Chapter 12, Problem 12.21P

Interpretation Introduction

Interpretation:

The evolution of phase and its compositions are to be described during the process along with the feasibility of carrying this process. The device needed to carry this process is to be determined. Also, the change in total volume of system during the process is to be estimated along with the composition at which the volume will reach its maximum value.

Concept Introduction:

In a $P-x-y$ diagram, the line $p-x$ represents the saturated liquid state and the region of subcooled liquid lies above this line. The curve $P-y$ represents the states of saturated vapor and the region below this curve is superheated vapor.

All the points that lies in between these curves are in two-phase region, where saturated liquid and saturated vapor coexist in equilibrium. At the edge of the $P-x-y$ diagram, the two curves meet where both saturated liquid and saturated vapor of pure species coexist at the vapor pressure $P_1^{\text{sat}}\text{ and }{P}_2^{\text{sat}}$ .

Answer to Problem 12.21P

For the given process, the phase evaluation is done along the horizontal straight line from point “a” to point “f”.

At point “a”, the system is in superheated vapor state with composition $y_1\left(\text{a}\right)=0.0\text{ and }{y}_2\left(\text{a}\right)=1.0$ .

At point “b” where first drop of saturated liquid appears and both saturated liquid and saturated vapor coexist in equilibrium. Phase composition at point “b” becomes:

  $\begin{array}{c}{y}_1\left(\text{b}\right)=0.12\\ y_2\left(\text{b}\right)=0.88\\ x_1\left(\text{c}\right)=0.205\\ x_2\left(\text{c}\right)=0.795\end{array}$

At point “c”, last bubble of the saturated vapor condenses, and system is in saturated liquid state.

The composition of the system at this point “c” will be:

  $\begin{array}{c}{x}_1\left(\text{c}\right)=0.205\\ x_2\left(\text{c}\right)=0.795\\ y_1\left(\text{b}\right)=0.12\\ y_2\left(\text{b}\right)=0.88\end{array}$

All the points lying on the straight-line connecting points “b” and “c” coexist in both the phases in equilibrium with total composition of the system varying from $x_1=0.12\text{ to 0}\text{.205}$ .

Between points “c” and “d” the system is in superheated vapor state with composition range of $x_1\left(\text{c}\right)=0.205\text{ to }{x}_1\left(\text{d}\right)=0.69$ .

At this point “d”, the system reaches the state of saturated liquid. At this point, first bubble of saturated vapor appears which in equilibrium with the saturated liquid.

The composition of the system at this point “d” will be:

  $\begin{array}{c}{x}_1\left(\text{d}\right)=0.69\\ x_2\left(\text{d}\right)=0.31\\ y_1\left(\text{e}\right)=0.79\\ y_2\left(\text{e}\right)=0.21\end{array}$

At point “e”, last drop of the saturated liquid evaporates, and system is in saturated vapor state.

The composition of the system at this point “e” will be:

  $\begin{array}{c}{y}_1\left(\text{e}\right)=0.69\\ y_2\left(\text{e}\right)=0.31\\ x_1\left(\text{d}\right)=0.79\\ x_2\left(\text{d}\right)=0.21\end{array}$

In between the points “d” and “e”, both the phases saturated vapor and saturated liquid coexist in equilibrium and overall composition of the system varies as moving from “d” and “e”.

At point “f” where there is $1\text{ mol}$ of tetrahydrofuran and $9\text{ mol}$ of chloroform present, the system is in superheated vapor state. The composition at this point is $y_1=0.9\text{ and }{y}_2=0.1$ .

The given process is not practically feasible.

There is no device which can be used to carry out such processes.

There is a proportional increase in the volume of the system with the increase in the total moles of the system.

The volume will reach maximum at the highest number of moles present in the system at which the composition of the system will be:

  $\begin{array}{c}{y}_1=0.9\\ y_2=0.1\end{array}$

Explanation of Solution

Given information:

$1\text{ mol}$ of pure tetrahydrofuran at ${50}^{\circ }\text{C and 52 kPa}$ is present in a closed vessel. In this vessel, pure chloroform is slowly added at constant temperature and pressure till it contains $1\text{ mol}$ tetrahydrofuran acetate and $9\text{ mol}$ chloroform.

Below shown is the $P-x-y$ diagram for the system containing chloroform(1)/tetrahydrofuran(2) at ${50}^{\circ }\text{C}$ .

Initially, there is $1\text{ mol}$ of tetrahydrofuran in a closed vessel at ${50}^{\circ }\text{C and 52 kPa}$ . Thus, there is no chloroform present in the vessel, so, $x_1=0$ . This is represented by point “a” on the $P-x-y$ diagram.

Now, keeping the temperature and pressure of the system constant, pure chloroform is slowly added to the vessel until there is $1\text{ mol}$ of tetrahydrofuran and $9\text{ mol}$ of chloroform present in the vessel. Thus, the final composition of the vessel becomes $x_1=0.9$ .

