Physical Transformations of Pure Substances

Physical transformations of pure substances The Stabilities of Phases (a) The number of phases A phase is a form of matter that is uniform throughout in chemical composition and physical state The number of phases in a system is denoted P. A gas, or a gaseous mixture, is a single phase (P = 1), a crystal of a substance is a single phase, and two fully mixed liquids form a single phase A solution of solid B in solid A - a homogeneous mixture of the two miscible substances - is uniform on a molecular scale A dispersion is uniform on a macroscopic scale but not on a microscopic scale, because it consists of grains or droplets of one substance in a matrix of the other (b) Phase transitions Temperature, T A phase transition, the spontaneous conversion of one phase into another phase, occurs at a characteristic transition temperature for a given pressure At the transition temperature the two phases are in equilibrium and the Gibbs energy of the system is a minimum at the prevailing pressure Thermal analysis, which takes advantage of the heat that is evolved or absorbed during a transition, can be used. Thus, if the phase transition is exothermic and the temperature of a sample is monitored as it cools, the presence of the transition can be recognized by a pause in the otherwise steady fall of the temperature A phase transition that is predicted by thermodynamics to be spontaneous might occur . too slowly to be significant in practice Liquid In gases and liquids the mobilities of the cooling molecules allow phase transitions to occur rapidly, but in solids thermodynamic instability Liquid may be frozen in freezing Thermodynamically unstable phases that persist I Solid because the transition is kinetically hindered are cooling called metastable phases (c) Thermodynamic criteria of phase stability Time, t In fact, this quantity plays such an important role in this focus and elsewhere in the text that is given a special name and symbol, the chemical potential, mu. For a system that consists of a single substance, the 'molar Gibbs energy' and the ‘chemical potential' are exactly the same The discussion in this topic is based on the following consequence of the second law: At equilibrium, the chemical potential of a substance is the same in and throughout every phase present in the system If the chemical potential at location I is higher than at location 2, the transfer is accompanied by A decrease in G, and so has a spontaneous tendency to occur Phase Boundaries The phase diagram of a pure substance shows the regions of pressure and temperature at which its various phases are thermodynamically stable. The lines separating the regions, which are called phase boundaries, show the values of p and T at which two phases coexist in equilibrium and their chemical potentials are equal (a) Characteristic properties related to phase transitions Pressure, p Therefore, the liquid-vapor phase boundary in a Critical point phase diagram shows how the vapor pressure of Liquid Solid the liquid varies with temperature. Similarly, the solid-vapor phase boundary shows the temperature Triple point . variation of the sublimation vapor pressure, the vapor pressure of the solid phase Vapor However, only when the temperature is such that the vapor pressure is equal to the external pressure To will it be possible for vaporization to occur Temperature, T throughout the bulk of the liquid and for the vapor to expand freely into the surroundings This condition of free vaporization throughout the liquid is called boiling. The temperature at which the vapor pressure of a liquid is equal to the external pressure is called the boiling temperature at that pressure Balling does not occur when a liquid is heated in a rigid, closed vessel. Instead, the vapor pressure, and hence the density of the vapor, rises as the temperature is raised. At the same time, the density of the liquid decreases slightly as a result of its expansion The temperature at which the surface disappears between liquid and vapor is the critical temperature of the substance. The vapor pressure at the critical temperature is called the critical pressure The temperature at which, under a specified pressure, the liquid and solid phases of a substance coexist in equilibrium is called the melting temperature. Because a substance melts at exactly the same temperature as it freezes, the melting temperature of a substance is the same as its freezing temperature There is a set of conditions under which three different phases of a substance (typically solid, liquid, and vapor) all simultaneously coexist in equilibrium. These conditions are represented by the triple point, a point at which the three phase boundaries meet (b) The phase rule The phase rule is a general relation between the variance, F, the number of components, C, and the number of phases at equilibrium, P, for a system of any composition. Each of these quantities has a precisely defined meaning: The variance (or number of degrees of freedom), F, of a system is the number of intensive variables that can be changed independently without disturbing the number of phases in