The vapour pressure of a liquid between 15C and 35C fits the expression log(p/Torr) =

Discuss the implications for phase stability of the variation of chemical potential with temperature and pressure.

Suggest a physical interpretation of the phenomena of superheating and supercooling.

Discuss what would be observed as a sample of water is taken along a path that encircles and is close to its critical point.

Use the phase diagram in Fig. 4.4 to state what would be observed when a sample of carbon dioxide, initially at 1.0 atm and 298 K, is subjected to the following cycle: (a) isobaric (constantpressure) heating to 320 K, (b) isothermal compression to 100 atm, (c) isobaric cooling to 210 K, (d) isothermal decompression to 1.0 atm, (e) isobaric heating to 298 K.

The use of supercritical fluids for the extraction of a component from a complicated mixture is not confined to the decaffeination of coffee. Consult library and internet resources and prepare a discussion of the principles, advantages, disadvantages, and current uses of supercritical fluid extraction technology.

Explain the significance of the Clapeyron equation and of the ClausiusClapeyron equation.

Distinguish between a first-order phase transition, a second-order phase transition, and a \(\lambda\)-transition at both molecular and macroscopic levels.

The vapour pressure of dichloromethane at 24.1C is 53.3 kPa and its enthalpy of vaporization is 28.7 kJ mol1. Estimate the temperature at which its vapour pressure is 70.0 kPa.

The vapour pressure of a substance at 20.0C is 58.0 kPa and its enthalpy of vaporization is 32.7 kJ mol1. Estimate the temperature at which its vapour pressure is 66.0 kPa.

The molar volume of a certain solid is 161.0 cm3 mol1 at 1.00 atm and 350.75 K, its melting temperature. The molar volume of the liquid at this temperature and pressure is 163.3 cm3 mol1. At 100 atm the melting temperature changes to 351.26 K. Calculate the enthalpy and entropy of fusion of the solid.

The molar volume of a certain solid is 142.0 cm3 mol1 at 1.00 atm and 427.15 K, its melting temperature. The molar volume of the liquid at this temperature and pressure is 152.6 cm3 mol1. At 1.2 MPa the melting temperature changes to 429.26 K. Calculate the enthalpy and entropy of fusion of the solid.

The vapour pressure of a liquid in the temperature range 200 K to 260 K was found to fit the expression ln(p/Torr) = 16.255 2501.8/(T/K). Calculate the enthalpy of vaporization of the liquid.

The vapour pressure of a liquid in the temperature range 200 K to 260 K was found to fit the expression ln(p/Torr) = 18.361 3036.8/(T/K). Calculate the enthalpy of vaporization of the liquid.

The vapour pressure of benzene between 10C and 30C fits the expression log(p/Torr) = 7.960 1780/(T/K). Calculate (a) the enthalpy of vaporization and (b) the normal boiling point of benzene.

The vapour pressure of a liquid between 15C and 35C fits the expression log(p/Torr) = 8.750 1625/(T/K). Calculate (a) the enthalpy of vaporization and (b) the normal boiling point of the liquid.

When benzene freezes at 5.5C its density changes from 0.879 g cm3 to 0.891 g cm3. Its enthalpy of fusion is 10.59 kJ mol1. Estimate the freezing point of benzene at 1000 atm.

When a certain liquid freezes at 3.65C its density changes from 0.789 g cm3 to 0.801 g cm3. Its enthalpy of fusion is 8.68 kJ mol1. Estimate the freezing point of the liquid at 100 MPa.

In July in Los Angeles, the incident sunlight at ground level has a power density of \(1.2 \mathrm{~kW} \mathrm{~m}^{-2}\) at noon. A swimming pool of area \(50 \mathrm{~m}^2\) is directly exposed to the sun. What is the maximum rate of loss of water? Assume that all the radiant energy is absorbed.

Suppose the incident sunlight at ground level has a power density of 0.87 kW m2 at noon. What is the maximum rate of loss of water from a lake of area 1.0 ha? (1 ha = 104 m2.) Assume that all the radiant energy is absorbed.

An open vessel containing (a) water, (b) benzene, (c) mercury stands in a laboratory measuring 5.0 m 5.0 m 3.0 m at 25C. What mass of each substance will be found in the air if there is no ventilation? (The vapour pressures are (a) 3.2 kPa, (b) 13.1 kPa, (c) 0.23 Pa.)

On a cold, dry morning after a frost, the temperature was 5C and the partial pressure of water in the atmosphere fell to 0.30 kPa. Will the frost sublime? What partial pressure of water would ensure that the frost remained?

