Auxiliary Functions Introduction • The study of thermodynamics provides the criteria for equilibrium in systems and the determination of the effect, on an equilibrium state, of changes in the external influences acting on the system. • This provision is determined by the practicality of the equations of state for the system. • To achieve this, it is desirable to develop simple expressions which contain convenient choice of independent variables which can be easily controlled and measured. • It’s is often found that some required thermodynamic expressions which are not easily amenable to experimental measurement are related in a simpler manner to some measurable quantities. • New thermodynamic functions are introduced and their properties and interrelationships are examined: Helmholtz and Gibbs Free Energies • The Helmholtz free energy, A, is defined as π΄ = π + ππ (5.1) • The Gibbs free energy, G, is defined as πΊ = π + ππ − ππ = π» − ππ (5.2) • For a system undergoing a change of state from 1 to 2, πΊ2 − πΊ1 = π»2 − π»1 − π2π2 − π1π1 = π2 − π1 + π2π2 − π1π1 − (π2π2 − π1π1) • For a closed system, the first law gives, π2 − π1 = π − π€ and thus πΊ2 − πΊ1 = π − π€ + π2π2 − π1π1 − (π2π2 − π1π1) (5.6) • If the processes is carried out such that π1 = π2 = π, the constant temperature of the heat reservoir which supplies or withdraws heat from the system, and also if π1 = π2 = π, the constant pressure at which the system undergoes a change in volume , then (πΊ2 − πΊ1) = π − π€ + π π2 − π1 − π(π2 − π1) • If the system preforms chemical or electrical work in addition to π − π work, the work terms are included in π€, i.e. π€ = π€′ + π π2 − π1 in which π€′ is the sum of all non π − π forms of work done. Substituting into the above eqn. gives, πΊ2 − πΊ1 = π − π€′ − π π2 − π1 and from the second law as, π ≤ π(π2 − π1) then, π€′ ≤ −(πΊ2 − πΊ1) (5.7) The equality can be written as, −π€ ′ = πΊ2 − πΊ1 + πβππππ • Thus, for an isothermal, isobaric process, during which no form of work other than P-V work is performed i.e. π€′ = 0, πΊ2 − πΊ1 + πβππππ = 0 (5.8) Such a process can only occur spontaneously (with an increase in entropy) if the Gibbs free energy decreases. As πππππ = 0 is a criterion for a reversible process, thermodynamic equilibrium requires that, ππΊ = 0 (5.9) • Thus for a system undergoing a process at constant π and constant π, the Gibbs free energy can only decrease or remain constant, and the attainment of equilibrium in the system coincides with the system having the minimum value of πΊ consistent with the values of π and π. • Since ππ = πππ − πππ and π» = π + ππ and thus ππ» = πππ + πππ (5.10) π΄ = π − ππ and thus ππ΄ = −πππ − πππ (5.11) πΊ = π» − ππ and thus ππΊ = −πππ + πππ (5.12) Variation of Composition and Size of System • If the size and composition of a system can vary during a process, the specification of only two variables will be no longer sufficient to fix the state of the system. • It is therefore necessary to specify the number of moles within the system. • Gibbs free energy is thus a function of π, π and the numbers of moles of all the species present in the system: πΊ = πΊ(π, π, ππ , ππ, ππ, … ) (5.13) Where ππ, ππ, ππ, … . are the number of moles of the species π, π, π, … present in the system and the state of the system is only