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 𝐴 = 𝑈 + 𝑇𝑆 (5.1) • The Gibbs free energy, G, is defined as 𝐺 = 𝑈 + 𝑃𝑉 − 𝑇𝑆 = 𝐻 − 𝑇𝑆 (5.2) • For a system undergoing a change of state from 1 to 2, 𝐺2 − 𝐺1 = 𝐻2 − 𝐻1 − 𝑇2𝑆2 − 𝑇1𝑆1 = 𝑈2 − 𝑈1 + 𝑃2𝑉2 − 𝑃1𝑉1 − (𝑇2𝑆2 − 𝑇1𝑆1) • For a closed system, the first law gives, 𝑈2 − 𝑈1 = 𝑞 − 𝑤 and thus 𝐺2 − 𝐺1 = 𝑞 − 𝑤 + 𝑃2𝑉2 − 𝑃1𝑉1 − (𝑇2𝑆2 − 𝑇1𝑆1) (5.6) • If the processes is carried out such that 𝑇1 = 𝑇2 = 𝑇, the constant temperature of the heat reservoir which supplies or withdraws heat from the system, and also if 𝑃1 = 𝑃2 = 𝑃, the constant pressure at which the system undergoes a change in volume , then (𝐺2 − 𝐺1) = 𝑞 − 𝑤 + 𝑃 𝑉2 − 𝑉1 − 𝑇(𝑆2 − 𝑆1) • If the system preforms chemical or electrical work in addition to 𝑃 − 𝑉 work, the work terms are included in 𝑤, i.e. 𝑤 = 𝑤′ + 𝑃 𝑉2 − 𝑉1 in which 𝑤′ is the sum of all non 𝑃 − 𝑉 forms of work done. Substituting into the above eqn. gives, 𝐺2 − 𝐺1 = 𝑞 − 𝑤′ − 𝑇 𝑆2 − 𝑆1 and from the second law as, 𝑞 ≤ 𝑇(𝑆2 − 𝑆1) then, 𝑤′ ≤ −(𝐺2 − 𝐺1) (5.7) The equality can be written as, −𝑤 ′ = 𝐺2 − 𝐺1 + 𝑇∆𝑆𝑖𝑟𝑟 • Thus, for an isothermal, isobaric process, during which no form of work other than P-V work is performed i.e. 𝑤′ = 0, 𝐺2 − 𝐺1 + 𝑇∆𝑆𝑖𝑟𝑟 = 0 (5.8) Such a process can only occur spontaneously (with an increase in entropy) if the Gibbs free energy decreases. As 𝑑𝑆𝑖𝑟𝑟 = 0 is a criterion for a reversible process, thermodynamic equilibrium requires that, 𝑑𝐺 = 0 (5.9) • Thus for a system undergoing a process at constant 𝑇 and constant 𝑃, 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 𝐺 consistent with the values of 𝑃 and 𝑇. • Since 𝑑𝑈 = 𝑇𝑑𝑆 − 𝑃𝑑𝑉 and 𝐻 = 𝑈 + 𝑃𝑉 and thus 𝑑𝐻 = 𝑇𝑑𝑆 + 𝑉𝑑𝑃 (5.10) 𝐴 = 𝑈 − 𝑇𝑆 and thus 𝑑𝐴 = −𝑆𝑑𝑇 − 𝑃𝑑𝑉 (5.11) 𝐺 = 𝐻 − 𝑇𝑆 and thus 𝑑𝐺 = −𝑆𝑑𝑇 + 𝑉𝑑𝑃 (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 𝑇, 𝑃 and the numbers of moles of all the species present in the system: 𝐺 = 𝐺(𝑇, 𝑃, 𝑛𝑖 , 