PPT 107 PHYSICAL CHEMISTRY

PPT 107 PHYSICAL CHEMISTRY Semester 2 Academic session 2012/2013

Chapter 4Material Equilibrium

CONTENT • Introduction to Material Equilibrium • Entropy and Equilibrium • The Gibbs and Helmholtz Energies • Thermodynamic Relations for a System Equilibrium • Calculation of Changes in State Function • Phase Equilibrium • Reaction Equilibrium

The zeroth, first, and second laws of thermodynamics give us the state functions: T, U, and S. The second law enables us to determine whether a given process is possible. A process that decreases Sunivis impossible; one that increases Sunivis possible and irreversible. Reversible processes have Suniv= 0. Such processes are possible in principle but hard to achieve in practice. Our aim in this chapter is to use this entropy criterion to derive specific conditions for material equilibrium in a nonisolated system. These conditions will be formulated in terms of state functions of the system

What is material equilibrium? In each phase of the closed system, the number of moles of each substances present remains constant in time No net chemical reactions are occurring in the system No net transfer of matter from one part of the system to another Concentration of chemical species in the various part of the system are constant

Material equilibrium Phase equilibrium Reaction equilibrium is equilibrium with respect to transport of matter between phases of the system without conversion of one species to another. is equilibrium with respect to conversion of one set of chemical species to another set.

Entropy and Equilibrium Consider an isolated system (not in material equilibrium) The spontaneous chemical reaction or transport of matter are irreversible process that increase the ENTROPY The process was continued until the system’s entropy is maximized. Once it is maximized, any further process can only decrease entropy (violate the second law)

Reaction equilibrium is ordinarily studied under one of two conditions. For reactions that involve gases, the chemicals are usually put in a container of fixed volume, and the system is allowed to reach equilibrium at constant T and V in a constant-temperature bath. For reactions in liquid solutions, the system is usually held at atmospheric pressure and allowed to reach equilibrium at constant T and P.

The system is not in material equilibrium but is in mechanical and thermal equilibrium • The surroundings are in material, mechanical and thermal equilibrium • System and surroundings can exchange energy (as heat and work) but not matter Let chemical reaction or transport of matter between phases or both be occurring in the system at rates small enough to maintain thermal and mechanical equilibrium.

Let heat dqsyst flow into the system as a result of the changes that occur in the system during an infinitesimal time period. For example, if an endothermic chemical reaction is occurring, dqsyst is positive. • Since system and surroundings are isolated , we have dqsurr= -dqsyst • Since, the chemical reaction or matter transport within the non equilibrium system is irreversible, dSuniv must be positive: dSuniv= dSsyst+ dSsurr > 0

The surroundings are in thermodynamic equilibrium throughout the process. Therefore, as far as the surroundings are concerned, the heat transfer is reversible dSsurr= dqsurr/T • The systems is not in thermodynamic equilibrium, and the process involves an irreversible change in the system, therefore dSsyst ≠dqsyst/T • This will lead to: dSsyst> -dSsurr = -dqsurr/T = dqsyst/T • ThereforedSsyst > dqsyst/T dS > dqirrev/T closed syst. in therm. and mech. equilib.

When the system has reached material equilibrium, any infinitesimal process is a change from a system at equilibrium to one infinitesimally close to equilibrium and hence is a reversible process. • Thus, at material equilibrium we have, ds = dqrev/T • Combiningthe reversible and irreversible will lead: ds ≥ dq/T material change, closed syst. in them & mech. Equilib • where the equality sign holds only when the system is in material equilibrium, and for a reversible process. • For an irreversible chemical reaction or phase change, dS is greater than dq/T because of the extra disorder created in the system by the irreversible material change.

The first law for a closed system is: dU = dq + dwdq = dU – dw • Since dq≤ T dS • Hence for a closed system in mechanical and thermal equilibrium we have dU – dw ≤ T dS dU ≤ T dS + dwmaterial change, closed syst. in mech. and therm. equilib. where the equality sign applies only at material equilibrium.

