# Determination of Hydrogen's Thermophysical Properties Using a Statistical Thermodynamic Method
## 3 Results and Discussion
### Accuracy of Statistical Thermodynamic Method
#### 3.1.2 Isobaric Heat Capacity and Enthalpy  
The ideal gas heat capacities of hydrogen are calculated and shown in Figure 8. The hydrogen pressure is set as 0.1 Mpa. The calculation results and REFPROP data are displayed in 1 K increments from 20 K to 300 K. [PERSON]'s [30], [PERSON]'s [21] and [PERSON]'s [19] data are randomly extracted from their studies. Figure 8a shows that the calculated isobaric heat capacities (\(c_{\rm p}\)) of para-H\({}_{2}\) agree well with all the data and deviate from the REFPROP data within 1% above 60 K. The calculated \(c_{\rm p}\) are consistent with [PERSON]'s [21] and [PERSON]'s [19] data within 20-60 K, but deviate from REFPROP and [PERSON]'s [30] data below 60 K. The maximum deviation between the calculation and REFPROP data is 14.14% at 20 K. As shown in Figure 8b, the calculated \(c_{\rm p}\) of ortho-H\({}_{2}\) is in good accordance with [PERSON]'s [21] and [PERSON]'s [19] data. The deviation between the present calculation and REFPROP data is within 1% above 80 K and within 5% above 27 K, but the deviation becomes larger when temperature approaches the normal boiling point. This is because the REFPROP data in Figure 8 are the real gas isobaric heat capacities, but the calculated data are the ideal gas isobaric heat capacities. Para-H\({}_{2}\) and ortho-H\({}_{2}\) cannot be regarded as ideal gases near their normal boiling points. The difference between the real gas and the ideal gas heat capacities becomes larger at lower temperatures. Consequently, there is a deviation between the calculated isobaric heat capacities and the REFPROP data. Figure 7: Para-ortho H\({}_{2}\) conversion heat versus temperature [4; 19; 29]. The para-H\({}_{2}\) and ortho-H\({}_{2}\) enthalpies are calculated based on the Helmholtz-type fundamental EOS. Figure 9a shows that the calculated para-H\({}_{2}\) enthalpies correspond well with [PERSON]'s [21] and [PERSON]'s [19] data and have the same tendency as [PERSON]'s [30] data. But the calculated enthalpies differ by about 300 kJ-kg\({}^{-1}\) from [PERSON]'s [30] data, which is probably the result of a different selection of the reference state. The deviation between the calculation and REFPROP data reaches its maximum of 2.22% at 20 K and is maintained at less than 0.1% above 25 K. As Figure 9b displays, the ortho-H\({}_{2}\) enthalpies obtained using statistical thermodynamics match [PERSON]'s [21] and [PERSON]'s [19] data well. However, the ortho-H\({}_{2}\) enthalpies from REFPROP cannot be directly used because REFPROP assumes a default reference state of a saturated liquid at the normal boiling point. The enthalpy of the saturated ortho-H\({}_{2}\) is assumed to be 0 kJ-kg\({}^{-1}\), which leads to an inaccurate prediction of ortho-H\({}_{2}\) enthalpies from the original REFPROP data. Ortho-H\({}_{2}\) molecules are restricted to odd energy levels (\(j=1\)) and para-H\({}_{2}\) molecules are restricted to even energy levels (\(j=0\)) at the normal boiling point. Consequently, ortho-H\({}_{2}\) has higher rotational energy than para-H\({}_{2}\). Based on this fact, the reference enthalpy of ortho-H\({}_{2}\) must be changed from 0 kJ-kg\({}^{-1}\) to 702.98 kJ-kg\({}^{-1}\) to show a correct energy relationship between ortho-H\({}_{2}\) and  
Figure 8: Isobaric heat capacity versus temperature: (**a**) para-H\({}_{2}\); (**b**) ortho-H\({}_{2}\)[19; 21; 30]. para-H\({}_{2}\)[19]. After the correction, the calculation is consistent with REFPROP data. The maximum deviation is 0.18% at 20 K.  
Table 3 shows the comparison between the calculated para-H\({}_{2}\) enthalpies and REFPROP data under different pressures from 0.5 Mpa to 20 Mpa. Temperatures are taken from 100 K to 1500 K in 100 K increments. The deviations are not more than 2.32% even though the pressure rises to 20 Mpa. This indicates that the statistical thermodynamic method is also applicable at higher temperatures and pressures.