On a $P-x-y$ diagram for chloroform(1)/tetrahydrofuran(2) system at $52{\text{ kPa and 50}}^{\circ }\text{C}$, this process is shown by a horizontal straight line from point “a” to point “f”.

Point “a” lies below the $P-y$ curve, so the system is in superheated vapor state at this point having composition,

  $\begin{array}{c}{y}_1\left(\text{a}\right)=0.0\\ y_2\left(\text{a}\right)=1.0\end{array}$

Slow addition of pure chloroform gradually shifts the system from point “a” to point “b” where is touches $P-y$ curve which represents saturated vapor state. At this point first drop of saturated liquid appears. Point “b” corresponds to the composition of the saturated vapor and point “c” corresponds to the composition of the first drop (saturated liquid) which is in equilibrium with this saturated vapor.

Therefore, the composition at this point “b” will be:

  $\begin{array}{c}{y}_1\left(\text{b}\right)=0.12\\ y_2\left(\text{b}\right)=0.88\\ x_1\left(\text{c}\right)=0.205\\ x_2\left(\text{c}\right)=0.795\end{array}$

At point “c”, last bubble of the saturated vapor condenses, and system is in saturated liquid state. Point “c” corresponds to the composition of the saturated liquid and point “b” corresponds to the composition of the last bubble (saturated vapor) which is in equilibrium with this saturated liquid.

The composition of the system at this point “c” will be:

  $\begin{array}{c}{x}_1\left(\text{c}\right)=0.205\\ x_2\left(\text{c}\right)=0.795\\ y_1\left(\text{b}\right)=0.12\\ y_2\left(\text{b}\right)=0.88\end{array}$

In between the points “b” and “c”, both the phases saturated vapor and saturated liquid coexist in equilibrium and overall composition of the system varies as moving from “b” and “c”.

Addition of chloroform beyond point “c” leads the system to point “d”

In this range the system is in subcooled liquid state with overall composition of the system varying from $x_1\left(\text{c}\right)=0.205\text{ to }{x}_1\left(\text{d}\right)=0.69$ .

At this point “d”, the system reaches the state of saturated liquid as it is on the curve $P-x$ . At this point, first bubble of saturated vapor appears which in equilibrium with the saturated liquid. Point “d” corresponds to the composition of the saturated liquid and point “e” corresponds to the composition of the first bubble (saturated vapor) which is in equilibrium with this saturated liquid.

The composition of the system at this point “d” will be:

  $\begin{array}{c}{x}_1\left(\text{d}\right)=0.69\\ x_2\left(\text{d}\right)=0.31\\ y_1\left(\text{e}\right)=0.79\\ y_2\left(\text{e}\right)=0.21\end{array}$

At point “e”, which lies on $P-y$ curve, last drop of the saturated liquid evaporates, and system is in saturated vapor state. Point “e” corresponds to the composition of the saturated vapor and point “d” corresponds to the composition of the last drop (saturated liquid) which is in equilibrium with this saturated vapor.

The composition of the system at this point “e” will be:

  $\begin{array}{c}{y}_1\left(\text{e}\right)=0.69\\ y_2\left(\text{e}\right)=0.31\\ x_1\left(\text{d}\right)=0.79\\ x_2\left(\text{d}\right)=0.21\end{array}$

In between the points “d” and “e”, both the phases saturated liquid and saturated vapor coexist in equilibrium and overall composition of the system varies as moving from “d” and “e”.

Further addition of pure chloroform to the vessel takes the system to point “f” where there is $1\text{ mol}$ of tetrahydrofuran and $9\text{ mol}$ of chloroform present. This point lies below the $P-y$ curve so it is in superheated vapor state. The composition at this point is $y_1=0.9\text{ and }{y}_2=0.1$ .

For a two-phase binary system, degree of freedom is $2$ . Thus, keeping temperature and pressure constant leads to fixed vapor and liquid composition of the system which contradicts the process described.

Therefore, the given process is not practically feasible.

Since, the given process is not practically feasible, there is no device which can be used to carry out such processes.

As temperature and pressure of the system is kept constant throughout the process, there will be proportional increase in the volume of the system as number of moles are increasing with the addition of pure ethanol in the vessel.

Since, the increase in the system volume is proportional to the moles added to the system, the maximum volume will appear at the highest number of moles present in the system, that is, $1\text{ mol}$ of tetrahydrofuran and $9\text{ mol}$ of chloroform. The composition of the system at this point is:

  $\begin{array}{c}{y}_1=0.9\\ y_2=0.1\end{array}$

Conclusion

The evolution of phase and its compositions are described during the process along with the feasibility of carrying this process.

The device needed to carry this process is determined.

The change in total volume of system during the process is estimated along with the composition at which the volume will reach its maximum value.