# Determination of Hydrogen's Thermophysical Properties Using a Statistical Thermodynamic Method
## 1 Introduction  
Hydrogen (H\({}_{2}\)) is widely used in the production of chemicals [1], petroleum refining [1], fuel cells [2], internal combustion engines [3] and aerospace engineering as rocket fuel [1]. Hydrogen has two spin isomers: ortho-hydrogen (ortho-H\({}_{2}\)) and para-hydrogen (para-H\({}_{2}\)), which have different rotational states. The equilibrium mixture ratio of the two forms changes with temperature [4]. They have considerably different rotational energies below 250 K. Their caloric properties, such as isobaric and isochoric heat capacity, thermal conductivity, enthalpy and entropy show significant differences [5]. Thus, accurate determination of hydrogen's properties is essential for product and process designs in hydrogen industries [6,7], like hydrogen liquefaction and cooling capacity utilization. Equations of state (EOSs) for hydrogen are the basis for determining the thermophysical properties of hydrogen. [PERSON] and [PERSON] [8] developed the Redlich-Kwong state equation of actual gas based on the van der Waals equation. [PERSON] [9] derived the second and third virial coefficients for para-H\({}_{2}\) from experimental P-V-T (pressure-volume-density) data [10]. [PERSON] [11] developed a new EOS for para-H\({}_{2}\), which employed one set of 24 coefficients for all fluid states. [PERSON] and [PERSON] [12] established a 16-term polynomial EOS for para-H\({}_{2}\) up to 2778 K and 69 MPa [7]. [PERSON] [13] proposed a 32-term modified Benedict-Webb-Rubin (MBWR) EOS for para-H\({}_{2}\) from the triple point to 2500 K at pressures up to 68.9 MPa (680 atm). In 1982, [PERSON] [14] developed models based on the MBWR equation to determine the thermophysical properties of para-H\({}_{2}\) and other fluids, such as argon, ethylene and nitrogen. These models were used in previousversions of the NIST (National Institute of Standards and Technology) standard property package, REFPROP (Reference Fluid Thermodynamic and Transport Properties Database). [PERSON] [15; 16] proposed an EOS explicit in the Helmholtz free energy, and extended the applicability of the Helmholtz-type EOS to 15 cryogenic fluids, including normal hydrogen. All thermophysical properties (compressibility factor, internal energy, enthalpy, entropy, speed of sound and isobaric/isochoric heat capacity) are derived via the differentiation of the Helmholtz free energy. [PERSON] [17; 18] developed Helmholtz-type equations (GERG-2004 and GERG-2008) of natural gas (21 components) by combining them with the EOS for pure fluids. GERG-2008 covers the gas phase, liquid phase, supercritical region and vapor-liquid equilibrium states, with a range of validity of 60-700 K and up to 70 MPa. [PERSON] [5; 19] created three new 14-term fundamental EOSs for para-H\({}_{2}\), ortho-H\({}_{2}\) and normal H\({}_{2}\). They are explicit in the reduced Helmholtz free energy. Despite the absence of the experimental data of normal H\({}_{2}\) and ortho-H\({}_{2}\), the quantum law of corresponding states was applied to improve their formulations. Consequently, thermophysical properties near the critical point and in liquid states can be accurately determined. The aforementioned three EOSs can be applied at pressures up to 2000 MPa and temperatures up to 1000 K.  
Early EOSs for hydrogen explicit in pressure are sometimes unavailable near two-phase or supercritical regions. Thermodynamic properties, such as entropy, enthalpy and isobaric heat capacity (\(c_{\rm p}\)), are derived from integrals of pressure or density. The continuity of an EOS has a great impact on the feasibility of integrals, and causes difficulties or instability in computation. However, all thermodynamic properties can be obtained via the differentiation of Helmholtz free energy from the fundamental EOS. In addition, ortho-H\({}_{2}\) properties should be considered in the design of processes, including para-ortho H\({}_{2}\) conversion. Hence, [PERSON]'s [5; 19] equations were selected for this study because they cover the EOS of ortho-H\({}_{2}\).