Material equilibrium NOORULNAJWA DIYANA YAACOB ERT 108 PHYSICAL CHEMISTRY

Subtopic  Thermodynamic properties of Nonequilibrium system  Entropy and Equilibrium  The Gibbs and Helmholtz Energies  Thermodynamic relations for a system equilibrium  Calculation of changes in state function  Phase equilibrium  Reaction equilibrium

What is material equilibrium 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 Nor 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 Reaction equilibrium Phase equilibrium

 Thermodynamics Properties of Nonequilibrium System Nonequilibrium system: chemical reaction or transport of matter from one phase to another is occurring.

1. We consider system not in phase equilibrium U= U sol + U NaCl S =S sol +S NaCl Partition is removed – removal done by reversibly and adiabatically q= 0 w=0 ΔS = 0 ΔU = 0

2. We consider system not in reaction equilibrium We can measure the ΔU (FIRST LAW) and ΔS if the system is in equilibrium When catalyst was added- system not in reaction equilibrium Changing mixture composition Can ascribe the new value to U and S Water H 2 O 2 Catalyst

 Entropy and Equilibrium Entropy, is a measure of the "disorder" of a system. What "disorder refers to is really the number of different microscopic states a system can be in, given that the system has a particular fixed composition, volume, energy, pressure, and temperature. While energy to strives to be minimal, entropy strives to be maximal Entropy wants to grow. Energy wants to shrink. Together, they make a compromise.

isolated systems: is one with rigid walls that has no communication (i.e., no heat, mass,or work transfer) with its surroundings. An example of an isolated system would be an insulated container, such as an insulated gas cylinder isolated (Insulated) System: U = constant Q = 0

Example: In 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. Any further process can only decrease entropy –(violate the second law)

Reaction equilibrium is ordinarily studied under two condition. Reaction involve gases Chemiccal are put in the container of fixed volumed and the system is allowedto reach equilibrium at constant T and V in a constant-temperature batch Reaction in liquid solution The system is held at atmospheric pressure and allowed to reach equilibrium at constant T and P

 To find equilibrium criteria, the system at temperature T is placed in a bath also at T.  The system is not in material equilibrium but is in mechanical and thermal equilibrium  The surrounding are in material, mechanical and thermal equilibrium  System and surroundings can exchanged energy (as heat and work) but not matter System at T Surroundings at T Rigid, adiabatic, impermeable wall Impermeable wall

Since system and surroundings are isolated, we have dq surr = -dq syst (Eq1) Since, the chemical reaction or matter transport within the non equilibrium system is irreversible, dS univ must be positive: dS univ = dq syst + dq surr > 0 (Eq2)

The surroundings are in thermodynamic equilibrium throughout the process. Therefore, the heat transfer is reversible, and d surr = dq surr /T (Eq3) The systems is not in thermodynamic equilibrium, and the process involves and irreversible change in the system, therefore D syst ≠dq syst /T (Eq4)

Equation (1) to (3) give dS syst > -dS surr = -dq surr /T = dq syst /T (Eq5) Therefore dS syst > dq syst /T (Eq6) dS syst > dq irrev /T (Eq7) closed syst. in them. 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,a material equilibrium we have, ds ≥ dq/T (Eq8) material change, closed syst. in them. and mech. equilib

The first law for a closed system is dq = dU – dw (Eq 9) Eq 7 gives dq≤ TdS Hence for a closed system in mechanical and thermal equilibrium we have dU – dw ≤ TdS or dU ≤ TdS + dw (Eq 10)

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

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

A  U - TS Helmholtz free energy dU  d(TS) – SdT – d(PV) + VdP d(H – TS)  – SdT - VdP d(U + PV – TS)  – SdT + VdP at constant T and P, dT=0, dP=0 d(H – TS)  0 P-V work only Consider for constant T & P, dw = -PdV into dU  TdS + dw dU  TdS + SdT – SdT + PdV + VdP - VdP

the state function H-TS, continually decrease (constant T and P) during the spontaneous, irreversible process of chemical reaction and matter transport until material equilibrium is reached

G  H – TS  U + PV – TS Gibbs free energy dA T,V  0 dG T,P  0 Equilibrium reached Constant T, P Time G G decreases during the approach at equilibrium Both A and G have units of energy (J or cal)

G  H – TS  U + PV – TS Gibbs free energy In a closed system capable of doing only P - V work, the constant-T-and-V material- equilibrium condition is the minimization of the Helmholtz function A, and the constant- T-and-P material-equilibrium condition is the minimization of the Gibbs function G. dA = 0at equilibrium, const. T, V dG = 0at equilibrium, const. T, P

G  H – TS  U + PV – TS Gibbs free energy Consider a system in mechanical and thermal equilibrium which undergoes an irreversible chemical reaction or phase change at constant T and P. closed syst., const. T, V, P-V work only  G = G 2 – G 1 = (H 2 – TS 2 ) – (H 1 – TS 1 ) =  H – T  S const. T

