB/P + (RT / 2P)²)

solving

Vm = (0.08206Latm/molK)x(400K) / (2x30atm) ±√((0.08206 Latm/molK x 400K x -0.0605 L/mol / 30atm) + (0.08206Latm/molK x 400K / (2x30atm))²)

reducing

.. Vm = 0.547L/mol ± 0.483L/mol

giving these 2 solutions

.. Vm = 1.030 L/mol

.. Vm = 0.064 L/mol

the first is the correct one

the second is the trivial one that we’ll ignore.

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now if want to include that C term

.. let C = -B² / 9

and you get to recalc all that stuff…. AND.. you’ll end up with a cubic equation to solve and I’d recommend doing that graphically.

and DO DOUBLE CHECK my work!.

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compared to

.. PV = nRT

.. V/n = RT/P = (0.08206Latm/molK) x (400K) / (30atm) = 1.09 L/mol

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and the % difference… ’cause this chemistry class and we like to use % difference in chem class

.. % difference = | 1.030 – 1.09 | / ( | 1.030 + 1.09 | / 2) x 100% = 5.7%

and you can see that at high pressures and high temps, the deviation from ideal may be too large for the process you’re trying to design.

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btw… if you calc the molar volume of CO2 at 400K and 30atm using VDW, the answer turns out to be…. 1.024 L/mol

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that help?