HeatCapacity - Maple Help

Heat Capacity of Real Gases

 Overview The molar specific heat capacity (or molar heat capacity) is the heat energy required to raise the temperature of 1 mol of a substance by 1° C.  In this lesson we will compute the molar heat capacity at constant volume for real diatomic gases and compare the results with those predicted for ideal gases.     Figure 1: Joules' Heat Apparatus 1845, Science Museum, London. (Public Domain)

Heat Capacity

For an ideal gas (monoatomic) the molar heat capacity at constant volume CV is given by

where R is the ideal gas constant.  For a diatomic ideal gas, after adjustment for the rotational motion, it is given by

All diatomic ideal gases have the same molar heat capacity at constant volume.

Real gases, in contrast, have non-ideal vibrational and rotational energies that create deviations from the ideal value of the molar heat capacity at constant volume.  In this lesson we use the Quantum Chemistry package to compute this variation in the gaseous diatomic molecule hydrogen fluoride.

First, we set the Digits to 15 and load the Quantum Chemistry package

 >
 $\left[{\mathrm{AOLabels}}{,}{\mathrm{ActiveSpaceCI}}{,}{\mathrm{ActiveSpaceSCF}}{,}{\mathrm{AtomicData}}{,}{\mathrm{BondAngles}}{,}{\mathrm{BondDistances}}{,}{\mathrm{Charges}}{,}{\mathrm{ChargesPlot}}{,}{\mathrm{CorrelationEnergy}}{,}{\mathrm{CoupledCluster}}{,}{\mathrm{DensityFunctional}}{,}{\mathrm{DensityPlot3D}}{,}{\mathrm{Dipole}}{,}{\mathrm{DipolePlot}}{,}{\mathrm{Energy}}{,}{\mathrm{ExcitationEnergies}}{,}{\mathrm{ExcitationSpectra}}{,}{\mathrm{ExcitationSpectraPlot}}{,}{\mathrm{ExcitedStateEnergies}}{,}{\mathrm{ExcitedStateSpins}}{,}{\mathrm{FullCI}}{,}{\mathrm{GeometryOptimization}}{,}{\mathrm{HartreeFock}}{,}{\mathrm{Interactive}}{,}{\mathrm{Isotopes}}{,}{\mathrm{MOCoefficients}}{,}{\mathrm{MODiagram}}{,}{\mathrm{MOEnergies}}{,}{\mathrm{MOIntegrals}}{,}{\mathrm{MOOccupations}}{,}{\mathrm{MOOccupationsPlot}}{,}{\mathrm{MOSymmetries}}{,}{\mathrm{MP2}}{,}{\mathrm{MolecularData}}{,}{\mathrm{MolecularGeometry}}{,}{\mathrm{NuclearEnergy}}{,}{\mathrm{NuclearGradient}}{,}{\mathrm{OscillatorStrengths}}{,}{\mathrm{Parametric2RDM}}{,}{\mathrm{PlotMolecule}}{,}{\mathrm{Populations}}{,}{\mathrm{RDM1}}{,}{\mathrm{RDM2}}{,}{\mathrm{RTM1}}{,}{\mathrm{ReadXYZ}}{,}{\mathrm{Restore}}{,}{\mathrm{Save}}{,}{\mathrm{SaveXYZ}}{,}{\mathrm{SearchBasisSets}}{,}{\mathrm{SearchFunctionals}}{,}{\mathrm{SkeletalStructure}}{,}{\mathrm{Thermodynamics}}{,}{\mathrm{TransitionDipolePlot}}{,}{\mathrm{TransitionDipoles}}{,}{\mathrm{TransitionOrbitalPlot}}{,}{\mathrm{TransitionOrbitals}}{,}{\mathrm{Variational2RDM}}{,}{\mathrm{VibrationalModeAnimation}}{,}{\mathrm{VibrationalModes}}{,}{\mathrm{Video}}\right]$ (2.1)

Second, we define the molecule hydrogen fluoride at its equilibrium bond length

 >
 ${\mathrm{mol_HF}}{≔}\left[\left[{"H"}{,}{0}{,}{0}{,}{0}\right]{,}\left[{"F"}{,}{0}{,}{0}{,}{0.92712350}\right]\right]$ (2.2)

Third, we compute a table of thermodynamic properties for HF including the enthalpy.  By default, the command uses a temperature of 298 K.

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 ${\mathrm{thermo_HF}}{≔}{table}{}\left(\left[{\mathrm{free_energy}}{=}{-}{2.63646607}{}{{10}}^{{8}}{}⟦\frac{{J}}{{\mathrm{mol}}}⟧{,}{\mathrm{heat_capacity}}{=}{20.78616201}{}⟦\frac{{J}}{{\mathrm{mol}}{}{K}}⟧{,}{\mathrm{electronic_energy}}{=}{-}{2.63594374}{}{{10}}^{{8}}{}⟦\frac{{J}}{{\mathrm{mol}}}⟧{,}{\mathrm{energy}}{=}{-}{2.63540286}{}{{10}}^{{8}}{}⟦\frac{{J}}{{\mathrm{mol}}}⟧{,}{{\mathrm{\theta }}}_{{C}}{=}{29.48020838}{}⟦{K}⟧{,}{\mathrm{entropy}}{=}{365.09759801}{}⟦\frac{{J}}{{\mathrm{mol}}{}{K}}⟧{,}{\mathrm{enthalpy}}{=}{-}{2.63537808}{}{{10}}^{{8}}{}⟦\frac{{J}}{{\mathrm{mol}}}⟧{,}{{\mathrm{\theta }}}_{{B}}{=}{29.48020838}{}⟦{K}⟧{,}{\mathrm{zpe}}{=}{23947.11548957}{}⟦\frac{{J}}{{\mathrm{mol}}}⟧\right]\right)$ (2.3)

(a) What is the computed (real-gas) molar heat capacity of hydrogen fluoride at constant volume and 298 K in J/mol/K?

(b) What is the computed (real-gas) molar heat capacity of hydrogen fluoride at constant volume and 2980 K in J/mol/K?
(Hint: Change the keyword temperature in the command Thermodynamics above to 2980.)

(c) What is the ideal (ideal-gas) molar heat capacity of hydrogen fluoride at constant volume in J/mol/K?
(Note that the ideal molar heat capacity is the same for all temperatures!)

(d) Compare (a) and (b) with (c).  Explain briefly.

Hint: The gas constant is known by Maple

 > $\mathrm{evalf}\left(\mathrm{ScientificConstants}\left[\mathrm{Constant}\right]\left(R\right)\right);$
 ${8.31445980}$ (2.4)
 >