Naphthalene, C10H8, melts at 80.2C. If the vapour pressure of the liquid is 1.3 kPa at 85.8C and 5.3 kPa at 119.3C, use the ClausiusClapeyron equation to calculate (a) the enthalpy of vaporization, (b) the normal boiling point, and (c) the enthalpy of vaporization at the boiling point.

The normal boiling point of hexane is 69.0C. Estimate (a) its enthalpy of vaporization and (b) its vapour pressure at 25C and 60C.

Calculate the melting point of ice under a pressure of 50 bar. Assume that the density of ice under these conditions is approximately \(0.92 \mathrm{~g} \mathrm{~cm}^{-3}\) and that of liquid water is \(1.00 \mathrm{~g} \mathrm{~cm}^{-3}\).

Calculate the melting point of ice under a pressure of 10 MPa. Assume that the density of ice under these conditions is approximately 0.915 g cm3 and that of liquid water is 0.998 g cm3.

What fraction of the enthalpy of vaporization of water is spent on expanding the water vapour?

What fraction of the enthalpy of vaporization of ethanol is spent on expanding its vapour?

The vapour pressure, p, of nitric acid varies with temperature as follows: /C 0 20 40 50 70 80 90 100 p/kPa 1.92 6.38 17.7 27.7 62.3 89.3 124.9 170.9 What are (a) the normal boiling point and (b) the enthalpy of vaporization of nitric acid?

The vapour pressure of the ketone carvone (M= 150.2 g mol1), a component of oil of spearmint, is as follows: /C 57.4 100.4 133.0 157.3 203.5 227.5 p/Torr 1.00 10.0 40.0 100 400 760 What are (a) the normal boiling point and (b) the enthalpy of vaporization of carvone?

Construct the phase diagram for benzene near its triple point at 36 Torr and 5.50C using the following data: fusH = 10.6 kJ mol1, vapH = 30.8 kJ mol1, (s) = 0.891 g cm3, (l) = 0.879 g cm3.

In an investigation of thermophysical properties of toluene (R.D. Goodwin J. Phys. Chem. Ref. Data 18, 1565 (1989)) presented expressions for two coexistence curves (phase boundaries). The solidliquid coexistence curve is given by p/bar = p3/bar + 1000 (5.60 + 11.727x)x where x = T/T3 1 and the triple point pressure and temperature are p3 = 0.4362 bar and T3 = 178.15 K. The liquidvapour curve is given by: ln(p/bar) = 10.418/y + 21.157 15.996y + 14.015y2 5.0120y3 + 4.7224(1 y)1.70 where y = T/Tc = T/(593.95 K). (a) Plot the solidliquid and liquidvapour phase boundaries. (b) Estimate the standard melting point of toluene. (c) Estimate the standard boiling point of toluene. (d) Compute the standard enthalpy of vaporization of toluene, given that the molar volumes of the liquid and vapour at the normal boiling point are 0.12 dm3 mol1 and 30.3 dm3 mol1, respectively.

In a study of the vapour pressure of chloromethane, A. Bah and N. Dupont-Pavlovsky (J. Chem. Eng. Data 40, 869 (1995)) presented data for the vapour pressure over solid chloromethane at low temperatures. Some of that data is shown below: T/K 145.94 147.96 149.93 151.94 153.97 154.94 p/Pa 13.07 18.49 25.99 36.76 50.86 59.56 Estimate the standard enthalpy of sublimation of chloromethane at 150 K. (Take the molar volume of the vapour to be that of a perfect gas, and that of the solid to be negligible.)

Show that, for a transition between two incompressible solid phases, G is independent of the pressure.

The change in enthalpy is given by dH = CpdT + Vdp. The Clapeyron equation relates dp and dT at equilibrium, and so in combination the two equations can be used to find how the enthalpy changes along a phase boundary as the temperature changes and the two phases remain in equilibrium. Show that d(H/T) = Cp d ln T.

In the gas saturation method for the measurement of vapour pressure, a volume V of gas (as measured at a temperature T and a pressure p) is bubbled slowly through the liquid that is maintained at the temperature T, and a mass loss m is measured. Show that the vapour pressure, p, of the liquid is related to its molar mass, M, by p = AmP/(1 + Am), where A = RT/MPV. The vapour pressure of geraniol (M= 154.2 g mol1), which is a component of oil of roses, was measured at 110C. It was found that, when 5.00 dm3 of nitrogen at 760 Torr was passed slowly through the heated liquid, the loss of mass was 0.32 g. Calculate the vapour pressure of geraniol.

Combine the barometric formula (stated in Impact I1.1) for the dependence of the pressure on altitude with the ClausiusClapeyron equation, and predict how the boiling temperature of a liquid depends on the altitude and the ambient temperature. Take the mean ambient temperature as 20C and predict the boiling temperature of water at 3000 m.