fixed when all the independent variables are fixed. Differentiation of (5.13) gives: ππΊ ππΊ ππΊ ππΊ = ππ + ππ + πππ + ππ π, ππ, ππ … ππ π, ππ, ππ, … πππ π, π, ππ, ππ, … ππΊ πππ πππ π, π , ππ, ππ, … +β― (5.14) • If the numbers of moles of all of the individual species remain constant during the process the above eqn. simplifies to eqn. (5.12) i.e. ππΊ = −πππ + πππ from which it is seen that ππΊ ππ π , ππ, ππ, … = −π and ππΊ ππ Substituting into eqn. (5.14) gives π, ππ, ππ, … π=π ππΊ = −πππ + πππ + π=1 =π ππΊ πππ π, π, ππ, … πππ (5.15) where π=π π=1 ππΊ πππ π, π, ππ, … πππ is the sum of π terms, one for each species of the π species, each of which is determined by partial differentiation of πΊ w.r.t the number of moles of the πth species at constant π, π, and ππ, where ππ represents the number of moles of every species other than the πth species. Chemical Potential • The term ππΊ πππ π, π, ππ… is called the chemical potential of the species π, designated οπ , i.e. ππΊ πππ π, π, ππ, … = οπ (5.16) • The chemical potential of π in a homogeneous phase is the rate of increase of πΊ with ππ when the species π is added to the system at constant π, π and number of moles of all the other species. • If the system is large enough that the addition of 1 mole of π, at constant π and π, does not measurably change the composition of the system, then οπ is the increase in πΊ of the system caused by the addition. • Eqn (5.15) can thus be written as ππΊ = −πππ + πππ + π1 πππππ (5.17) • • Eqn (5.17) can be applied to open systems which exchange matter and heat with their surroundings, and to closed systems which undergo changes in composition caused by chemical reactions. Equations (3.12), ππ = πππ − πππ, (5.10), ππ» = πππ + πππ, and (5.11), ππ΄ = −πππ − πππ, can be made applicable to open systems by including the terms describing the dependences on composition of, respectively, π, π» and π΄: π ππ = πππ − πππ + 1 ππ πππ π , π , ππ, … πππ (5.18) π ππ» = πππ − πππ + 1 ππ» πππ π , π , ππ, … πππ (5.19) π ππ΄ = −πππ + πππ + 1 ππ΄ πππ π, π , ππ, … πππ (5.20) • Inspection of Eqns. (5.16), (5.18), (5.19) and (5.20) shows that: ππΊ ππ ππ» ππ΄ = ππ = = = πππ π, π, ππ πππ π, π , ππ πππ π, π, ππ πππ π, π , ππ (5.21) and hence the complete set of equations is: ππ = πππ − πππ + πππππ (5.22) ππ» = πππ + πππ + πππππ (5.23) ππ΄ = −πππ − πππ + πππππ (5.24) ππΊ = −πππ + πππ + πππππ (5.25) • The first law gives ππ = πΏπ − πΏπ€ which on comparison with equation (5.22), ππ = πππ − πππ + πππππ , indicates that, for a closed system undergoing a process involving a reversible change in composition (e.g. a reversible chemical reaction), πΏπ = πππ and πΏπ€ = πππ + π’ππππ The term π’ππππ is thus the chemical work done by the system which was denoted as π€′ in Eq. (5.7), π€′ ≤ −(πΊ2 − πΊ1) . Thermodynamic Relations The following relations are obtained from Eqns. (5.22)-(5.25): π= ππ ππ π , ππππ = ππ» ππ π , ππππ (5.26) ππ π=− ππ ππ΄ =− ππ π, ππππ π, ππππ (5.27) ππ» π= ππ ππΊ = ππ π, ππππ π, ππππ (5.28) π=− ππ΄ ππ π , ππππ =− ππΊ ππ π , ππππ (5.29) Maxwell’s Equations • If π is a state function and π₯ and π¦ are chosen as the independent variables in a closed system of fixed composition then, π = π(π₯, π¦) and by differentiation, ππ ππ ππ = ππ₯ + ππ¦ ππ₯ π¦ ππ¦ π₯ • If the partial derivative ππ ππ₯ π¦ is itself a function of π₯ and π¦, being given by ππ ππ₯ π¦ = πΏ(π₯, π¦), and similarly ππ ππ¦ π₯ = π π₯, π¦ then ππ = πΏππ₯ + πππ¦ Thus π ππ ππ¦ ππ₯ and π¦ π₯ ππΏ = ππ¦ π₯ π ππ ππ = ππ₯ ππ¦ π₯ π¦ ππ₯ π¦ But, as π is a state function, the change in π is independent of the order of differentiation, and thus, π ππ π ππ π2π = = ππ¦ ππ₯ π¦ π₯ ππ₯ ππ¦ π₯ π¦ ππ₯ππ¦ and hence ππΏ ππ = ππ¦ π₯ ππ₯ π¦ (5.30) • Application of equation (5.30) to Eqns. (3.12) and (5.10)-(5.12) gives a set of relations which are referred to as the Maxwell’s Equations: ππ ππ ππ =− ππ π π (5.31) ππ ππ ππ = ππ π π (5.32) ππ ππ ππ = ππ π π (5.33) ππ ππ ππ =− ππ π π (5.34) • Consider the dependence of entropy of an ideal gas on π and π: π = π π, π and by differentiation ππ = ππ ππ ππ + π ππ ππ ππ π (π) Combination of the definition of the constant volume heat capacity (Eqn. πΏπ (2.6), ππ = πΆπ£ ππ and Eqn. (3.8), ππ = πππ£ applied to a reversible constant π volume process gives, πππ = πΏππ£ = ππ = πππ£ ππ the first P.D on the right-hand side of eqn. (i) is obtained as ππ πππ£ = ππ π π and the second P.D in eqn. (i) is obtained from Maxwell’s equation (5.33). • Thus eqn. (i) can be written as, πππ£ ππ ππ = ππ + π ππ ππ π (ii) From the ideal gas law ππ ππ π ππ = π (iii) and thus eqn. (ii) can be written as πππ£ ππ ππ = ππ + ππ π π (iv) integration of which between two states gives π2 π2 π2 − π1 = πππ£ πΌπ + ππ πΌπ π1 π1 • Considering a closed system of fixed composition Eqn. (3.12) gives, ππ = πππ − πππ and thus ππ ππ =π −π ππ π ππ π Using Maxwell’s eqn. (5.33) allows this to be written as ππ ππ =π −π ππ π ππ π (v) which is an equation of state relating the internal energy of a closed system of fixed composition to measurable quantities π, π and π. If the system is 1 mole of an ideal gas, substitution of eqn. (iii) into eqn. (v) gives ππ =0 ππ π Therefore the internal energy of an ideal gas is independent of the volume of the gas. • Similarly for a closed system of fixed composition, eqn. (5.10) gives, ππ» = πππ + πππ in which case ππ» ππ ππ =π ππ π π +π Substituting Maxwell’s eqn. (5.34) gives ππ» ππ = −π +π ππ π ππ π which is an equation of state which gives the dependence of enthalpy on π, π and π. If the system is an ideal gas, this equation of state shows that the enthalpy of an ideal gas is independent of its pressure. The Gibbs-Helmholtz Eqn.(5.2) gives πΊ = π» − ππ Eqn. (5.12) gives ππΊ = −πππ + πππ and therefore, ππΊ = −π ππ π Thus at constant pressure and composition, ππΊ πΊ =π»+π ππ or πΊππ = π»ππ + πππΊ Dividing throughout by π2 and rearranging gives, πππΊ − πΊππ π»ππ =− 2 π2 π which on comparison with the identity π₯ π¦ππ₯ − π₯ππ¦ π = π¦ π¦2 shows that π πΊ π π» =− 2 ππ π (5.36) The above equation is known as the Gibbs-Helmholtz equation and is applicable to a closed system of fixed composition undergoing processes at constant composition. • For an isobaric change of state of a closed system of fixed composition, eqn. (5.36) gives the relation of the change in πΊ to the change in π» as π(βπΊ π) βπ» =− 2 ππ π (5.36a) • This equation is used in experimental thermodynamics since it allows βπ», the heat of reaction, to be obtained from a measurement of βπΊ, the free energy change for the reaction, with temperature, or, conversely βπΊ to be obtained from a measurement of βπ».