equilibrium A constituent of a system is any chemical species that is present A component is a chemically independent constituent of a system The number of components, C, in a system is the minimum number of types of independent species (ions or molecules) necessary to define the composition of all the phases present in the system The relation between these quantities, which is called the phase rule, is established by considering the conditions for equilibrium to exist between the phases in terms of the chemical potentials of all the constituents When only one phase is present in a one-component system, F=2 and both p and T can be varied independently (at least over a small range) without changing the number of phases. The system is said to be bivariant, meaning having two degrees of freedom When two phases are in equilibrium F = 1, which implies that pressure is not freely variable if the temperature is set; indeed, at a given temperature, a liquid has A characteristic vapor pressure When three phases are in equilibrium, F = 0 and the system is invariant, meaning that it has no degrees of freedom. Four phases cannot be in equilibrium in a one-component system because F cannot be negative Three Representative Phase Diagrams (a) Carbon dioxide The features to notice include the positive slope (up from left to right) of the solid-liquid phase boundary; the direction of this line is characteristic of most substances. This slope indicates that the melting temperature of solid carbon dioxide rises as the pressure is increased Cylinders of carbon dioxide generally contain the liquid or compressed gas; at 25 degrees Celsius that implies a vapor pressure of 67 atm if both gas and liquid are present in equilibrium When the gas is released through a tap (which acts as a throttle) the gas cools by the Joule-Thompson effect, so when it emerges into a region where the pressure is only 1 atm, it condenses into a finely divided snow-like solid That carbon dioxide gas cannot be liquefied except by applying high pressure reflects the weakness of the intermolecular forces between the non-polar carbon dioxide molecules (b) Water The liquid-vapor boundary in the phase diagram summarizes how the vapor pressure of liquid water varies with temperature. It also summarizes how the boiling temperature varies with pressure: simply read off the temperature at which the vapor pressure is equal to the prevailing atmospheric pressure The very steep slope is the solid-liquid boundary indicates that enormous pressures are needed to bring about significant changes The reason for this almost unique behavior can be traced to the decrease in volume that occurs on melting: it is more favorable for the solid to transform into the liquid as the pressure is raised The decrease in volume is a result of the very open structure of ice: the water molecules are held apart, as well as together, by the hydrogen bonds between them, but the hydrogen-bonded structure partially collapses on melting and the liquid is denser than the solid The solid phases of ice differ in the arrangement of the water molecules: under the influence of very high pressures, hydrogen bonds buckle and the H2O molecules adopt different arrangements (c) Helium Helium behaves unusually at low temperatures because the mass of its atoms is so low and there are only very weak interactions between neighbors At 1 atm, the solid and gas phases of helium are never in equilibrium however low the temperature: the atoms are so light that they vibrate with a large-amplitude motion even at very low temperatures and the solid simply shakes itself apart Checklist of concepts 1. A phase is a form of matter that is uniform throughout in chemical composition and physical state 2. A phase transition is the spontaneous conversion of one phase into another 3. The thermodynamic analysis of phases is based on the fact that at equilibrium, the chemical potential of a substance is the same throughout a sample 4. A phase diagram indicates the values of the pressure and temperature at which a particular phase is most stable, or is in equilibrium with other phases 5. The phase rule relates the number of variables that may be changed while the phases of a system remain in mutual equilibrium Topic 4B µ ¶ @¹ = ¡Sm @T p µ ¶ @¹ = Vm @p T Variation of chemical potential with T [constant p] Variation of chemical potential with p [constant T] By combining the equality of chemical potentials of a substance in each phase with these expressions for the variation of mu with temperature and pressure it is possible to deduce how phase equilibrium respond to changes in the conditions The dependence of stability on the conditions At sufficiently low temperatures the solid phase of a substance commonly has the lowest chemical potential and is therefore the most stable phase However, the chemical potentials of different phases depend on temperature to different extents (because the molar entropy of each phase is different), and above a certain temperature the chemical potential of another phase (perhaps another sold phase, a liquid, or a gas) might turn out to be lower (a) The temperature dependence of phase