Figure 4.9 gives a schematic representation of how the chemical potentials of the solid, liquid, and gaseous phases of a substance vary with temperature. All have a negative slope, but it is unlikely that they are truly straight lines as indicated in the illustration. Derive an expression for the curvatures (specifically, the second derivatives with respect to temperature) of these lines. Is there a restriction on the curvature of these lines? Which state of matter shows the greatest curvature?

The Clapeyron equation does not apply to second-order phase transitions, but there are two analogous equations, the Ehrenfest equations, that do. They are: = = where is the expansion coefficient, T the isothermal compressibility, and the subscripts 1 and 2 refer to two different phases. Derive these two equations. Why does the Clapeyron equation not apply to second-order transitions?

For a first-order phase transition, to which the Clapeyron equation does apply, prove the relation CS = Cp where CS = (q/T)S is the heat capacity along the coexistence curve of two phases.

Proteins are polypeptides, polymers of amino acids that can exist in ordered structures stabilized by a variety of molecular interactions. However, when certain conditions are changed, the compact structure of a polypeptide chain may collapse into a random coil. This structural change may be regarded as a phase transition occurring at a characteristic transition temperature, the melting temperature, Tm, which increases with the strength and number of intermolecular interactions in the chain. A thermodynamic treatment allows predictions to be made of the temperature Tm for the unfolding of a helical polypeptide held together by hydrogen bonds into a random coil. If a polypeptide has n amino acids, n 4 hydrogen bonds are formed to form an -helix, the most common type of helix in naturally occurring proteins (see Chapter 19). Because the first and last residues in the chain are free to move, n 2 residues form the compact helix and have restricted motion. Based on these ideas, the molar Gibbs energy of unfolding of a polypeptide with n 5 may be written as Gm = (n 4)hbHm (n 2)ThbSm where hbHm and hbSm are, respectively, the molar enthalpy and entropy of dissociation of hydrogen bonds in the polypeptide. (a) Justify the form of the equation for the Gibbs energy of unfolding. That is, why are the enthalpy and entropy terms written as (n 4)hbHm and (n 2)hbSm, respectively? (b) Show that Tm may be written as Tm = (c) Plot Tm/(hbHm/hbSm) for 5 n 20. At what value of n does Tm change by less than 1% when n increases by one?

The use of supercritical fluids as mobile phases in SFC depends on their properties as nonpolar solvents. The solubility parameter, , is defined as (Ucohesive /Vm)1/2, where Ucohesive is the cohesive energy of the solvent, the energy per mole needed to increase the volume isothermally to an infinite value. Diethyl ether, carbon tetrachloride, and dioxane have solubility parameter ranges of 78, 89, and 1011, respectively. (a) Derive a practical equation for the computation of the isotherms for the reduced internal energy change, Ur(Tr,Vr) defined as Ur(Tr,Vr) = (b) Draw a graph of Ur against pr for the isotherms Tr = 1,1.2, and 1.5 in the reduced pressure range for which 0.7 Vr 2. (c) Draw a graph of against pr for the carbon dioxide isotherms Tr = 1 and 1.5 in the reduced pressure range for which 1 Vr 3. In what pressure range at Tf = 1 will carbon dioxide have solvent properties similar to those of liquid carbon tetrachloride? Hint. Use mathematical software or a spreadsheet.

A substance as wellknown as methane still receives research attention because it is an important component of natural gas, a commonly used fossil fuel . Friend et al. have published a review of thermophysical properties of methane (D.G. Friend, J.F. Ely, and H. Ingham, J. Phys. Chem. Ref. Data 18, 583 (1989)), which included the following data describing the liquidvapour phase boundary. T/K 100 108 110 112 114 120 130 140 150 160 170 190 p/MPa 0.034 0.074 0.088 0.104 0.122 0.192 0.368 0.642 1.041 1.593 2.329 4.521 (a) Plot the liquidvapour phase boundary. (b) Estimate the standard boiling point of methane. (c) Compute the standard enthalpy of vaporization of methane, given that the molar volumes of the liquid and vapour at the standard boiling point are 3.80 102 and 8.89 dm3 mol1, respectively.

Diamond, an allotrope of carbon, is the hardest substance and the best conductor of heat yet characterized. For these reasons, diamond is used widely in industrial applications that require a strong abrasive. Unfortunately, it is difficult to synthesize diamond from the more readily available allotropes of carbon, such as graphite. To illustrate this point, calculate the pressure required to convert graphite into diamond at 25C. The following data apply to 25C and 100 kPa. Assume the specific volume, Vs, and T are constant with respect to pressure changes. Graphite Diamond fG7/(kJ mol1) 0 +2.8678 Vs/(cm3 g1) 0.444 0.284 T/kPa 3.04 108 0.187 108