Chapter 6, Problem 6.66E

Interpretation Introduction

Interpretation:

Each line in the phase diagram, in terms of the derivative it represents, is to be labeled.

Concept introduction:

Phase diagram represents the different physical states of a substance at different values of temperature and pressure. In the water, the molar volume of solid is greater than the molar volume of the liquid.

Answer to Problem 6.66E

The phase transition between solid phase to liquid phase is represented by the derivative,

$\left(\frac{dp}{dT}\right)=\frac{{\Delta }_{\text{fus}}H}{T\Delta V}$

The phase transition between solid phase to liquid phase is represented by the derivative,

$\left(\frac{dp}{dT}\right)=\frac{{\Delta }_{\text{sub}}H}{T\Delta V}\sim \frac{{\Delta }_{\text{sub}}H}{T{V}_{\text{gas}}}$

The phase transition between solid phase to liquid phase is represented by the derivative,

$\left(\frac{dp}{dT}\right)=\frac{{\Delta }_{\text{vap}}H}{T\Delta V}\sim \frac{{\Delta }_{\text{vap}}H}{T{V}_{\text{gas}}}$

The phase transition between solid phases is represented by the derivative,

$\left(\frac{dp}{dT}\right)=\frac{{\Delta }_{\text{trans}}H}{T\Delta V}$

Explanation of Solution

The phase diagram of water shown in Figure 6.6 is shown below.

Figure 1

In the given diagram, the total numbers of phase transitions represented are,

1. Gas phase to liquid phase

2. Gas phase to ice phase $\text{I}$

3. Liquid phase to ice phase $\text{I}$

4. Liquid phase to ice phase $\text{III}$

5. Liquid phase to ice phase $\text{V}$

6. Liquid phase to ice phase $\text{VI}$

7. Ice phase $\text{I}$ to ice phase $\text{II}$

8. Ice phase $\text{I}$ to ice phase $\text{III}$

9. Ice phase $\text{II}$ to ice phase $\text{III}$

10. Ice phase $\text{II}$ to ice phase $\text{V}$

11. Ice phase $\text{III}$ to ice phase $\text{V}$

12. Ice phase $\text{V}$ to ice phase $\text{VI}$

Thus, the total number of phase transitions are $12$.

The phase transition between solid phase to liquid phase, that is, ice phase $\text{I}$ to liquid phase, ice phase $\text{III}$ to liquid phase, ice phase $\text{V}$ to liquid phase and ice phase $\text{VI}$ to liquid phase is represented by the derivative,

$\left(\frac{dp}{dT}\right)=\frac{{\Delta }_{\text{fus}}H}{T\Delta V}$

Where,

• ${\Delta }_{\text{fus}}H$ is the enthalpy of fusion.

• $\Delta V$ is the change in molar volume.

The phase transition between gas phase to ice phase $\text{I}$ is represented by the derivative,

$\left(\frac{dp}{dT}\right)=\frac{{\Delta }_{\text{sub}}H}{T\Delta V}\sim \frac{{\Delta }_{\text{sub}}H}{T{V}_{\text{gas}}}$

Where,

• ${\Delta }_{\text{fus}}H$ is the enthalpy of sublimation.

• $\Delta V$ is the change in molar volume.

The volume of gas is much greater than the volume of solid. Therefore, it can be neglected.

The phase transition between solid phase and liquid phase is represented by the derivative,

$\left(\frac{dp}{dT}\right)=\frac{{\Delta }_{\text{vap}}H}{T\Delta V}\sim \frac{{\Delta }_{\text{vap}}H}{T{V}_{\text{gas}}}$

Where,

• ${\Delta }_{\text{vap}}H$ is the enthalpy of vaporization.

• $\Delta V$ is the change in molar volume.

The volume of gas is much greater than the volume of liquid. Therefore, it can be neglected.

The phase transition between solid phases is represented by the derivative,

$\left(\frac{dp}{dT}\right)=\frac{{\Delta }_{\text{trans}}H}{T\Delta V}$

Where,

• ${\Delta }_{\text{trans}}H$ is the enthalpy of phase transition.

• $\Delta V$ is the change in molar volume.

Conclusion

The phase transition between solid phase to liquid phase is represented by the derivative,

$\left(\frac{dp}{dT}\right)=\frac{{\Delta }_{\text{fus}}H}{T\Delta V}$

The phase transition between solid phase to liquid phase is represented by the derivative,

$\left(\frac{dp}{dT}\right)=\frac{{\Delta }_{\text{sub}}H}{T\Delta V}\sim \frac{{\Delta }_{\text{sub}}H}{T{V}_{\text{gas}}}$

The phase transition between solid phase to liquid phase is represented by the derivative

$\left(\frac{dp}{dT}\right)=\frac{{\Delta }_{\text{vap}}H}{T\Delta V}\sim \frac{{\Delta }_{\text{vap}}H}{T{V}_{\text{gas}}}$

The phase transition between solid phases is represented by the derivative,

$\left(\frac{dp}{dT}\right)=\frac{{\Delta }_{\text{trans}}H}{T\Delta V}$