Chapter 7, Problem 7.1CP

Interpretation Introduction

Interpretation:Two gases in corresponding states expected to have the same value of z should be explained by using the concept of intermolecular potential.

Concept introduction:Interactions between molecules and the behaviors of P, V and T in many fluids and fluid mixtures are referred to as the intermolecular potential concept.

Gas compression factor $z=\frac{P{V}_m}{RT}$

Here, P is pressure, $V_m$ is molar volume, R is Universal gas constant and T is temperature.

Answer to Problem 7.1CP

Intermolecular potential concept shown that; two different gases can have different values for the distance at which the potential becomes positive and for the depth of the intermolecular potential. The differences in the intermolecular potentials are to a significant extent can be normalize by normalizing P,T and V to their critical values.

Explanation of Solution

The volume factor for a van der Waals gas can be obtained from the concept of intermolecular potential can be shown below:

  $b=$ Intermolecular repulsion potential

  $a=$ Intermolecular attraction potential

Assumption: A and B are gases which are in corresponding states and the respective intermolecular potentials for that gases are $a_A,a_B,b_A\&b_B$ .

Reduced parameters for temperature (T), pressure (P) and volumes (V) are as below for component A;

  $\begin{array}{l}{T}_r{}^A=\frac{T^A}{T_c{}^A}\mathrm{..........}(1)\\ T^A=T_r{}^A{T}_c{}^A\\ P_r{}^A=\frac{P^A}{P_c{}^A}\mathrm{..........}(2)\\ P^A=P_r{}^A{P}_c{}^A\\ V_{mr}{}^A=\frac{V_{mc}{}^A}{V_{mc}{}^A}\\ V_m{}^A=V_{mr}{}^A{V}_{mc}{}^A\mathrm{..........}(3)\end{array}$

Reduced parameters for temperature (T), pressure (P) and volumes (V) are as below for component B;

  $\begin{array}{l}{T}_r{}^B=\frac{T^B}{T_c{}^B}\mathrm{..........}(4)\\ T^B=T_r{}^B{T}_c{}^B\\ P_r{}^B=\frac{P^B}{P_c{}^B}\mathrm{..........}(5)\\ P^B=P_r{}^B{P}_c{}^B\\ V_{mr}{}^B=\frac{V_{mc}{}^B}{V_{mc}{}^B}\\ V_m{}^B=V_{mr}{}^B{V}_{mc}{}^B\mathrm{..........}(6)\end{array}$

Gas compression factor; $z=\frac{P{V}_m}{RT}$ ,

Hence; $z_A=\frac{P^A{V}_m{}^A}{R{T}^A}=\frac{P_r{}^A{P}_c{}^A{V}_{mr}{}^A{V}_{mc}{}^A}{R{T}_r{}^A{T}_c{}^A}$

From equation $(1),(2)and(3)$

  $z_A=\left(\frac{P_r{}^A{V}_{mr}{}^A}{T_r{}^A}\right)\left(\frac{P_c{}^A{V}_{mc}{}^A}{R{T}_c{}^A}\right)\mathrm{..........}(7)$

  $\begin{array}{c}{z}_B=\frac{P^B{V}_m{}^B}{R{T}^B}\\ =\frac{P_r{}^B{P}_c{}^B{V}_{mr}{}^B{V}_{mc}{}^B}{R{T}_r{}^B{T}_c{}^B}\end{array}$

From equation $(4),(5)and(6)$

  $z_B=\left(\frac{P_r{}^B{V}_{mr}{}^B}{T_r{}^B}\right)\left(\frac{P_c{}^B{V}_{mc}{}^B}{R{T}_c{}^B}\right)\mathrm{..........}(8)$

Here, a and b are van der Waals constants.

The critical pressure is; $P_c=\frac{a}{27b^2}$

The critical volume is; $V_{mc}=3b$

The critical temperature is; $T_c=\frac{8a}{27Rb}$

Hence.

  $\begin{array}{c}\frac{P_c{V}_{mc}}{R{T}_c}=\frac{1}{R}\times \frac{a}{27b^2}\times 3b\times \frac{27Rb}{8a}\\ =\frac{3}{8}\end{array}$

Furthermore, two gases are in the corresponding states thus,

  $\begin{array}{l}{T}_r{}^A=T_r{}^B\\ P_r{}^A=P_r{}^B\\ V_{mr}{}^A=V_{mr}{}^B\end{array}$

Or,

  $\frac{P_r{}^A{V}_{mr}{}^A}{T_r{}^A}=\frac{P_r{}^B{V}_{mr}{}^B}{T_r{}^B}$

  $\begin{array}{l}{z}_A=\frac{3}{8}\left(\frac{P_r{}^A{V}_{mr}{}^A}{T_r{}^A}\right)\\ \left(\frac{P_r{}^A{V}_{mr}{}^A}{T_r{}^A}\right)=\frac{8}{3}{z}_A\end{array}$

  $\begin{array}{l}{z}_B=\frac{3}{8}\left(\frac{P_r{}^B{V}_{mr}{}^B}{T_r{}^B}\right)\\ \left(\frac{P_r{}^B{V}_{mr}{}^B}{T_r{}^B}\right)=\frac{8}{3}{z}_B\end{array}$

Hence,

  $\begin{array}{l}\frac{8}{3}{z}_A=\frac{8}{3}{z}_B\\ z_A=z_B\end{array}$

Thus, intermolecular potential concept shows that the two different gases can have different values for the distance at which the potential becomes positive and for the depth of the intermolecular potential. The differences in the intermolecular potentials are to a significant extent can be normalize by normalizing P,T and V to their critical values.

Conclusion

Therefore, intermolecular potential concept shown that; two different gases can have different values for the distance at which the potential becomes positive and for the depth of the intermolecular potential. The differences in the intermolecular potentials are to a significant extent can be normalize by normalizing P,T and V to their critical values.