Calculate Z and V for ethylene at 25°C and 12 bar by the following equations: (a) The truncated virial equation [Eq. (3.38)] with the following experimental values of virial coefficients: B = −140 cm3·mol−1 C = 7200 cm6·mol−2 (b) The truncated virial equation [Eq. (3.36)], with a value of B from the generalized Pitzer correlation [Eqs. (3.58)–(3.62)] (c) The Redlich/Kwong equation (d) The Soave/Redlich/Kwong equation (e) The Peng/Robinson equation

[tex]T_C = 282.3 \ K \\\\T = 298.15 \ K \\\\ T_r = \frac{T}{T_c} \\\\ T_r= 1.056 \\\\P_c=50.4 \bar \\\\ P = \bar{12} \\\\ P_r= \frac{P}{P_c} \\\\P_r= 0.238\\\\[/tex]

To find:

points=?

Solution:

a)

[tex]B=-140 \ \frac{cm^3}{mol} \\\\C = 7200 \ \frac{cm^6}{mol^2}\\\\ V=\frac{RT}{P}=2066 \ \frac{cm^3}{mol}\\\\[/tex]

Given:

[tex]\frac{PV}{RT}= 1+ \frac{B}{V}+\frac{C}{V^2}\\\\ V=?\\\\ V= 1919 \ \frac{cm^3}{mol}\\\\Z=\frac{P.V}{R.T}= 0.929\\\\[/tex]

(b)

[tex]B_0 = 0.083 -\frac{0.422}{T^{1.6}_{r}}= -0.304\\\\ B_1= 0.139 -\frac{0.172}{T^{4.2}_{r}}= 2.262 \times 10^{-3}\\\\ z=1+(B_0+\omega B_1 \ \frac{P_r}{T_r} = 0.932 \\\\ V= \frac{Z\cdot R \cdot T}{P} = 1924 \frac{cm^3}{mol}[/tex]

(c)

For Redlich/Kwong EOS:

[tex]\sigma=1\\\\ \varepsilon = 0\\\\ \Omega = 0.08664\\\\ \Psi = 0.42748 \\\\[/tex]

[tex]\alpha (T_r) = T_{r}^{-0.5}\\\\ q\ T_r= \frac{\Psi \alpha \ T_r}{\Omega \ T_r}\\\\\beta\ T_r , P_r= \frac{\Omega \ P_r}{ T_r}\\\\[/tex]

Calculating the value of Z and by assume: [tex]Z=0.9\\\\[/tex]

Given:

[tex]\to z = 1+B(T_r), P_r - q (T_r) \beta (T_r), P_r \frac{Z-\beta(T_r) , P_r}{(Z)+\varepsilon \beta (T_r), P_r (Z)+\sigma \beta (T_r), P_r }[/tex]

Finding the value of z:

[tex]Z=0.928\\\\ V=\frac{ZRT}{P} = 1916.5 \ \frac{cm^3}{mol}\\\\[/tex]

(d)

For SRK EOS:

[tex]\sigma =1\\\\\varepsilon = 0\\\\ \Omega= 0.08664\\\\ \Psi= 0.42748 \\\\\alpha (T_r) \omega =[ 1+ 0.480 + 1.574 \omega -0.176 \omega^2 (1-T_r^{\frac{1}{2}})]^2\\\\ \alpha (T_r)=\frac{\Psi \alpha (T_r) , \omega}{\Omega \ T_r}\\\\\beta (T_r), P_r=\frac{\omega\ P_r}{T_r}\\\\[/tex]

Given:

[tex]z = 1+B(T_r), P_r - q (T_r) \beta (T_r), P_r \frac{Z-\beta(T_r) , P_r}{(Z)+\varepsilon \beta (T_r), P_r (Z)+\sigma \beta (T_r), P_r }[/tex]

Finding the value of z:

[tex]z=0.928\\\\V=\frac{ZRT}{P} = 1918 \ \frac{cm^3}{mol}\\\\[/tex]

(e)

For Peng/Robinson EOS:

[tex]\sigma =1+\sqrt{2}\\\\\varepsilon = 1-\sqrt{2}\\\\ \Omega= 0.07779\\\\ \Psi= 0.45724 \\\\\alpha (T_r) \omega =[ 1+ 0.37464 + 1.5422 \omega -0.26992 \omega^2 (1-T_r^{\frac{1}{2}})]^2\\\\ \alpha (T_r)=\frac{\Psi \alpha (T_r) , \omega}{\Omega \ T_r}\\\\\beta (T_r), P_r=\frac{\omega\ P_r}{T_r}\\\\[/tex]

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