THE ESTABLISHMENT OF THE NOVEL M-H EOS
The basis of arithmetic solution of the Hou’s modified M-H EOS
The M-H EOS was originally proposed as an empirical equation by Martin and Hou in 1955 on the basis of a large number of P-V-T data analysis about many kinds of substances. The initial formula can be expressed as follows9:
\(\mathrm{P=}\)\(\sum_{i=1}^{5}\frac{f_{i}\left(T\right)}{\left(V-b\right)^{i}}\)(1)
with\(\mathrm{f}_{\mathrm{i}}\left(\mathrm{T}\right)\mathrm{=}\mathrm{A}_{\mathrm{i}}\mathrm{+}\mathrm{B}_{\mathrm{i}}\mathrm{T+}\mathrm{C}_{\mathrm{i}}\mathrm{e}^{\mathrm{-k}\frac{\mathrm{T}}{\mathrm{T}_{\mathrm{c}}}}\)(2)
where k = 5.475, Ai, Bi, Ci , and b are the characteristic constants for a given substance.
Based on the universal physical properties of substances, there are 10 initial value conditions for the EOS, however, the unknown characteristic constants are up to 169. Fortunately, not every characteristic constant is of equal importance or necessary, which means those less important characteristic constants can be omitted. Ref. (9) gave the detailed reasons for the simplification of the related unknown characteristic constants and the theory foundations of the initial value conditions. In this work, we just exhibit the result of the simplified fi (T ) and the initial value conditions of the M-H EOS.
The simplified fi (T ) can be expressed as follows9:
\(\mathrm{f}_{\mathrm{1}}\left(\mathrm{T}\right)\mathrm{=}\mathrm{B}_{\mathrm{1}}\mathrm{T}\)(3)
\(\mathrm{f}_{\mathrm{2}}\left(\mathrm{T}\right)\mathrm{=}\mathrm{A}_{\mathrm{2}}\mathrm{+}\mathrm{B}_{\mathrm{2}}\mathrm{T+}\mathrm{C}_{\mathrm{2}}\mathrm{e}^{\mathrm{-5.475}\frac{\mathrm{T}}{\mathrm{T}_{\mathrm{c}}}}\)(4)
\(\mathrm{f}_{\mathrm{3}}\left(\mathrm{T}\right)\mathrm{=}\mathrm{A}_{\mathrm{3}}\mathrm{+}\mathrm{B}_{\mathrm{3}}\mathrm{T}\mathrm{+}\mathrm{C}_{\mathrm{3}}\mathrm{e}^{\mathrm{-5.475}\frac{\mathrm{T}}{\mathrm{T}_{\mathrm{c}}}}\)(5)
\(\mathrm{f}_{\mathrm{4}}\left(\mathrm{T}\right)\mathrm{=}\mathrm{A}_{\mathrm{4}}\)(6)
\(\mathrm{f}_{\mathrm{5}}\left(\mathrm{T}\right)\mathrm{=}\mathrm{B}_{\mathrm{5}}\mathrm{T}\)(7)
The number of the unknown characteristic constants is 10:A 1, A 2,B 2, C 2,A 3, B 3,C 3, A 4,B 5, b .
