The generation of
this plot from an interactive MathCad™ program (its pdf) exemplifies the computation of the specific
heat of the rotator H2. Because of nuclear spin 1/2 (h/2π) of the proton (a fermion!) the molecule H2 exists in two modifications called para-H2 with nuclear spins antiparallel ↑↓ and ortho-H2 with nuclear spins parallel ↑↑. For the
former there is only one state (a singlet ), for the latter there are three states (a triplet
[+1, 0, -1](h/2π)). para-H2 occupies only even numbered, ortho-H2 only odd rotational quantumstates. At low temperature para-H2 is more stable, because it can reach the rotationless lowest state J=0, whereas ortho-H2 must end at J=1 when the temperature is lowered towards zero Kelvin (see the retarded onset of the specific heat curve Cpo(rtho) compared to Cpp(ara), and Cpm(ixture)). Therefore, p-H2 is the predominant species when T → 0 K. At 'high' temperatures the two modifications form the
'normal' hydrogen gas with p-H2/o-H2 = 1 : 3, according to the ratio of the number of spin states. The computation of the specific heat has to take into account this complex behaviour which is
borne out exactly by experiment, see the points measured for 95% p-H2 by K.Clusius and K.Hiller, Z.physik.Chem B4,158(1929). The magenta curve is computed for 100% p-H2, difficult to make experimentally. It was essential to show, that the value of R = 1.987 cal mol-1 K-1 (red horizontal line) for two classical rotational degrees of freedom is surpassed by the measurements as predicted by quantum mechanics. The green curve Cpm is for normal hydrogen gas. It shows the expected behaviour without the complication by nuclear spin. Among others, A.Eucken & K.Hiller, Z.physik.Chem B4,142(1929), have measured this curve. Cpo(T) cannot be observed directly, because there is no known bulk-method to prepare (nearly) pure o-H2. The blue curve Cpg is the most dramatic: It can be measured with hydrogen gas having the equilibrium ratio of the two modifications at every temperature. K.F. Bonhoeffer and P. Harteck, Z.physik.Chem B4,113(1929), discovered that activated charcoal (from blood) with paramagnetic dangling bonds on and iron ions in it, is a catalyst that accelerates the conversion enormously. The high rise of Cpg is caused by the reaction enthalpy of 339.22 cal/mol necessary to form o-H2 from p-H2. Curve z shows the contribution of conversion to the specific heat of rotation.|
The StatTher package offers the same MathCad™ program for D2, T2, and 14N2. Note the changes in the temperature scale and explain the differences of the plots!
By the way, Werner Heisenberg won a Nobel Prize in 1932 for the
detailed prediction of the physics underlying this plot by his Quantum Mechanics and Pauli's exclusion principle. The first computation of the above plot seems to have been made by W.F.Giauque, J.Amer.Chem.Soc. 52,4808,4816(1930), who had neither MathCad nor a computer at that time ... but also earned a Nobel Prize in 1949 for his pioneering work in low temperature thermodynamics.