# Determination of Hydrogen's Thermophysical Properties Using a Statistical Thermodynamic Method
## 2 Theoretical Model and Implementation
### Statistical Thermodynamic Model for Hydrogen
#### 2.1.4 Fundamental EOSs for Hydrogen  
[PERSON] [1; 5; 16] has conducted many studies on the fundamental EOSs of hydrogen, and deduced the expressions of properties such as enthalpy, entropy and heat capacity from these EOSs. The fundamental EOSs for para-H\({}_{2}\), ortho-H\({}_{2}\) and normal H\({}_{2}\) are explicit in the reduced Helmholtz free energy [5; 19]:  
\[\alpha(\delta,\tau)=\alpha^{0}(\delta,\tau)+\alpha^{\tau}(\delta,\tau) \tag{6}\]  
\[\delta=\frac{\rho_{\rm m}}{\rho_{\rm m,c}}\ \ \ \ \tau=\frac{T_{\rm c}}{T}\]  
where \(\alpha(\delta,\tau)\) is the reduced Helmholtz free energy; \(\alpha^{0}(\delta,\tau)\) is the ideal gas Helmholtz free energy; \(\alpha^{\tau}(\delta,\tau)\) is the residual gas Helmholtz free energy; \(\delta\) and \(\tau\) are the reduced density and reciprocal reduced temperature. The expression of the molar density can be deduced from the ideal gas EOS:  
\[\rho_{\rm m}=\frac{n}{V}=\frac{p}{RT} \tag{7}\]  
where \(n\), \(V\), \(p\) and \(T\) are the amount of substance in mol, volume in m\({}^{3}\), pressure in Pa and temperature in K, respectively.  
A computationally convenient parameterized form of the ideal gas Helmholtz free energy is [5]  
\[\alpha^{0}(\delta,\tau)=\ln\delta+1.5\ln\tau+a_{1}+a_{2}\tau+\sum\limits_{k=3 }^{N}a_{k}\ln[1-\exp(b_{k}\tau)] \tag{8}\]  
where the subscript \(k\) is the index of each term in the ideal gas heat capacity equation. The coefficients \(a_{k}\) and \(b_{k}\) in the equation of the ideal gas Helmholtz free energy are reported in Reference [5].  
The residual gas Helmholtz free energy is [5]  
\[\begin{array}{rl}\alpha^{\tau}(\delta,\tau)&=\sum\limits_{i=1}^{l}N_{i} \delta^{d_{i}}\tau^{t_{i}}+\sum\limits_{i=l+1}^{m}N_{i}\delta^{d_{i}}\tau^{t_{ i}}\exp(-\delta^{p_{i}})\\ &+\sum\limits_{i=m+1}^{m}N_{i}\delta^{d_{i}}\tau^{t_{i}}\exp\Bigl{[}\vartheta_ {i}(\delta-D_{i})^{2}+\beta_{i}(\tau-\gamma_{i})^{2}\Bigr{]}\end{array} \tag{9}\]  
where \(l\) = 7, \(m\) = 9, \(n\) = 14. All parameters and coefficients of the residual gas Helmholtz free energy are given in Reference [5].  
Enthalpy can be derived from the reduced Helmholtz free energy [16]:  
\[h=\frac{RT}{M}\Bigl{[}1+\tau\Bigl{(}\alpha_{\tau}^{0}+\alpha_{\tau}^{\tau} \Bigr{)}+\delta\alpha_{\delta}^{\tau}\Bigr{]} \tag{10}\]  
where \(\alpha_{\tau}^{0}\) is the partial derivative of the ideal gas Helmholtz free energy with respect to \(\tau\); \(\alpha_{\delta}^{\tau}\) is the partial derivative of the residual gas Helmholtz free energy with respect to \(\tau\); \(\alpha_{\delta}^{\tau}\) is the partial derivative of the residual gas Helmholtz free energy with respect to \(\delta\).  
Considering that ortho-H\({}_{2}\) has higher energy than para-H\({}_{2}\), the reference enthalpy of ortho-H\({}_{2}\) at its standard boiling point should be adjusted from 0 kJ\(\cdot\)kg\({}^{-1}\) to 702.98 kJ\(\cdot\)kg\({}^{-1}\) to show an accurate energy relationship between ortho-H\({}_{2}\) and para-H\({}_{2}\)[5]. Therefore, a correction term 702.98 kJ\(\cdot\)kg\({}^{-1}\) must be added after the ortho-H\({}_{2}\) enthalpy is calculated by Equation (10).