Relationship between the minimization-of-G-equilibrium condition (T & P constant) and the maximization-of-S univ Consider a system in mechanical and thermal equilibrium which undergoes an irreversible chemical reaction or phase change at constant T and P. closed syst., const. T, V, P-V work only The decrease in G syst as the system proceeds to equilibrium at constant T and P corresponds toa proportional increase in S univ

const. T const. T, closed syst. It turns out that A carries a greater significance than being simply a signpost of spontaneous change: The change in the Helmholtz energy is equal to the maximum work the system can do: Closed system, in thermal &mechanic. equilibrium

G  H – TS  U + PV – TS G  U– TS + PV  A + PV const. T and P, closed syst. If the P - V work is done in a mechanically reversible manner, then const. T and P, closed syst. or

When the change is reversible The maximum non-expansion work from a process at constant P and T is given by the value of  G H  U + PVG  H – TS (const. T, P) and

 Thermodynamic Reactions for a System in Equilibrium Basic Equations dU = TdS - PdV H  U + PV A  U – TS G  H - TS closed syst., rev. proc., P-V work only closed syst., in equilib., P-V work only

The rates of change of U, H, and S with respect to T can be determined from the heat capacities C P and C V. Key properties closed syst., in equilib. Basic Equations Heat capacities (C P C V )

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

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

dG = -SdT + VdP dA = -SdT - PdV dH = TdS + VdP dU = TdS - PdV The Gibbs Equations closed syst., rev. proc., P-V work only First Law Definitions

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

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

The Gibbs equation (4.33) for dU is dU＝TdS－PdVdU＝TdS－PdV dS＝0dS＝0 dV＝0dV＝0 ↓ V is held constant ↓ S is held constant = dU = TdS - PdV The Maxwell Relations

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, andκ

Volume dependence of U The Gibbs equation gives dU ＝ TdS － PdV From (4.45) Divided above equation by dV T, the infinitesimal volume change at constant T, to give For an isothermal process dU T ＝ TdS T － PdV T T subscripts indicate that the infinitesimal changes dU, dS, and dV are for a constant- T process

from (4.34) dH ＝ TdS ＋ VdP Pressure dependence of H Temperature dependence of U Temperature dependence of H

Temperature and Pressure dependence of G Temperature dependence of S Pressure dependence of S From (4.31) The Gibbs equation (4.36) for dG is dG ＝ -SdT + VdP dT＝0dT＝0dP＝0dP＝0

Reminder: The equations of this section apply to a closed system of fixed composition and also to a closed system where the composition changes reversibly

Joule-Thomson Coefficient from (2.65)

Heat-Capacity Difference

1.As T  0, C P  C V Heat-Capacity Difference 2.C P  C V (since  > 0) 3.C P = C V (if  = 0)

Example:

Ideal gases Solids 300 J/cm 3 (25 o C, 1 atm) Internal Pressure Liquids 300 J/cm 3 (25 o C, 1 atm) Strong intermolecular forces in solids and liquids.

Example

 Calculation of Changes in State Function 1.Calculation of ΔS Suppose a closed system of constant composition goes from state (P 1,T 1 ) to state (P 2,T 2 ), the system’s entropy is a function of T and P

Integration gives: Since S is a state function, ΔS is independent of the path used to connect states 1 and 2. A convenient path (Figure 4.5) is first to hold P constant at P1 and change T from T1 to T2. Then T is held constant at T2, and P is changed from P1 to P2. For step (a), dP=0 and gives For step (b), dT=0 and gives

Example

2. Calculation of ΔH and ΔU ΔU can be easily found from ΔH using : ΔU = ΔH – Δ (PV) Alternatively we can write down the equation for ΔU similar to:

3. Calculation of ΔG For isothermal process: Alternatively, ΔG for an isothermal process that does not involve an irreversible composition change can be found as: A special case:

 Phase Equilibrium A phase equilbrium involves the same chemical species present in different phase. - - Phase equilib, in closed syst, P-V work only

For the spontaneous flow of moles of j from phase to phase - Closed syst that has not yet reached phase equilibrium

One EXCEPTION to the phase equilibrium, Then, j cannot flow out of (since it is absent from ). The system will therefore unchanged with time and hence in equilibrium. So the equilibrium condition becomes: Phase equilib, j absent from

 Reaction Equilibrium A reaction equilibrium involves different chemical species present in the same phase. Let the reaction be: reactants products a, b,…..e, f….. Are the coefficients

Adopt the convention of of transporting the reactant to the right side of equation: are negative for reactant and positive for products During a chemical reaction, the change Δn in the no. of moles of each substance is proportional to its stoichometric coefficient v. This proportionality constant is called the extent of reaction (xi) For general chemical reaction undergoing a definite amount of reaction, the change in moles of species i,, equals multiplied by the proportionality constant :

The condition for chemical-reaction equilibrium in a closed system is Reaction equilib, in closed system., P-V work only

Example