Obviously dG = dE + PdV +VdP – TdS – SdTBut from the first Law of ThermodynamicsFor water, CRC gives: A = 10999.4, B = 9.183837Free Energy, Vapor Pressure and the Equilibrium Betweena Vapor and Condensed Phase(Entirely plagiarized from Physics 5719, “The Physics of Nuts and Bolts”, aka “Lab Survival Skills”)References: Thermodynamics, G. N. Lewis and M. Randall, rev. K. Pitzer and L. Brewer, McGraw Hill, New York (1961).Chemical Thermodynamics, I. Klotz, Benjamin, New York (1964).The Gibbs Free Energy is generally agreed to be the “weapon of choice” for describing (a) chemical reactions and (b) equilibria between phases. It is defined as:G = H – TS = E + PV – TS (1)Where H = EnthalpyE = Total internal energyT = [Absolute] TemperatureS = EntropyObviously dG = dE + PdV +VdP – TdS – SdTBut from the first Law of ThermodynamicsdE = TdS – PdV since dS = δQ/T and the mechanical work done on a system when it expands is –PdV.Hence: dG = -SdT + VdPLet us apply this to a closed system containing a pure substance consisting of its vapor and a condensed phase co-existing in equilibrium. The objective will be to see how the pressure of thevapor depends on the temperature of the system. Clearly we may write the above equation twice, once for each phase:dGc = -ScdT + VcdP where c refers to the condensed phasedGv = -SvdT + VvdP where v refers to the vapor phaseThe objective of the exercise is to change the temperature of the system (obviously the vapor and condensed phase are in thermal equilibrium, at the same temperature) and see how the pressure of the vapor changes. The system remains in thermal equilibrium if the molar free energy of the system, also known as the chemical potential, remains constant. The definition of chemical equilibrium between two phases is that the free energy is the same in both phases: Gc = Gv,. Hence:dGc = dGv Changes in free energy when some independent variable is changed must be the same if they are to remain in equilibrium.-ScdT + VcdP = -SvdT + VvdP(Sv - Sc )dT = (Vv- Vc)dP dP/dT = (Sv – Sc)/(Vv – Vc) = ΔHv/(TΔV)Here we are talking about the entropy change associated with the reversible, isothermal evaporation of a small quantity of material from a condensed phase to the vapor phase. The entropy change insuch a process is simply the heat of vaporization, ΔHv, divided by the temperature at which the vaporization took place, the transitiontemperature.This is the Clapeyron equation (E. Clapeyron, J. cole polytech. ẻ(Paris) 14(23), 153 (1834). It relates the change in pressure of a vapor to the temperature in a closed, mono-component system to the heat of vaporization, system temperature and molar volume change of the material on vaporization. For lack of a better model, we treat most vapors as ideal gases, whose molar volume is given by:V/n = RT/PUnder most circumstances, the molar volume of the vapor is about three orders of magnitude larger than the molar volume of the condensed phase. Hence the required volume change can be approximated as the volume of the vapor alone, which is a constantfor the isothermal transfer of mass at equilibrium we are discussingand is given by the above expression when calculating equations ofstate. dP = (ΔHv/Vv)dT/T = (PΔHv/RT)dT/T dP/P = ΔHv/R)dT/T2ln(P/ P0) = -(ΔHv/R)(1/T – 1/T0)P = P0 exp(-ΔHv/R(1/T – 1/T0))The vapor pressure in equilibrium with a condensed phase increases exponentially (sort of: exp(-1/T) isn’t exactly an exponential!) with temperature from zero up to the critical temperature. This is borne out in the vapor pressure charts and generating functions for them. Deviations from linearity on the log-log plot reflect the temperature dependence of the heat ofvaporization and the fact that exp (-1/T) isn’t really linear in the exponent.Vapor pressure of some elements; stolen from VEECO website.Vapor pressure data are given in the CRC Handbook in the form Log10p(Torr) = -0.2185*A/T + BFor water, CRC gives: A = 10999.4, B = 9.183837Plotting this gives:Vapor Pressure of WaterTemperature (C)-20020406080100120Vapor Pressure (Torr)0.1110100100010000"Normal boiling point"The first thing we see is that the generating function is imperfect: the normal boiling point of water on the plot is about 108.2 C. (I cheated; I looked at the data!)Kubaschewski, Evans and Alcock, Metallurgical Thermochemistry,Pergammon, Oxford (1967), use the higher order expression:Log10p(Torr) = A/T + BlogT + CT + DVapor Pressure of WaterTemperature (C)-20020406080100120Vapor Pressure (Torr)0.1110100100010000"Normal boiling point"Kubaschewski et al.In this case the vapor pressure at 100 C is 758 Torr, a much better approximation! Caveat emptor!Great resource tool: the “RCA Charts”, which live in 327 JFB. Treat them with great respect: they are much older than you are and irreplaceable.Phase diagram of water.So what does all this mean? - Vapor pressure is a continuous function of temperature for alltemperatures from absolute zero to the critical point, above which only a single phase is defined. - The melting point on a vapor pressure curve conveys no significance whatsoever. The vapor pressure of water at 0 C is 4.6 Torr; it’s just another point on an otherwise smooth curve. Actually this isn’t quite true: because the solid and liquid have different heat capacities, the vapor pressure curvechanges slope at the melting point. The vapor pressure still changes continuously across the melting point and is of no particular significance there. Homework: Find or calculate the vapor pressures of CO2 and gallium at their melting points.- The Clapeyron Equation can be used equally well to determine the decrease in the freezing point of water with increasing pressure. This is an interesting calculation that apparently gives the wrong answer to the question: “what makes ice skates work?”. Note that this is not the normal case: the freezing point decreases with increasing pressure only because water expands on freezing; the solid is less dense than the liquid! (You know this: ice floats on water.)- Conversely, the freezing point of water increases with decreasing pressure. Hence the Triple Point, at which all three phases, solid, liquid and vapor, coexist is 0.01 C. (Actually this is now a “defined temperature”.) - “Normal boiling point”. At 95 C, the generating function says the vapor pressure of water is 633 Torr; at 97 C the generating function says 657