THE GIBBS AND HELMHOLTZ ENERGIES We now use the previous relation to deduce conditions for material equilibrium in terms of state functions of the system. We first examine material equilibrium in a system held at constant T and V Irreversible, P-V work.

d(U – TS)  – SdT + dw d(U – TS) – SdT - PdV Consider material equilibrium at constant T and V dU TdS + dw dU TdS +SdT – SdT+ dw dU d(TS) – SdT + dw dw = -P dV for P-V work only at constant T and V, dT=0, dV=0 d(U – TS)  0 Equality sign holds at material equilibrium

Therefore, for a closed system held at constant T and V, the state function (U – TS) continually decreases during the spontaneous, irreversible processes of chemical reaction and matter transport between phases until material equilibrium is reached. At material equilibrium, d(U - TS) equals 0, and (U – TS) has reached a minimum. Any spontaneous change at constant T and V away from equilibrium (in either direction) would mean an increase in (U – TS), which, working back through the preceding equations, would mean a decrease in (Suniv = Ssyst + Ssurr). This decrease would violate the second law.

The condition for material equilibrium in a closed system capable of doing only P-V work and held at constant T and V is minimization of the system’s state function (U – TS).This state function is called the Helmholtz free energy, the Helmholtz energy, the Helmholtz function, or the work function and is symbolized by A: A  U - TS

Helmholtz free energy For a closed system (T & V constant), the state function U-TS, continually decrease during the spontaneous, irreversible process of chemical reaction and matter transport until material equilibrium is reached d(U-TS)=0 at equilibrium

Gibbs free energy Now consider material equilibrium for constant T and P conditions, dP = 0, dT= 0. To introduce dP and dT into: dU TdS + dw with dw= -PdV, we add and subtract SdT and VdP

d(H – TS)  0 dU T dS+ S dT– S dT+ P dV+ V dP – V dP dU d(TS) – SdT – d(PV) + VdP d(U + PV – TS)  – SdT + VdP d(H – TS) – SdT+ VdP at constant T and P, dT=0, dP=0 const. T, P where the equality sign holds at material equilibrium.

Thus, the state function H – TS continually decreases during material changes at constant T and P until equilibrium is reached. The condition for material equilibrium at constant T and P in a closed system doing P-V work only is minimization of the system’s state function H – TS. This state function is called the Gibbs function, the Gibbs energy, or the Gibbs free energy and is symbolized by G: G  H – TS  U + PV – TS G = A + PV

G Constant T, P Equilibrium reached Time G decreases during the approach to equilibrium at constant T and P, reaching a minimum at equilibrium. As G of the system decreases at constant T and P, • Suniv increases. Since U, V, and S are extensive, G is extensive. • Both A and G have units of energy (J or cal). However, they are not energies in the sense of being conserved

THE GIBSS & HELMHOLTZ ENERGIES In a closed system capable of doing only P-V work A spontaneous process at constant-T-and-V is accompanied by a decrease in the Helmholtz energy,A. A spontaneous process at constant-T-and-P is accompanied by a decrease in the Gibbs energy, G. dA = 0 at equilibrium, const. T, V dG = 0 at equilibrium, const. T, P

ρ= .958 g/cm3

closed syst., const. T, V, P-V work only What is the relation between the minimization-of-G equilibrium condition at constant T and P and the maximization-of-Suniv equilibrium condition? Consider a system in mechanical and thermal equilibrium which undergoes an irreversible chemical reaction or phase change at constant T and P Since the surroundings undergo a reversible isothermal process The decrease in Gsyst as the system proceeds to equilibrium at constant T and P corresponds to a proportional increase in S univ

The names “work function” and “Gibbs free energy” arise as follows. Let us drop the restriction that only P-V work be performed. For a constant-temperature process in such a system: For afiniteisothermal process The work wby done by the system on its surroundings is ∆A ≤ -wby for an isothermal process. Multiplication of an inequality by (-1) reverses the direction of the inequality; therefore const. T, closed syst. The equality sign is holds for a reversible process

Now consider G G  A + PV dG = dA + PdV + VdP const. T and P, closed syst. Let us divide the work into P-V work and non-P-V work wnon-P-V If the P-V work is done in a mechanically reversible manner, then:

For a reversible change The maximum non-expansion work from a process at constant P and T is given by the value of -G The change in the Helmholtz energy is equal to the maximum work the system can do:

THERMODYNAMIC RELATIONS FOR A SYSTEM IN EQUILIBRIUM All thermodynamic state-function relations can be derived from six basic equations. The first basic equation; it combines the first and second laws. The next three basic equations are the definitions of H, A, and G. Finally, the two equations are for the CP and CV equations.

