Chemistry $^{3}$ : Introducing inorganic, organic and physical chemistry

Andrew Burrows, John Holman, Andrew Parsons

Chapter 17 Phase equilibrium and solutions - all with Video Answers

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Chapter Questions

Problem 1

The figure shows a generalized phase diagram. (Section 17.1)(a) What phase(s) is/are present at W, $X, Y$, and Z?
(b) What is the significance of points $X, Y, T$, and $C$ ?
(c) What phase change would occur if you started at point $\mathrm{U}$ and:
(i) increased the temperature with the pressure constant;
(ii) increased the pressure with the temperature constant?

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Problem 2

For the phase diagram below (Section 17,1$)$ :(a) Which area of the diagram $(A, B$, or $C)$ represents the liquid phase?
(b) Which phase(s) can exist at SATP?
(c) If a sample of this substance is cooled from $250 \mathrm{~K}$ to $50 \mathrm{~K}$ at a pressure of $20 \mathrm{bar}$, what phase changes will occur?
(d) What name is given to the change from phase $A$ to phase $\mathrm{C}$ ?

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Problem 3

Use the phase diagrams in Figures $17.5$ to $17.8$ (pp.769-772) to answer the following questions. (Section 17.1)
(a) $\mathrm{CO}_{2}$ fire extinguishers contain liquid $\mathrm{CO}_{2}$ in equilibrium with gas. At $298 \mathrm{~K}$, ostimate the pressure of $\mathrm{CO}_{2}$ (g) above the liquid.
(b) Which form of sulfur will crystallize from liquid sulfur at normal atmospheric pressure?
(c) Which has the lower density: Ice I or lce II?
(d) What will happen when water is:
(i) heated from $250 \mathrm{~K}$ to $300 \mathrm{~K}$ at a pressure of $0.005 \mathrm{bar}$; then
(ii) compressed to 10 bar with the temperature held at $300 \mathrm{~K}$; then
(iii) cooled to $250 \mathrm{~K}$ with the pressure held at 10 bar?

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Problem 4

The conditions at the triple point of methanol are $-98.15^{\circ} \mathrm{C}$ and $0.5$ atm while those at its critical point are $240^{\circ} \mathrm{C}$ and $78.5 \mathrm{~atm}$.
Use this information to sketch (not to scale) and fully label a phase diagram for methanol. (Section 17.1)

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Problem 5

Mercury has a melting point of $234.3 \mathrm{~K}$. At this temperature, the density of $\mathrm{Hg}$ (i) is $13.690 \mathrm{gcm}^{-3}$ and that of $\mathrm{Hg}(\mathrm{s})$ is $14.193 \mathrm{gcm}^{-3}$. The enthalpy change of fusion is $9.75 \mathrm{Jg}^{-1}$. Find the pressure needed to change the melting temperature by 1K. (Section 17.2)

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Problem 6

Sulfur melts at a temperature of $115.2^{\circ} \mathrm{C}$ under a pressure of 1 atm. In one process for its mining, sulfur is melted and pumped from underground deposits as a liquid under a pressure of 6 bar. Use the Clapeyron equation to estimate the lowest temperature at which sulfur is a liquid at this pressure. (Section 17.2)
(The enthalpy change of fusion at $115.2^{\circ} \mathrm{C}$ is $53.67 \mathrm{~J} \mathrm{~g}^{-1}$ and the densities of solid and liquid sulfur are $2.15 \mathrm{~g} \mathrm{~cm}^{-3}$ and $1.811 \mathrm{gcm}^{-3}$, respectively.)

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Problem 7

Explain the observation that, if a liquid evaporates from an open container, the temperature remains about constant but, if it evaporates from an insulated flask, the temperature falls. (Section 17.2)

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Problem 8

The vapour pressure of tetrachloromethane $\left(\mathrm{CCl}_{4}\right)$ at $0^{\circ} \mathrm{C}$ is $44.0 \mathrm{mbar}$ and at $50^{\circ} \mathrm{C}$ it is $422.0 \mathrm{mbar}$. Use these data to estimate the enthalpy change of vaporization of $\mathrm{CCl}_{4}$ and to estimate its standard boiling point. (Section 17.2)

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Problem 9

Mount Everest is the highest mountain on Earth with a height of $8850 \mathrm{~m}$. At this altitude, the atmospheric. pressure is about one-third that at sea-level. Find the melting temperature and boiling temperature of water at this pressure. (Section 17.2) (The densities of ice and water at $0^{\circ} \mathrm{C}$ are $0.92 \mathrm{gcm}^{-3}$ and $1.00 \mathrm{~g} \mathrm{~cm}^{-3}$, respectively, and the enthalpy change of fusion of ice is $+6.01 \mathrm{~kJ} \mathrm{~mol}^{-1}$. The enthalpy change of vaporization of $\mathrm{H}_{2} \mathrm{O}(\mathrm{l})$ at the normal boiling point is $+40.7 \mathrm{~kJ} \mathrm{~mol}^{-1}$.)