𝑛𝑗, 𝑛𝑘, … ) (5.13) Where 𝑛𝑖, 𝑛𝑗, 𝑛𝑘, … . are the number of moles of the species 𝑖, 𝑗, 𝑘, … 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: 𝜕𝐺 𝜕𝐺 𝜕𝐺 𝑑𝐺 = 𝑑𝑇 + 𝑑𝑃 + 𝑑𝑛𝑖 + 𝜕𝑇 𝑃, 𝑛𝑖, 𝑛𝑗 … 𝜕𝑃 𝑇, 𝑛𝑖, 𝑛𝑗, … 𝜕𝑛𝑖 𝑇, 𝑃, 𝑛𝑗, 𝑛𝑘, … 𝜕𝐺 𝑑𝑛𝑗 𝜕𝑛𝑗 𝑇, 𝑃 , 𝑛𝑖, 𝑛𝑘, … +⋯ (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. 𝑑𝐺 = −𝑆𝑑𝑇 + 𝑉𝑑𝑃 from which it is seen that 𝜕𝐺 𝜕𝑇 𝑃 , 𝑛𝑖, 𝑛𝑗, … = −𝑆 and 𝜕𝐺 𝜕𝑃 Substituting into eqn. (5.14) gives 𝑇, 𝑛𝑖, 𝑛𝑗, … 𝑖=𝑘 𝑑𝐺 = −𝑆𝑑𝑇 + 𝑉𝑑𝑃 + 𝑖=1 =𝑉 𝜕𝐺 𝜕𝑛𝑖 𝑇, 𝑃, 𝑛𝑗, … 𝑑𝑛𝑖 (5.15) where 𝑖=𝑘 𝑖=1 𝜕𝐺 𝜕𝑛𝑖 𝑇, 𝑃, 𝑛𝑗, … 𝑑𝑛𝑖 is the sum of 𝑘 terms, one for each species of the 𝑘 species, each of which is determined by partial differentiation of 𝐺 w.r.t the number of moles of the 𝑖th species at constant 𝑇, 𝑃, and 𝑛𝑗, where 𝑛𝑗 represents the number of moles of every species other than the 𝑖th species. Chemical Potential • The term 𝜕𝐺 𝜕𝑛𝑖 𝑇, 𝑃, 𝑛𝑗… is called the chemical potential of the species 𝑖, designated 𝑖 , i.e. 𝜕𝐺 𝜕𝑛𝑖 𝑇, 𝑃, 𝑛𝑗, … = 𝑖 (5.16) • The chemical potential of 𝑖 in a homogeneous phase is the rate of increase of 𝐺 with 𝑛𝑖 when the species 𝑖 is added to the system at constant 𝑇, 𝑃 and number of moles of all the other species. • If the system is large enough that the addition of 1 mole of 𝑖, at constant 𝑇 and 𝑃, does not measurably change the composition of the system, then 𝑖 is the increase in 𝐺 of the system caused by the addition. • Eqn (5.15) can thus be written as 𝑑𝐺 = −𝑆𝑑𝑇 + 𝑉𝑑𝑃 + 𝑘1 𝜇𝑖𝑑𝑛𝑖 (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), 𝑑𝑈 = 𝑇𝑑𝑆 − 𝑃𝑑𝑉, (5.10), 𝑑𝐻 = 𝑇𝑑𝑆 + 𝑉𝑑𝑃, and (5.11), 𝑑𝐴 = −𝑆𝑑𝑇 − 𝑃𝑑𝑉, can be made applicable to open systems by including the terms describing the dependences on composition of, respectively, 𝑈, 𝐻 and 𝐴: 𝑘 𝑑𝑈 = 𝑇𝑑𝑆 − 𝑃𝑑𝑉 + 1 𝜕𝑈 𝜕𝑛𝑖 𝑆 , 𝑉 , 𝑛𝑗, … 𝑑𝑛𝑖 (5.18) 𝑘 𝑑𝐻 = 𝑇𝑑𝑆 − 𝑉𝑑𝑃 + 1 𝜕𝐻 𝜕𝑛𝑖 𝑆 , 𝑃 , 𝑛𝑗, … 𝑑𝑛𝑖 (5.19) 𝑘 𝑑𝐴 = −𝑆𝑑𝑇 + 𝑃𝑑𝑉 + 1 𝜕𝐴 𝜕𝑛𝑖 𝑇, 𝑉 , 𝑛𝑗, … 𝑑𝑛𝑖 (5.20) • Inspection of Eqns. (5.16), (5.18), (5.19) and (5.20) shows