stability Because molar entropy is greater than 0 for all substances above T = 0, the chemical potential of a pure substance decreases as the temperature is raised It also implies that because the molar entropy of gas is greater than the molar entropy of liquid, the slope is steeper for gases than for liquids. Because it is almost always the case that molar entropy of a liquid is greater than the molar entropy of a solid, the slope is also steeper for a liquid than the corresponding solid The chemical potential of the gas phase plunges steeply downwards is raised (because the molar entropy of the vapor is so high), and there comes a temperature at which it lies below that of the liquid (b) The response of melting to applied pressure In most cases, molar volume of a liquid is greater than molar volume of a solid, so an increase in pressure increases the chemical potential of the liquid phase of a substance more than that of its solid phase. The effect of pressure in such a case is to raise the freezing temperature slightly For water, however, the molar volume of a liquid is less than the molar volume of a solid, and an increase in pressure increases the chemical potential of the solid more than that of the liquid. In this case, the freezing temperature is lowered slightly (c) The vapor pressure of a liquid subjected to pressure Pressure can be exerted on the condensed phase mechanically or by subjecting it to the applied pressure of an inert gas. In the later case, the partial vapor pressure is the partial pressure of the vapor in equilibrium with the condensed phase When pressure is applied to a condensed phase, its vapor pressure rises: in effect, molecules are squeezed out of the phase and escape as a gas One complication that has been ignored is that, if the condensed phase is a liquid, then the pressurizing gas might dissolve and change its properties Another complication is that the gas-phase molecules might attract molecules out of the liquid by the process of gas salvation, the attachment of molecules to gas-phase species The location of phase boundaries The precise locations of the phase boundaries - the pressures and temperatures at which two phases can coexist - can be found by making use once again of the fact that, when two phases are in equilibrium, their chemical potentials must be equal (a) The slopes of the phase boundaries Now p and T are changed infinitesimally, but in such a way that the phases remain in equilibrium: after these changes, the chemical potentials of the two phases change but remain equal . Vm (®) dp ¡ Sm (®) dT = Vm (¯) dp ¡ Sm (¯) dT Pressure, p Phase ® dp b dp ¢trsS = dT ¢trsV a Clapeyron equation Phase ¯ The Clapeyron equation is an exact expression for the slope of the tangent to the phase boundary at any point at and applies to any phase equilibrium of any pure substance Temperature, T A more practical application is to the prediction of the response of freezing and boiling points to the application of pressure, when it can be used in the form obtained by inverting both sides (b) The solid-liquid boundary Melting (fusion) is accompanied by a molar enthalpy change and if it occurs at a temperature T the molar entropy of melting is divided by T; all points on the phase boundary correspond to equilibrium, so T is in fact a transition temperature Slope of solid-liquid boundary dp ¢f usH dT = T ¢f usV The enthalpy of melting is positive (the only exception is helium-3); the change in molar volume is usually positive and always small p ¼ p¤ + ¢Hf usH (T ¡ T ¤ ) T ¤ ¢f usV (c) The liquid-vapor boundary The entropy of vaporization at a temperature T is equal to the change of enthalpy of vaporization divided by temperature (as before, all points on the phase boundary correspond to equilibrium, so T is a transition temperature), so... dP ¢vapH = dT T ¢vapV Consequently dT/dp is large, and the boiling temperature is more responsive to pressure than the freezing temperature Clausius-Clapeyron equation d ln p ¢vapH dT = RT 2 Like the Clapeyron equation (which is exact), the Clausius-Clapeyron equation (which is an approximation) is important for understanding the appearance of phase diagrams, particularly the location and shape of the liquid-vapor and solid-vapor phase boundaries (d) The solid-vapor boundary The only difference between the solid-vapor and the liquid-vapor boundary is the replacement of the enthalpy of vaporization by the enthalpy of sublimation Because the enthalpy of sublimation is greater than the enthalpy of vaporization, at similar temperatures the equation predicts a steeper slope for the sublimation curve than for the vaporization curve Checklist of Concepts 1. The chemical potential of a substance decreases with increasing temperature in proportion to its molar entropy 2. The chemical potential of a substance increases with increasing pressure in proportion to its molar volume 3. The vapor pressure of a condensed phase increases when pressure is applied 4. The Clapeyron equation is an exact expression for the slope of a phase boundary 5. The Clausius-Clapeyron is an approximate expression for the boundary between A condensed phase and its vapor