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Problem 10

Determine the melting point of ice under a pressure of 500 bar. The density of ice at $0^{\circ} \mathrm{C}$ is $0.9168 \mathrm{gcm}^{-3}$ and that of water is $0.9987 \mathrm{gcm}^{-3}$. The enthalpy of fusion of ice is $6.008 \mathrm{~kJ} \mathrm{~mol}^{-1}$. (Section 17.2)

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Problem 11

The vapour pressure exerted by carbon disulfide $\left(\mathrm{CS}_{2}\right)$ was measured at different temperatures.Determine the onthalpy and entropy changes of vaporization of $\mathrm{CS}_{2}$. (Section 17.2)

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Problem 12

The vapour pressures of sulfur dioxide in the solid and liquid states are given by:
Solid: $\operatorname{In}(p / \mathrm{Pa})=29.28-\left(\frac{4308}{T / K}\right)$
Liquid: $\ln (p / P a)=24.055-\left(\frac{3284}{T / K}\right)$
Calculate (i) the temperature and pressure of the triple point of sulfur dioxide and (ii) the enthalpy and entropy changes of: fusion at the triple point. (Section 17.2)

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Problem 13

The vapour pressure of a liquid is $2.026 \mathrm{kPa}$ at $20.0^{\circ} \mathrm{C}$ and $44,209 \mathrm{kPa}$ at $80^{\circ} \mathrm{C}$.
(a) Find the enthalpy change of vaporization of the liquid.
(b) Estimate the normal boiling point of the liquid.
(c) Trouton's rule states that $\mathrm{A}_{\mathrm{vap}} \mathrm{S}$ for liquids $\mathrm{r} 88 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$. Comment on the liquid under investigation here.
(Section 17.2)

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Problem 14

At 1 bar, two solid forms of tin (grey tin and white tin) are in equilibrium at $291 \mathrm{~K}$. The densities of the two forms are $5750 \mathrm{kgm}^{-3}$ and $7280 \mathrm{~kg} \mathrm{~m}^{-3}$ respectively. The entropy change for the transition is $8.8 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}$. At what temperature will the two forms be in equilibrium at a pressure of 200 bar? (Section 17.2).

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Problem 15

Calculate the energy of attraction between a $\mathrm{Mg}^{2+}$ ion and a $\mathrm{Cl}^{\text {- ion separated by }} 0.2 \mathrm{~nm}$ in a vacuum. (Section 17.3)

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Problem 16

Compare the energy of interaction between two singly charged ions separated by a distance of $0.1 \mathrm{~nm}$ in (a) vacuum, (b) water (relative permittivity $=78$ ), (c) a hydrocarbon oil (relative permittivity = 2). (Section 17.3)

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Problem 17

- Show that for a molecule following the Lennard-Jones potential, the equilibrium separation, $r_{0}$, is given by $r_{0}=2^{1 / 6} \sigma$.

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Problem 18

For each of the following pairs, which compound would you expect to have the higher enthalpy change of vaporization? Give the reason for your answer in each case. (Section 17.3)
(a) Ethanol and methoxymethane
(b) Propane and methoxymethane
(c) Butane and propanone
(d) (E) $-1,2$-dichloroethene and $(Z)-1,2$ - dichloroethene.

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Problem 19

Arrange the following in order of increasing boiling point, giving detailed reasons for your answer. (Section 17.3)
$\mathrm{He}, \mathrm{H}_{2} \mathrm{O}, \mathrm{Ar}, \mathrm{N}_{2}, \mathrm{HCl}, \mathrm{NaCl}$

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Problem 20

The figure below shows the Lennard-Jones potential for three noble gases. Explain the variation in shapes of the three curves. (Section 17.3)

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Problem 21

A diver descends to a depth where the pressure is $5 \mathrm{~atm}$. The diver's body contains around $5 \mathrm{dm}^{3}$ of blood. The Henry's law constant for $\mathrm{N}_{2}(\mathrm{~g})$ in water at $310 \mathrm{~K}$ is $1.07 \times 10^{5} \mathrm{~atm}$. (Section 17.4)
(a) Calculate the amount of nitrogen gas absorbed from the air in the diver's blood at a pressure of $1 \mathrm{~atm}$ and at $5 \mathrm{~atm}$. (Assume that the solubility of nitrogen in the blood is the same as in water.)
(b) If all the gas dissolved at 5 atm was suddenly released, what volume would it occupy at $1 \mathrm{~atm}$ and $298 \mathrm{~K}$ ?