that: 𝜕𝐺 𝜕𝑈 𝜕𝐻 𝜕𝐴 = 𝜇𝑖 = = = 𝜕𝑛𝑖 𝑇, 𝑃, 𝑛𝑗 𝜕𝑛𝑖 𝑆, 𝑉 , 𝑛𝑗 𝜕𝑛𝑖 𝑆, 𝑃, 𝑛𝑗 𝜕𝑛𝑖 𝑇, 𝑉 , 𝑛𝑗 (5.21) and hence the complete set of equations is: 𝑑𝑈 = 𝑇𝑑𝑆 − 𝑃𝑑𝑉 + 𝜇𝑖𝑑𝑛𝑖 (5.22) 𝑑𝐻 = 𝑇𝑑𝑆 + 𝑉𝑑𝑃 + 𝜇𝑖𝑑𝑛𝑖 (5.23) 𝑑𝐴 = −𝑆𝑑𝑇 − 𝑃𝑑𝑉 + 𝜇𝑖𝑑𝑛𝑖 (5.24) 𝑑𝐺 = −𝑆𝑑𝑇 + 𝑉𝑑𝑃 + 𝜇𝑖𝑑𝑛𝑖 (5.25) • The first law gives 𝑑𝑈 = 𝛿𝑞 − 𝛿𝑤 which on comparison with equation (5.22), 𝑑𝑈 = 𝑇𝑑𝑆 − 𝑃𝑑𝑉 + 𝜇𝑖𝑑𝑛𝑖 , indicates that, for a closed system undergoing a process involving a reversible change in composition (e.g. a reversible chemical reaction), 𝛿𝑞 = 𝑇𝑑𝑆 and 𝛿𝑤 = 𝑃𝑑𝑉 + 𝑢𝑖𝑑𝑛𝑖 The term 𝑢𝑖𝑑𝑛𝑖 is thus the chemical work done by the system which was denoted as 𝑤′ in Eq. (5.7), 𝑤′ ≤ −(𝐺2 − 𝐺1) . Thermodynamic Relations The following relations are obtained from Eqns. (5.22)-(5.25): 𝑇= 𝜕𝑈 𝜕𝑆 𝑉 , 𝑐𝑜𝑚𝑝 = 𝜕𝐻 𝜕𝑆 𝑃 , 𝑐𝑜𝑚𝑝 (5.26) 𝜕𝑈 𝑃=− 𝜕𝑉 𝜕𝐴 =− 𝜕𝑉 𝑆, 𝑐𝑜𝑚𝑝 𝑇, 𝑐𝑜𝑚𝑝 (5.27) 𝜕𝐻 𝑉= 𝜕𝑃 𝜕𝐺 = 𝜕𝑃 𝑆, 𝑐𝑜𝑚𝑝 𝑇, 𝑐𝑜𝑚𝑝 (5.28) 𝑆=− 𝜕𝐴 𝜕𝑇 𝑉 , 𝑐𝑜𝑚𝑝 =− 𝜕𝐺 𝜕𝑇 𝑃 , 𝑐𝑜𝑚𝑝 (5.29) Maxwell’s Equations • If 𝑍 is a state function and 𝑥 and 𝑦 are chosen as the independent variables in a closed system of fixed composition then, 𝑍 = 𝑍(𝑥, 𝑦) and by differentiation, 𝜕𝑍 𝜕𝑍 𝑑𝑍 = 𝑑𝑥 + 𝑑𝑦 𝜕𝑥 𝑦 𝜕𝑦 𝑥 • If the partial derivative 𝜕𝑍 𝜕𝑥 𝑦 is itself a function of 𝑥 and 𝑦, being given by 𝜕𝑍 𝜕𝑥 𝑦 = 𝐿(𝑥, 𝑦), and similarly 𝜕𝑍 𝜕𝑦 𝑥 = 𝑀 𝑥, 𝑦 then 𝑑𝑍 = 𝐿𝑑𝑥 + 𝑀𝑑𝑦 Thus 𝜕 𝜕𝑍 𝜕𝑦 𝜕𝑥 and 𝑦 𝑥 𝜕𝐿 = 𝜕𝑦 𝑥 𝜕 𝜕𝑍 𝜕𝑀 = 𝜕𝑥 𝜕𝑦 𝑥 𝑦 𝜕𝑥 𝑦 But, as 𝑍 is a state function, the change in 𝑍 is independent of the order of differentiation, and thus, 𝜕 𝜕𝑍 𝜕 𝜕𝑍 𝜕2𝑍 = = 𝜕𝑦 𝜕𝑥 𝑦 𝑥 𝜕𝑥 𝜕𝑦 𝑥 𝑦 𝜕𝑥𝜕𝑦 and hence 𝜕𝐿 𝜕𝑀 = 𝜕𝑦 𝑥 𝜕𝑥 𝑦 (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: 𝜕𝑇 𝜕𝑉 𝜕𝑃 =− 𝜕𝑆 𝑆 𝑉 (5.31) 𝜕𝑇 𝜕𝑃 𝜕𝑉 = 𝜕𝑆 𝑆 𝑃 (5.32) 𝜕𝑆 𝜕𝑉 𝜕𝑃 = 𝜕𝑇 𝑇 𝑉 (5.33) 𝜕𝑆 𝜕𝑃 𝜕𝑉 =− 𝜕𝑇 𝑇 𝑃 (5.34) • Consider the dependence of entropy of an ideal gas on 𝑇 and 𝑉: 𝑆 = 𝑆 𝑇, 𝑉 and by differentiation 𝑑𝑆 = 𝜕𝑆 𝜕𝑇 𝑑𝑇 + 𝑉 𝜕𝑆 𝜕𝑉 𝑑𝑉 𝑇 (𝑖) Combination of the definition of the constant volume heat capacity (Eqn. 𝛿𝑞 (2.6), 𝑑𝑈 = 𝐶𝑣 𝑑𝑇 and Eqn. (3.8), 𝑑𝑆 = 𝑟𝑒𝑣 applied to a reversible constant 𝑇 volume process gives, 𝑇𝑑𝑆 = 𝛿𝑞𝑣 = 𝑑𝑈 = 𝑛𝑐𝑣 𝑑𝑇 the first P.D on the right-hand side of eqn. (i) is obtained as 𝜕𝑆 𝑛𝑐𝑣 = 𝜕𝑇 𝑉 𝑇 and the second P.D in eqn. (i) is obtained from Maxwell’s equation (5.33). • Thus eqn. (i) can be written as, 𝑛𝑐𝑣 𝜕𝑃 𝑑𝑆 = 𝑑𝑇 + 𝑇 𝜕𝑇 𝑑𝑉 𝑉 (ii) From the ideal gas law 𝜕𝑃 𝜕𝑇 𝑉 𝑛𝑅 = 𝑉 (iii) and thus eqn. (ii) can be written as 𝑛𝑐𝑣 𝑛𝑅 𝑑𝑆 = 𝑑𝑇 + 𝑑𝑉 𝑇 𝑉 (iv) integration of which between two states gives 𝑇2 𝑉2 𝑆2 − 𝑆1 = 𝑛𝑐𝑣 𝐼𝑛 + 𝑛𝑅𝐼𝑛 𝑇1 𝑉1 • Considering a closed system of fixed composition Eqn. (3.12) gives, 𝑑𝑈 = 𝑇𝑑𝑆 − 𝑃𝑑𝑉 and thus 𝜕𝑈 𝜕𝑆 =𝑇 −𝑃 𝜕𝑉 𝑇 𝜕𝑉 𝑇 Using Maxwell’s eqn. (5.33) allows this to be written as 𝜕𝑈 𝜕𝑃 =𝑇 −𝑃 𝜕𝑉 𝑇 𝜕𝑇 𝑉 (v) which is an equation of state relating the internal energy of a closed system of fixed composition to measurable quantities 𝑇, 𝑉 and 𝑃. If the system is 1 mole of an ideal gas, substitution of eqn. (iii) into eqn. (v) gives 𝜕𝑈 =0 𝜕𝑉 𝑇 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, 𝑑𝐻 = 𝑇𝑑𝑆 + 𝑉𝑑𝑃 in which case 𝜕𝐻 𝜕𝑃 𝜕𝑆 =𝑇 𝜕𝑃 𝑇 𝑇 +𝑉 Substituting Maxwell’s eqn. (5.34) gives 𝜕𝐻 𝜕𝑉 = −𝑇 +𝑉 𝜕𝑃 𝑇 𝜕𝑇 𝑃 which is an equation of state which gives the dependence of enthalpy on 𝑇, 𝑃 and 𝑉. 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 𝐺 = 𝐻 − 𝑇𝑆 Eqn. (5.12) gives 𝑑𝐺 = −𝑆𝑑𝑇 + 𝑉𝑑𝑃 and therefore, 𝜕𝐺 = −𝑆 𝜕𝑇 𝑃 Thus at constant pressure and composition, 𝑑𝐺 𝐺 =𝐻+𝑇 𝑑𝑇 or 𝐺𝑑𝑇 = 𝐻𝑑𝑇 + 𝑇𝑑𝐺 Dividing throughout by 𝑇2 and rearranging gives, 𝑇𝑑𝐺 − 𝐺𝑑𝑇 𝐻𝑑𝑇 =− 2 𝑇2 𝑇 which on comparison with the identity 𝑥 𝑦𝑑𝑥 − 𝑥𝑑𝑦 𝑑 = 𝑦 𝑦2 shows that 𝑑 𝐺 𝑇 𝐻 =− 2 𝑑𝑇 𝑇 (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 𝐺 to the change in 𝐻 as 𝑑(∆𝐺 𝑇) ∆𝐻 =− 2 𝑑𝑇 𝑇 (5.36a) • This equation is used in experimental thermodynamics since it allows ∆𝐻, the heat of reaction, to be obtained from a measurement of ∆𝐺, the free energy change for the reaction, with temperature, or, conversely ∆𝐺 to be obtained from a measurement of ∆𝐻.