The heat capacities CV and CP have alternative expressions that are also basic equations. Consider a reversible flow of heat accompanied by a temperature change dT. By definition,: CX = dqX / dT where X (P or V) dqrev = TdS CX = TdS/ dT where dS/dT is for constant X. Putting X equal to V and P, we have closed syst., in equilib. Key properties The rates of change of U, H, and S with respect to T can be determined from the heat capacities CPandCV.

dU = TdS - PdV dH = TdS + VdP dA = -SdT - PdV dG = -SdT + VdP The Gibbs Equations closed syst., rev. proc., P-V work only How to derive dH, dA and dG?

H  U + PV The Gibbs Equations dH= ? dH = d(U + PV) dU = TdS - PdV = dU + d(PV) = dU + PdV + VdP = (TdS - PdV) + PdV + VdP dH = TdS + VdP

dA = -SdT - PdV dG = -SdT + VdP dA= ? A U - TS dU = TdS - PdV dA = d(U - TS) = dU - d(TS) = dU - TdS - SdT = (TdS - PdV) - TdS - SdT dG= ? G H - TS dH = TdS+VdP dG = d(H - TS) = dH - d(TS) = dH - TdS - SdT = (TdS + VdP) - TdS - SdT

The Gibbs equation dU= T dS – P dVimplies that U is being considered a function of the variables S and V. From U= U (S,V) we have (dG= -SdT + VdP) The Power of thermodynamics: Difficultly measured properties to be expressed in terms of easily measured properties.

The Euler Reciprocity Relations If Z＝f(x，y)，and Z has continuous second partial derivatives, then That is

The Maxwell Relations (Application of Euler relation to Gibss equations) dU = T dS – P dV = M dx + N dy Where M ≡ T, N ≡ P, x ≡ S, y≡ V The Euler relation Gives

The Maxwell Relations (Application of Euler relation to Gibss equations) dU = TdS - PdV The Gibbs equation (4.33) for dU is dU＝TdS－PdV dV＝0 dS＝0 Applying Euler Reciprocity,

These are the Maxwell Relations The first two are little used. The last two are extremely valuable. The equations relate the isothermal pressure and volume variations of entropy to measurable properties.

Dependence of State Functions on T, P, and V We now find the dependence of U, H, S and G on the variables of the system. The most common independent variables are T and P. We can relate the temperature and pressure variations of H, S, and G to the measurable Cp,α, and κ. The variations of U we shall find as a function to the temperature and volume.

U== U(T, V) H==H(T, P) G==G(T, P) S==S(T, P)

Volume dependence of U We want . The Gibbs equation gives dU=TdS-PdV. The partial derivative corresponds to an isothermal process. For an isothermal process, the equation dU=TdS-PdV becomes The Gibbs equation gives dU＝TdS－PdV For an isothermal process dUT＝TdST－PdVT Divided above equation by dVT, the infinitesimal volume change at constant T, to give T subscripts indicate that the infinitesimal changes dU, dS, and dV are for a constant-T process From Maxwell Relations

Temperature dependence of U From Basic Equations Temperature dependence of H

Pressure dependence of H from Gibbs equations, dH＝TdS＋VdP From Maxwell Relations

Temperature dependence of S The equations of this section apply to a closed system of fixed composition and also to a closed system where the composition changes reversibly From Basic Equations Pressure dependence of S The Basic Equation dG = SdT – VdP With the Euler reciprocity relation to the above equation From Maxwell Relations

The equations of this section apply to a closed system of fixed composition and also to a closed system where the composition changes reversibly Temperature and Pressure dependence of G dG＝ -S dT+ V dP dP＝0 dT＝0

For solids and liquids, temperature changes usually have significant effects on thermodynamic properties, but pressure effects are small unless very large pressure changes are involved. For gases not at high pressure, temperature changes usually have significant effects on thermodynamic properties and pressure changes have significant effects on properties that involve the entropy (for example, S, A, G) but usually have only slight effects on properties not involving S(for example, U, H, CP).

Joule-Thomson Coefficient (easily measured quantities) from (2.65) From pressure dependence of H

Heat-Capacity Difference (easily measured quantities) From volume dependence of U

Heat-Capacity Difference • As T 0, CP CV • CP  CV (since  > 0) • CP= CV (if  = 0)

PPT 107 PHYSICAL CHEMISTRY