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Problem 22

The vapour pressure of pure $\mathrm{CHCl}_{3}$ at $318 \mathrm{~K}$ is $58 \mathrm{kPa}$. What would be the partial vapour pressure of $\mathrm{CHCl}_{3}$ above a mixture of $1 \mathrm{~mol}$ of $\mathrm{CHCl}_{3}$ with $1 \mathrm{~mol}$ of ethanol, assuming the mixture behaves as an ideal solution? Comment on the fact that the measured value of the partial vapour pressure of $\mathrm{CHCl}_{3}$ above the mixture is $42 \mathrm{kPa}$. (Section 17.4)

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Problem 23

Ethanol and methanol form nearly ideal solutions. At $20^{\circ} \mathrm{C}$, the vapour pressure of pure ethanol is $5930 \mathrm{~Pa}$ and of methanol is 11830 Pa. (Section 17.4) Calculate(a) the vapour pressure of each compound and
(b) the total vapour pressure over a solution formed by mixing $50 \mathrm{~g}$ of each.
$$
\left.\left(\mathrm{M}_{\mathrm{r}}\left(\mathrm{CH}_{3} \mathrm{OH}\right)=32.04 ; \mathrm{M}_{r} \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)=46.07 .\right)
$$

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Problem 24

In a solution of ethanol $\left(p^{\circ}=0.174\right.$ bar) and 2-methyihexane $\left(p^{\circ}=0.059 \mathrm{bar}\right)$, the mole fraction of ethanol, $x_{\text {ehand }}$, is $0.90$. The vapour in equilibrium with the solution has a total vapour pressure of $0.248 \mathrm{bar}$. The mole fraction of ethanol in the vapour is $0.67$. Calculate the activity coefficients of each component in the solution. (Section 17.4)

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Problem 25

A protein has a molar mass of $69000 \mathrm{gmol}^{-1}$. Assuming ideal behaviour, calculate the osmotic pressure of a solution containing $20 \mathrm{~g} \mathrm{dm}^{-3}$ of the protein at $298 \mathrm{~K}$. (Section 17.4).

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Problem 26

The figure shows a sketch of the vapour-liquid phase diagram for mixtures of butanone and dichloromethane. (Section 17.4)

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Problem 27

A solution of iodoethane, I, and propanone, P, with a mole. fraction of $I, x_{1}=0.55$, had a partial vapour pressure of $I$ of $28.44 \mathrm{kPa}$ and a partial vapour pressure of $\mathrm{P}$ of $19.21 \mathrm{kPa}$ at $50^{\circ} \mathrm{C}$. At this temperature, the saturated vapour pressure of I is $47.12 \mathrm{kPa}$ and of $\mathrm{P}$ is $37.38 \mathrm{kPa}$. Calculate the activity coefficients of both components in the solution. (Section 17.4)

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Problem 28

When a patient suffers severe dehydration, why will medical treatment involve giving 'saline' solution rather than pure water? (Section 17.4)

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Problem 29

Use the Phase Rule to calculate the number of degrees of freedom at the triple point of water. Comment on your answer. (Section 17.4)

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Problem 30

The vapour pressure over a mixture of $2 \mathrm{~mol}$ of hexane and $1 \mathrm{~mol}$ of octane is $9.6 \mathrm{kPa}$ at $40^{\circ} \mathrm{C}$. An eq uimolar solution has a vapour pressure of $8.2 \mathrm{kPa}$ at the same temperature. What are the vapour pressures of the two pure liquids? (Section 17.4)

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Problem 31

The vapour pressure of puro toluene is $0.0285$ bar at $20^{\circ} \mathrm{C}$ and that of benzene is $0.0974$ bar. One mole of each compound were mixed and formed an ideal solution. Calculate the mole fraction of each compound in the vapour over the solution. $($ Section $17.4)$

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Problem 32

The transport of water up a tree occurs partially through osmosis; the concentration of sugar in the tree sap is higher than the water around the tree roots. The sap in a certain species of tree can be represented as a $30 \mathrm{~g} \mathrm{dm}^{-3}$ solution of sucrose in water. Find the osmotic pressure generated by this solution. (Section 17.4)

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