The initial value conditions of the M-H EOS are expressed as follows9:
\(\mathrm{P}_{\mathrm{c}}\mathrm{=}\mathrm{P}\left(\mathrm{T}_{\mathrm{c}}\mathrm{,}\mathrm{V}_{\mathrm{c}}\right)\)(8)
\(\mathrm{PV=RT}\mathrm{\text{\ as\ }}\mathrm{P}\mathrm{\rightarrow 0}\)(9)
\(\left(\mathrm{\text{dP}}\mathrm{/}\mathrm{\text{dV}}\right)_{\mathrm{T}_{\mathrm{c}}}\mathrm{=0}\)(10)
\(\left(\frac{\mathrm{d}^{\mathrm{2}}\mathrm{P}}{\mathrm{d}\mathrm{V}^{\mathrm{2}}}\right)_{\mathrm{T}_{\mathrm{c}}}\mathrm{=0}\)(11)
\(\left(\frac{\mathrm{d}^{\mathrm{3}}\mathrm{P}}{\mathrm{d}\mathrm{V}^{\mathrm{3}}}\right)_{\mathrm{T}_{\mathrm{c}}}\mathrm{=0}\)(12)
\(\left(\frac{\mathrm{d}^{\mathrm{4}}\mathrm{P}}{\mathrm{d}\mathrm{V}^{\mathrm{4}}}\right)_{\mathrm{T}_{\mathrm{c}}}\mathrm{=0}\)(13)
\(\left[\left(\frac{\mathrm{\text{dZ}}}{\mathrm{d}\mathrm{P}_{\mathrm{r}}}\right)_{\mathrm{T}_{\mathrm{r}}}\right]_{\mathrm{P}_{\mathrm{r}}\mathrm{=0}}\mathrm{=\ -}\left(\mathrm{1-}\mathrm{Z}_{\mathrm{c}}\right)\mathrm{\text{\ \ \ \ \ \ }}\mathrm{\text{at}}\mathrm{\text{\ \ \ \ }}\mathrm{T}^{\mathrm{{}^{\prime}}}\mathrm{\cong}\mathrm{\ 0.8}\mathrm{T}_{\mathrm{c}}\)(14)
\(\left[\left(\frac{\mathrm{\text{dZ}}}{\mathrm{d}\mathrm{P}_{\mathrm{r}}}\right)_{\mathrm{T}_{\mathrm{r}}}\right]_{\mathrm{P}_{\mathrm{r}}\mathrm{=0}}\mathrm{=0}\mathrm{\text{\ \ \ \ \ }}\mathrm{at\ \ \ \ Boyle-Point\ Temperature}\mathrm{\ }\mathrm{T}_{\mathrm{B}}\)(15)
\(\left(\mathrm{\text{dP}}\mathrm{/}\mathrm{\text{dT}}\right)_{\mathrm{V}}\mathrm{=}\mathrm{m}\mathrm{=\ -}\mathrm{M}\frac{\mathrm{P}_{\mathrm{c}}}{\mathrm{T}_{\mathrm{c}}}\mathrm{\text{\ \ \ \ \ }}\mathrm{\text{at}}\mathrm{\text{\ \ \ V}}\mathrm{=}\mathrm{V}_{\mathrm{c}}\)(16)
\(\left(\frac{\mathrm{d}^{\mathrm{2}}\mathrm{P}}{\mathrm{d}\mathrm{T}^{\mathrm{2}}}\right)_{\mathrm{V}}\mathrm{=0}\mathrm{\text{\ \ \ \ \ \ }}\mathrm{\text{at\ }}\mathrm{\text{\ V}}\mathrm{=}\mathrm{V}_{\mathrm{c}}\)(17)
Here, the number of the initial value conditions of the M-H EOS is also 10.
Although the arithmetic solutions of the unknown characteristic constants can be obtained by using the initial value conditions above, there still is a major defect that the precision calculated by the M-H EOS in liquid-phase state is too low in practical applications9,12. In fact, the original EOS was mostly suitable for the calculation in gas-phase state.
After the original M-H EOS was put forward, Martin10,11 and Hou12 independently revised the original M-H EOS in order to extend application range to liquid-phase state. Among them, the new formula modified by Hou is more prominent12. According to the constraint condition of gas-liquid equilibrium, Hou introduced a new initial value condition and a new characteristic constant B 4 into the original M-H EOS. Hou’s modified M-H EOS expands the range of applications to liquid-phase state12.
The formula is similar as the original M-H EOS except that f4 (T) has a change which adds a new characteristic constant B4. The new f4 (T) is expressed as follows12: