Materials Chemistry II – Kinetics – Lecture 2 PDF

Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz Topics and questions 1) What have we learnt so far ? 2) Thermodynamics square / Guggenheim scheme 3) Relationship between statistical physics (partition functions - q, Q) and macroscopic properti...

Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz Topics and questions 1) What have we learnt so far ? 2) Thermodynamics square / Guggenheim scheme 3) Relationship between statistical physics (partition functions - q, Q) and macroscopic properties (thermodynamic state functions) - internal energy U - entropy S - Helmholtz energy A - pressure p - Gibbs energy G 2 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 What have we learnt so far ? We have considered a system. A system can be anything, e.g. molecules. 𝜀𝜀 A system can be in a configuration. 0 A statistical ensemble is a collection of configurations a system can be in. 𝐸𝐸 = 0 𝑊𝑊 = 1 W is weight of a configuration 𝐸𝐸 = 𝜀𝜀 𝑊𝑊 = 3 𝐸𝐸 = 2𝜀𝜀 𝑊𝑊 = 3 𝑊𝑊 = 3 𝑊𝑊 = 6 𝑊𝑊 = 3𝑥𝑥𝑥𝑥𝑥𝑥 = 3! 𝐸𝐸 = 3𝜀𝜀 𝑊𝑊 = 1 𝑁𝑁! 𝐸𝐸 = 4𝜀𝜀 𝑊𝑊 = 3 𝑊𝑊 = 3 𝑊𝑊 = 𝑛𝑛0 ! 𝑛𝑛1 ! 𝑛𝑛2 ! … 𝐸𝐸 = 5𝜀𝜀 𝑊𝑊 = 3 A system will be found in the configuration 𝐸𝐸 = 6𝜀𝜀 𝑊𝑊 = 1 with the largest weight 3 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 What have we learnt so far ? … continues We have looked for a maximum of weight W of a configuration ! ! 𝑑𝑑𝑑𝑑 = 0 𝑑𝑑 𝑙𝑙𝑙𝑙 𝑊𝑊 = 0 by a method of undetermined (Lagrange) multipliers 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑊𝑊 𝑑𝑑 𝑙𝑙𝑙𝑙 𝑊𝑊 = 𝑑𝑑𝑛𝑛𝑖𝑖 = 0 𝜕𝜕 𝑛𝑛𝑖𝑖 𝑖𝑖 constant total number molecules of a system 𝑁𝑁 = 𝑛𝑛𝑖𝑖 𝑑𝑑𝑛𝑛𝑖𝑖 = 0 𝑖𝑖 𝑖𝑖 constant total energy of a system 𝐸𝐸 = 𝑛𝑛𝑖𝑖 𝜀𝜀𝑖𝑖 𝜀𝜀𝑖𝑖 𝑑𝑑𝑛𝑛𝑖𝑖 = 0 𝑖𝑖 𝑖𝑖 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑊𝑊 Lagrange multipliers 𝛼𝛼 and − 𝛽𝛽 𝑑𝑑𝑛𝑛𝑖𝑖 + 𝛼𝛼 𝑑𝑑𝑛𝑛𝑖𝑖 − 𝛽𝛽 𝜀𝜀𝑖𝑖 𝑑𝑑𝑛𝑛𝑖𝑖 = 0 𝜕𝜕 𝑛𝑛𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 … Stirling’s approximation 𝑙𝑙𝑙𝑙 𝑛𝑛! ≈ 𝑛𝑛 𝑙𝑙𝑙𝑙 𝑛𝑛 − 𝑛𝑛 … … Boltzmann distribution (pi) pi – probability of the occupation of the i-th energy level 𝑛𝑛𝑖𝑖 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 𝑝𝑝𝑖𝑖 = = 𝑞𝑞 = 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑗𝑗 q - molecular (microcanonical) partition function 𝑁𝑁 ∑𝑗𝑗 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑗𝑗 𝑗𝑗 4 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 What have we learnt so far ? … end statistical ensembles microcanonical ensemble 𝑞𝑞 = 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 q – molecular partition function 𝑖𝑖 εi – energy of the i-th energy level - isolated system with constant energy E - characteristic thermodynamic state function S (entropy) canonical ensemble 𝑄𝑄 = 𝑒𝑒 −𝛽𝛽𝐸𝐸𝑖𝑖 Ei – energy of the state i of the system 𝑖𝑖 - constant temperature T - characteristic thermodynamic state function A (Helmholtz energy) for independent molecules distinguishable - different in chemical nature – Ar vs. Ne 𝑄𝑄 = 𝑞𝑞 𝑁𝑁 - Ti in a solid state with a set of coordinates 𝑞𝑞 𝑁𝑁 indistinguishable - particles free to move – Ar gas 𝑄𝑄 = 𝑁𝑁! - Ti vapor (gas phase) Gibbs paradox 5 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Thermodynamic square / Guggenheim scheme Good Physicists Have Studied Under Very Ambitious Teachers (*) -S U V H A -p G T thermodynamic state functions variables V = constant for U – internal energy S – entropy microcanonical and H – enthalpy V – volume √ canonical ensembles A – Helmholtz energy p – pressure T = constant for a G – Gibbs energy T – temperature √ canonical ensemble *hint: search for other mnemonics in internet 6 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Internal energy U -S U V H A -p G T Any individual molecule can be in states with various energies 𝜀𝜀2 The lowest energy level ε 0 is often taken as 0 𝜀𝜀1 and all other energies will be measured relative to ε 0 𝜀𝜀0 = 0 The total energy of a system is given by We can use Boltzmann distribution 𝐸𝐸 = 𝑛𝑛𝑖𝑖 𝜀𝜀𝑖𝑖 𝑛𝑛𝑖𝑖 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 1 −𝛽𝛽𝜀𝜀 𝑝𝑝𝑖𝑖 = = −𝛽𝛽𝜀𝜀 = 𝑒𝑒 𝑖𝑖 𝑖𝑖 𝑁𝑁 ∑𝑗𝑗 𝑒𝑒 𝑗𝑗 𝑞𝑞 𝑞𝑞 = 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 𝑖𝑖 𝑁𝑁 −𝛽𝛽𝜀𝜀 𝑛𝑛𝑖𝑖 = 𝑒𝑒 𝑖𝑖 𝑞𝑞 7 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Internal energy U -S U V H A -p G T Any individual molecule can be in states with various energies 𝜀𝜀2 The lowest energy level ε 0 is often taken as 0 𝜀𝜀1 and all other energies will be measured relative to ε 0 𝜀𝜀0 = 0 The total energy of a system is given by We can use Boltzmann distribution 𝐸𝐸 = 𝑛𝑛𝑖𝑖 𝜀𝜀𝑖𝑖 𝑛𝑛𝑖𝑖 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 1 −𝛽𝛽𝜀𝜀 𝑝𝑝𝑖𝑖 = = −𝛽𝛽𝜀𝜀 = 𝑒𝑒 𝑖𝑖 𝑖𝑖 𝑁𝑁 ∑𝑗𝑗 𝑒𝑒 𝑗𝑗 𝑞𝑞 𝑁𝑁 𝐸𝐸 = 𝜀𝜀𝑖𝑖 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 𝑞𝑞 𝑞𝑞 = 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑁𝑁 −𝛽𝛽𝜀𝜀 𝑛𝑛𝑖𝑖 = 𝑒𝑒 𝑖𝑖 𝑞𝑞 8 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Internal energy U -S U V H A -p G T Any individual molecule can be in states with various energies 𝜀𝜀2 The lowest energy level ε 0 is often taken as 0 𝜀𝜀1 and all other energies will be measured relative to ε 0 𝜀𝜀0 = 0 The total energy of a system is given by We can use Boltzmann distribution 𝐸𝐸 = 𝑛𝑛𝑖𝑖 𝜀𝜀𝑖𝑖 𝑛𝑛𝑖𝑖 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 1 −𝛽𝛽𝜀𝜀 𝑝𝑝𝑖𝑖 = = −𝛽𝛽𝜀𝜀 = 𝑒𝑒 𝑖𝑖 𝑖𝑖 𝑁𝑁 ∑𝑗𝑗 𝑒𝑒 𝑗𝑗 𝑞𝑞 𝑁𝑁 −𝛽𝛽𝜀𝜀𝑖𝑖 𝑑𝑑 𝐸𝐸 = 𝜀𝜀𝑖𝑖 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 𝜀𝜀𝑖𝑖 𝑒𝑒 =− 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 𝑞𝑞 𝑑𝑑𝛽𝛽 𝑞𝑞 = 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 𝑖𝑖 𝑁𝑁 𝑑𝑑 −𝛽𝛽𝜀𝜀 𝑁𝑁 𝑑𝑑 𝑖𝑖 𝐸𝐸 = − 𝑒𝑒 𝑖𝑖 =− 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 𝑁𝑁 −𝛽𝛽𝜀𝜀 𝑞𝑞 𝑑𝑑𝛽𝛽 𝑞𝑞 𝑑𝑑𝛽𝛽 𝑛𝑛𝑖𝑖 = 𝑒𝑒 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑞𝑞 9 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Internal energy U -S U V H A -p G T Any individual molecule can be in states with various energies 𝜀𝜀2 The lowest energy level ε 0 is often taken as 0 𝜀𝜀1 and all other energies will be measured relative to ε 0 𝜀𝜀0 = 0 The total energy of a system is given by We can use Boltzmann distribution 𝐸𝐸 = 𝑛𝑛𝑖𝑖 𝜀𝜀𝑖𝑖 𝑛𝑛𝑖𝑖 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 1 −𝛽𝛽𝜀𝜀 𝑝𝑝𝑖𝑖 = = −𝛽𝛽𝜀𝜀 = 𝑒𝑒 𝑖𝑖 𝑖𝑖 𝑁𝑁 ∑𝑗𝑗 𝑒𝑒 𝑗𝑗 𝑞𝑞 𝑁𝑁 −𝛽𝛽𝜀𝜀𝑖𝑖 𝑑𝑑 𝐸𝐸 = 𝜀𝜀𝑖𝑖 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 𝜀𝜀𝑖𝑖 𝑒𝑒 =− 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 𝑞𝑞 𝑑𝑑𝛽𝛽 𝑞𝑞 = 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 𝑖𝑖 𝑁𝑁 𝑑𝑑 −𝛽𝛽𝜀𝜀 𝑁𝑁 𝑑𝑑 𝑖𝑖 𝐸𝐸 = − 𝑒𝑒 𝑖𝑖 =− 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 𝑁𝑁 −𝛽𝛽𝜀𝜀 𝑞𝑞 𝑑𝑑𝛽𝛽 𝑞𝑞 𝑑𝑑𝛽𝛽 𝑛𝑛𝑖𝑖 = 𝑒𝑒 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑞𝑞 𝑁𝑁 𝑑𝑑𝑞𝑞 𝑑𝑑 𝑙𝑙𝑙𝑙 𝑞𝑞 𝐸𝐸 = − = −𝑁𝑁 𝑞𝑞 𝑑𝑑𝛽𝛽 𝑑𝑑𝛽𝛽 In order to calculate the internal energy U of a system, we may have to add a constant U0 𝑈𝑈 = 𝑈𝑈0 + 𝐸𝐸 10 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Internal energy U -S U V H A -p G T Any individual molecule can be in states with various energies 𝜀𝜀2 The lowest energy level ε 0 is often taken as 0 𝜀𝜀1 and all other energies will be measured relative to ε 0 𝜀𝜀0 = 0 The total energy of a system is given by We can use Boltzmann distribution 𝐸𝐸 = 𝑛𝑛𝑖𝑖 𝜀𝜀𝑖𝑖 𝑛𝑛𝑖𝑖 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 1 −𝛽𝛽𝜀𝜀 𝑝𝑝𝑖𝑖 = = −𝛽𝛽𝜀𝜀 = 𝑒𝑒 𝑖𝑖 𝑖𝑖 𝑁𝑁 ∑𝑗𝑗 𝑒𝑒 𝑗𝑗 𝑞𝑞 𝑁𝑁 −𝛽𝛽𝜀𝜀𝑖𝑖 𝑑𝑑 𝐸𝐸 = 𝜀𝜀𝑖𝑖 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 𝜀𝜀𝑖𝑖 𝑒𝑒 =− 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 𝑞𝑞 𝑑𝑑𝛽𝛽 𝑞𝑞 = 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 𝑖𝑖 𝑁𝑁 𝑑𝑑 −𝛽𝛽𝜀𝜀 𝑁𝑁 𝑑𝑑 𝑖𝑖 𝐸𝐸 = − 𝑒𝑒 𝑖𝑖 =− 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 𝑁𝑁 −𝛽𝛽𝜀𝜀 𝑞𝑞 𝑑𝑑𝛽𝛽 𝑞𝑞 𝑑𝑑𝛽𝛽 𝑛𝑛𝑖𝑖 = 𝑒𝑒 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑞𝑞 𝑁𝑁 𝑑𝑑𝑞𝑞 𝑑𝑑 𝑙𝑙𝑙𝑙 𝑞𝑞 𝐸𝐸 = − = −𝑁𝑁 𝑞𝑞 𝑑𝑑𝛽𝛽 𝑑𝑑𝛽𝛽 In order to calculate the internal energy U of a system, we may have to add a constant U0 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑞𝑞 𝑈𝑈 = 𝑈𝑈0 − 𝑁𝑁 𝜕𝜕𝜕𝜕 𝑁𝑁𝑁𝑁 11 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Entropy S -S U V H A -p G T 1st thermodynamic law 𝑑𝑑𝑑𝑑 = 𝛿𝛿𝑞𝑞 + 𝛿𝛿𝑤𝑤 the change of the internal energy is given by sum of - the energy transferred to the system as heat 𝛿𝛿𝑞𝑞 - the work done on the system 𝛿𝛿𝑤𝑤 = −𝑝𝑝𝑝𝑝𝑝𝑝 = 0 (𝑉𝑉 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐.) 2nd thermodynamic law 𝛿𝛿𝑞𝑞 - thermodynamic definition of entropy 𝑑𝑑𝑑𝑑 = → 𝛿𝛿𝛿𝛿 = 𝑇𝑇 𝑑𝑑𝑑𝑑 𝑇𝑇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 = 𝑇𝑇 𝑑𝑑𝑑𝑑 → 𝑑𝑑𝑑𝑑 = 𝑇𝑇 𝑈𝑈 = 𝑈𝑈0 + 𝐸𝐸 𝐸𝐸 = 𝑛𝑛𝑖𝑖 𝜀𝜀𝑖𝑖 𝑑𝑑𝑑𝑑 = 𝑛𝑛𝑖𝑖 𝑑𝑑𝜀𝜀𝑖𝑖 + 𝜀𝜀𝑖𝑖 𝑑𝑑𝑛𝑛𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑑𝑑𝑑𝑑 = 0 + 𝑑𝑑𝑑𝑑 𝑑𝑑𝜀𝜀𝑖𝑖 = 0 the energy levels do not change at V = const. 𝑑𝑑𝑑𝑑 = 𝜀𝜀𝑖𝑖 𝑑𝑑𝑛𝑛𝑖𝑖 𝑑𝑑𝑑𝑑 = 𝜀𝜀𝑖𝑖 𝑑𝑑𝑛𝑛𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑑𝑑𝑑𝑑 1 𝑑𝑑𝑑𝑑 = = 𝜀𝜀𝑖𝑖 𝑑𝑑𝑛𝑛𝑖𝑖 = 𝑘𝑘𝛽𝛽 𝜀𝜀𝑖𝑖 𝑑𝑑𝑛𝑛𝑖𝑖 𝑇𝑇 𝑇𝑇 𝑖𝑖 𝑖𝑖 12 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Entropy S -S U V H A -p G T 1st thermodynamic law 𝑑𝑑𝑑𝑑 = 𝛿𝛿𝑞𝑞 + 𝛿𝛿𝑤𝑤 the change of the internal energy is given by sum of - the energy transferred to the system as heat 𝛿𝛿𝑞𝑞 - the work done on the system 𝛿𝛿𝑤𝑤 = −𝑝𝑝𝑝𝑝𝑝𝑝 = 0 (𝑉𝑉 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐.) 2nd thermodynamic law 𝛿𝛿𝑞𝑞 - thermodynamic definition of entropy 𝑑𝑑𝑑𝑑 = → 𝛿𝛿𝛿𝛿 = 𝑇𝑇 𝑑𝑑𝑑𝑑 𝑇𝑇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 = 𝑇𝑇 𝑑𝑑𝑑𝑑 → 𝑑𝑑𝑑𝑑 = 𝑇𝑇 𝑈𝑈 = 𝑈𝑈0 + 𝐸𝐸 𝐸𝐸 = 𝑛𝑛𝑖𝑖 𝜀𝜀𝑖𝑖 𝑑𝑑𝑑𝑑 = 𝑛𝑛𝑖𝑖 𝑑𝑑𝜀𝜀𝑖𝑖 + 𝜀𝜀𝑖𝑖 𝑑𝑑𝑛𝑛𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑑𝑑𝑑𝑑 = 0 + 𝑑𝑑𝑑𝑑 𝑑𝑑𝜀𝜀𝑖𝑖 = 0 the energy levels do not change at V = const. 𝑑𝑑𝑑𝑑 = 𝜀𝜀𝑖𝑖 𝑑𝑑𝑛𝑛𝑖𝑖 𝑑𝑑𝑑𝑑 = 𝜀𝜀𝑖𝑖 𝑑𝑑𝑛𝑛𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑑𝑑𝑑𝑑 1 𝑑𝑑𝑑𝑑 = = 𝜀𝜀𝑖𝑖 𝑑𝑑𝑛𝑛𝑖𝑖 = 𝑘𝑘𝛽𝛽 𝜀𝜀𝑖𝑖 𝑑𝑑𝑛𝑛𝑖𝑖 𝑇𝑇 𝑇𝑇 𝑖𝑖 𝑖𝑖 13 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Entropy S - … continues (1) … -S U V H A -p G T 𝑑𝑑𝑑𝑑 1 𝑑𝑑𝑑𝑑 = = 𝜀𝜀𝑖𝑖 𝑑𝑑𝑛𝑛𝑖𝑖 = 𝑘𝑘𝛽𝛽 𝜀𝜀𝑖𝑖 𝑑𝑑𝑛𝑛𝑖𝑖 = 𝑘𝑘 𝛽𝛽𝛽𝛽𝑖𝑖 𝑑𝑑𝑛𝑛𝑖𝑖 𝑇𝑇 𝑇𝑇 𝑖𝑖 𝑖𝑖 𝑖𝑖 In order to determine the most probable system configuration W, we have solved a set of equations 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑊𝑊 + 𝛼𝛼 − 𝛽𝛽𝜀𝜀𝑖𝑖 = 0 𝜕𝜕 𝑛𝑛𝑖𝑖 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑊𝑊 𝛽𝛽𝜀𝜀𝑖𝑖 = + 𝛼𝛼 𝜕𝜕 𝑛𝑛𝑖𝑖 𝑑𝑑𝑑𝑑 1 𝑑𝑑𝑑𝑑 = = 𝜀𝜀𝑖𝑖 𝑑𝑑𝑛𝑛𝑖𝑖 = 𝑘𝑘𝛽𝛽 𝜀𝜀𝑖𝑖 𝑑𝑑𝑛𝑛𝑖𝑖 𝑇𝑇 𝑇𝑇 𝑖𝑖 𝑖𝑖 14 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Entropy S - … continues (1) … -S U V H A -p G T 𝑑𝑑𝑑𝑑 1 𝑑𝑑𝑑𝑑 = = 𝜀𝜀𝑖𝑖 𝑑𝑑𝑛𝑛𝑖𝑖 = 𝑘𝑘𝛽𝛽 𝜀𝜀𝑖𝑖 𝑑𝑑𝑛𝑛𝑖𝑖 = 𝑘𝑘 𝛽𝛽𝛽𝛽𝑖𝑖 𝑑𝑑𝑛𝑛𝑖𝑖 𝑇𝑇 𝑇𝑇 𝑖𝑖 𝑖𝑖 𝑖𝑖 In order to determine the most probable system configuration W, we have solved a set of equations 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑊𝑊 + 𝛼𝛼 − 𝛽𝛽𝜀𝜀𝑖𝑖 = 0 𝜕𝜕 𝑛𝑛𝑖𝑖 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑊𝑊 𝛽𝛽𝜀𝜀𝑖𝑖 = + 𝛼𝛼 𝜕𝜕 𝑛𝑛𝑖𝑖 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑊𝑊 𝑑𝑑𝑑𝑑 = 𝑘𝑘 + 𝛼𝛼 𝑑𝑑𝑛𝑛𝑖𝑖 𝜕𝜕 𝑛𝑛𝑖𝑖 𝑖𝑖 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑊𝑊 𝑑𝑑𝑑𝑑 = 𝑘𝑘 𝑑𝑑𝑛𝑛𝑖𝑖 + 𝑘𝑘𝛼𝛼 𝑑𝑑𝑛𝑛𝑖𝑖 𝑁𝑁 = 𝑛𝑛𝑖𝑖 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐. 𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑛𝑛𝑖𝑖 = 0 𝜕𝜕 𝑛𝑛𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑊𝑊 𝑑𝑑𝑑𝑑 = 𝑘𝑘 𝑑𝑑𝑛𝑛𝑖𝑖 𝜕𝜕 𝑛𝑛𝑖𝑖 𝑖𝑖 𝑑𝑑𝑑𝑑 = 𝑘𝑘 𝑑𝑑 𝑙𝑙𝑙𝑙 𝑊𝑊 𝑆𝑆 = 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑊𝑊 Boltzmann equation 15 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Boltzmann equation 𝑆𝑆 = 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑊𝑊 W is a weight of a configuration 𝑇𝑇 → 0 𝑊𝑊 = 1 𝑙𝑙𝑙𝑙 𝑊𝑊 = 0 𝑆𝑆 = 0 3rd thermodynamic law 𝑆𝑆 = 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑊𝑊 Boltzmann equation 16 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Entropy S = function (q) - … continues (2) … -S U V H A -p G T microcanonical ensemble q – molecular partition function 𝑁𝑁! 𝑆𝑆 = 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑊𝑊 𝑊𝑊 = Boltzmann distribution 𝑛𝑛0 ! 𝑛𝑛1 ! 𝑛𝑛2 ! … 𝑁𝑁! 𝑛𝑛𝑖𝑖 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 𝑙𝑙𝑙𝑙 𝑊𝑊 = 𝑙𝑙𝑙𝑙 𝑛𝑛0 ! 𝑛𝑛1 ! 𝑛𝑛2 ! … ≈ 𝑁𝑁 𝑙𝑙𝑙𝑙 𝑁𝑁 − 𝑛𝑛𝑖𝑖 𝑙𝑙𝑙𝑙 𝑛𝑛𝑖𝑖 𝑝𝑝𝑖𝑖 = = 𝑁𝑁 ∑𝑗𝑗 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑗𝑗 = 𝑞𝑞 𝑖𝑖 ≈ 𝑛𝑛𝑖𝑖 𝑙𝑙𝑙𝑙 𝑁𝑁 − 𝑛𝑛𝑖𝑖 𝑙𝑙𝑙𝑙 𝑛𝑛𝑖𝑖 𝑛𝑛𝑖𝑖 = 𝑁𝑁 𝑝𝑝𝑖𝑖 𝑖𝑖 𝑛𝑛𝑖𝑖 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 ≈ − 𝑛𝑛𝑖𝑖 𝑙𝑙𝑙𝑙 = −𝑁𝑁 𝑝𝑝𝑖𝑖 𝑙𝑙𝑙𝑙 𝑝𝑝𝑖𝑖 𝑙𝑙𝑙𝑙 𝑝𝑝𝑖𝑖 = 𝑙𝑙𝑙𝑙 𝑁𝑁 𝑞𝑞 𝑖𝑖 𝑖𝑖 = −𝑁𝑁 𝑝𝑝𝑖𝑖 −𝛽𝛽𝜀𝜀𝑖𝑖 − 𝑙𝑙𝑙𝑙 𝑞𝑞 = 𝑙𝑙𝑙𝑙 𝑒𝑒 −𝛽𝛽𝜀𝜀𝑖𝑖 − 𝑙𝑙𝑙𝑙 𝑞𝑞 𝑖𝑖 = −𝛽𝛽𝜀𝜀𝑖𝑖 − 𝑙𝑙𝑙𝑙 𝑞𝑞 𝑆𝑆 = −𝑁𝑁𝑁𝑁 𝑝𝑝𝑖𝑖 −𝛽𝛽𝜀𝜀𝑖𝑖 − 𝑙𝑙𝑙𝑙 𝑞𝑞 = −𝑁𝑁𝑁𝑁 −𝛽𝛽 𝑝𝑝𝑖𝑖 𝜀𝜀𝑖𝑖 − 𝑝𝑝𝑖𝑖 𝑙𝑙𝑙𝑙 𝑞𝑞 𝑖𝑖 𝑖𝑖 𝑖𝑖 17 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Entropy S = function (q) - … continues (2) … -S U V H A -p G T microcanonical ensemble q – molecular partition function 𝑁𝑁! 𝑆𝑆 = 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑊𝑊 𝑊𝑊 = boundary conditions 𝑛𝑛0 ! 𝑛𝑛1 ! 𝑛𝑛2 ! … 𝑁𝑁! 𝑙𝑙𝑙𝑙 𝑊𝑊 = 𝑙𝑙𝑙𝑙 𝐸𝐸 = 𝑛𝑛𝑖𝑖 𝜀𝜀𝑖𝑖 𝑛𝑛0 ! 𝑛𝑛1 ! 𝑛𝑛2 ! … ≈ 𝑁𝑁 𝑙𝑙𝑙𝑙 𝑁𝑁 − 𝑛𝑛𝑖𝑖 𝑙𝑙𝑙𝑙 𝑛𝑛𝑖𝑖 𝑖𝑖 𝑖𝑖 𝐸𝐸 𝑛𝑛𝑖𝑖 ≈ 𝑛𝑛𝑖𝑖 𝑙𝑙𝑙𝑙 𝑁𝑁 − 𝑛𝑛𝑖𝑖 𝑙𝑙𝑙𝑙 𝑛𝑛𝑖𝑖 = 𝜀𝜀 = 𝑝𝑝𝑖𝑖 𝜀𝜀𝑖𝑖 𝑖𝑖 𝑁𝑁 𝑁𝑁 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑛𝑛𝑖𝑖 ≈ − 𝑛𝑛𝑖𝑖 𝑙𝑙𝑙𝑙 = −𝑁𝑁 𝑝𝑝𝑖𝑖 𝑙𝑙𝑙𝑙 𝑝𝑝𝑖𝑖 𝑁𝑁 𝑖𝑖 𝑖𝑖 𝑁𝑁 = 𝑛𝑛𝑖𝑖 = −𝑁𝑁 𝑝𝑝𝑖𝑖 −𝛽𝛽𝜀𝜀𝑖𝑖 − 𝑙𝑙𝑙𝑙 𝑞𝑞 𝑖𝑖 𝑁𝑁 𝑛𝑛𝑖𝑖 𝑖𝑖 = = 𝑝𝑝𝑖𝑖 = 1 𝑁𝑁 𝑁𝑁 𝑆𝑆 = −𝑁𝑁𝑁𝑁 𝑝𝑝𝑖𝑖 −𝛽𝛽𝜀𝜀𝑖𝑖 − 𝑙𝑙𝑙𝑙 𝑞𝑞 = −𝑁𝑁𝑁𝑁 −𝛽𝛽 𝑝𝑝𝑖𝑖 𝜀𝜀𝑖𝑖 − 𝑝𝑝𝑖𝑖 𝑙𝑙𝑙𝑙 𝑞𝑞 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 18 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Entropy S = function (q) - … continues (2) … -S U V H A -p G T microcanonical ensemble q – molecular partition function 𝑁𝑁! 𝑆𝑆 = 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑊𝑊 𝑊𝑊 = boundary conditions 𝑛𝑛0 ! 𝑛𝑛1 ! 𝑛𝑛2 ! … 𝑁𝑁! 𝑙𝑙𝑙𝑙 𝑊𝑊 = 𝑙𝑙𝑙𝑙 𝐸𝐸 = 𝑛𝑛𝑖𝑖 𝜀𝜀𝑖𝑖 𝑛𝑛0 ! 𝑛𝑛1 ! 𝑛𝑛2 ! … ≈ 𝑁𝑁 𝑙𝑙𝑙𝑙 𝑁𝑁 − 𝑛𝑛𝑖𝑖 𝑙𝑙𝑙𝑙 𝑛𝑛𝑖𝑖 𝑖𝑖 𝑖𝑖 𝐸𝐸 𝑛𝑛𝑖𝑖 ≈ 𝑛𝑛𝑖𝑖 𝑙𝑙𝑙𝑙 𝑁𝑁 − 𝑛𝑛𝑖𝑖 𝑙𝑙𝑙𝑙 𝑛𝑛𝑖𝑖 = 𝜀𝜀 = 𝑝𝑝𝑖𝑖 𝜀𝜀𝑖𝑖 𝑖𝑖 𝑁𝑁 𝑁𝑁 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑛𝑛𝑖𝑖 ≈ − 𝑛𝑛𝑖𝑖 𝑙𝑙𝑙𝑙 = −𝑁𝑁 𝑝𝑝𝑖𝑖 𝑙𝑙𝑙𝑙 𝑝𝑝𝑖𝑖 𝑁𝑁 𝑖𝑖 𝑖𝑖 𝑁𝑁 = 𝑛𝑛𝑖𝑖 = −𝑁𝑁 𝑝𝑝𝑖𝑖 −𝛽𝛽𝜀𝜀𝑖𝑖 − 𝑙𝑙𝑙𝑙 𝑞𝑞 𝑖𝑖 𝑁𝑁 𝑛𝑛𝑖𝑖 𝑖𝑖 = = 𝑝𝑝𝑖𝑖 = 1 𝑁𝑁 𝑁𝑁 𝑆𝑆 = −𝑁𝑁𝑁𝑁 𝑝𝑝𝑖𝑖 −𝛽𝛽𝜀𝜀𝑖𝑖 − 𝑙𝑙𝑙𝑙 𝑞𝑞 = −𝑁𝑁𝑁𝑁 −𝛽𝛽 𝑝𝑝𝑖𝑖 𝜀𝜀𝑖𝑖 − 𝑝𝑝𝑖𝑖 𝑙𝑙𝑙𝑙 𝑞𝑞 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 we have defined 𝑆𝑆 = 𝑘𝑘𝛽𝛽 𝐸𝐸 + 𝑁𝑁𝑁𝑁 𝑙𝑙𝑙𝑙 𝑞𝑞 𝑈𝑈 = 𝑈𝑈0 + 𝐸𝐸 𝐸𝐸 = 𝑈𝑈 − 𝑈𝑈0 19 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Entropy S = function (q) - end -S U V H A -p G T microcanonical ensemble q – molecular partition function 𝑁𝑁! 𝑆𝑆 = 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑊𝑊 𝑊𝑊 = boundary conditions 𝑛𝑛0 ! 𝑛𝑛1 ! 𝑛𝑛2 ! … 𝑁𝑁! 𝑙𝑙𝑙𝑙 𝑊𝑊 = 𝑙𝑙𝑙𝑙 𝐸𝐸 = 𝑛𝑛𝑖𝑖 𝜀𝜀𝑖𝑖 𝑛𝑛0 ! 𝑛𝑛1 ! 𝑛𝑛2 ! … ≈ 𝑁𝑁 𝑙𝑙𝑙𝑙 𝑁𝑁 − 𝑛𝑛𝑖𝑖 𝑙𝑙𝑙𝑙 𝑛𝑛𝑖𝑖 𝑖𝑖 𝑖𝑖 𝐸𝐸 𝑛𝑛𝑖𝑖 ≈ 𝑛𝑛𝑖𝑖 𝑙𝑙𝑙𝑙 𝑁𝑁 − 𝑛𝑛𝑖𝑖 𝑙𝑙𝑙𝑙 𝑛𝑛𝑖𝑖 = 𝜀𝜀 = 𝑝𝑝𝑖𝑖 𝜀𝜀𝑖𝑖 𝑖𝑖 𝑁𝑁 𝑁𝑁 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑛𝑛𝑖𝑖 ≈ − 𝑛𝑛𝑖𝑖 𝑙𝑙𝑙𝑙 = −𝑁𝑁 𝑝𝑝𝑖𝑖 𝑙𝑙𝑙𝑙 𝑝𝑝𝑖𝑖 𝑁𝑁 𝑖𝑖 𝑖𝑖 𝑁𝑁 = 𝑛𝑛𝑖𝑖 = −𝑁𝑁 𝑝𝑝𝑖𝑖 −𝛽𝛽𝜀𝜀𝑖𝑖 − 𝑙𝑙𝑙𝑙 𝑞𝑞 𝑖𝑖 𝑁𝑁 𝑛𝑛𝑖𝑖 𝑖𝑖 = = 𝑝𝑝𝑖𝑖 = 1 𝑁𝑁 𝑁𝑁 𝑆𝑆 = −𝑁𝑁𝑁𝑁 𝑝𝑝𝑖𝑖 −𝛽𝛽𝜀𝜀𝑖𝑖 − 𝑙𝑙𝑙𝑙 𝑞𝑞 = −𝑁𝑁𝑁𝑁 −𝛽𝛽 𝑝𝑝𝑖𝑖 𝜀𝜀𝑖𝑖 − 𝑝𝑝𝑖𝑖 𝑙𝑙𝑙𝑙 𝑞𝑞 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 we have defined 𝑆𝑆 = 𝑘𝑘𝛽𝛽 𝐸𝐸 + 𝑁𝑁𝑁𝑁 𝑙𝑙𝑙𝑙 𝑞𝑞 𝑈𝑈 = 𝑈𝑈0 + 𝐸𝐸 𝑆𝑆 = 𝑘𝑘𝛽𝛽 𝑈𝑈 − 𝑈𝑈0 + 𝑁𝑁𝑁𝑁 𝑙𝑙𝑙𝑙 𝑞𝑞 𝐸𝐸 = 𝑈𝑈 − 𝑈𝑈0 20 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Microcanonical and canonical ensemble ensemble microcanonical canonical constant N, V, E N, V, T characteristic state function S (entropy) A (Helmholtz energy) partition function q (molecular) Q (canonical) for independent 𝑞𝑞 molecules 𝑄𝑄 = 𝑞𝑞 𝑁𝑁 (distinguishable) 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑞𝑞 𝜕𝜕 𝑁𝑁 𝑙𝑙𝑙𝑙 𝑞𝑞 internal energy U 𝑈𝑈 = 𝑈𝑈0 − 𝑁𝑁 𝑈𝑈 = 𝑈𝑈0 − 𝜕𝜕𝜕𝜕 𝑁𝑁𝑁𝑁 𝜕𝜕𝜕𝜕 𝑁𝑁𝑁𝑁 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑞𝑞 𝑁𝑁 𝑈𝑈 = 𝑈𝑈0 − 𝜕𝜕𝜕𝜕 𝑁𝑁𝑁𝑁 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑈𝑈 = 𝑈𝑈0 − 𝜕𝜕𝜕𝜕 𝑁𝑁𝑁𝑁 entropy S 𝑆𝑆 = 𝑘𝑘𝛽𝛽 𝑈𝑈 − 𝑈𝑈0 + 𝑁𝑁𝑁𝑁 𝑙𝑙𝑙𝑙 𝑞𝑞 𝑆𝑆 = 𝑘𝑘𝛽𝛽 𝑈𝑈 − 𝑈𝑈0 + 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑞𝑞 𝑁𝑁 𝑆𝑆 = 𝑘𝑘𝛽𝛽 𝑈𝑈 − 𝑈𝑈0 + 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 21 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Helmholtz energy A -S U V H A -p G T The Helmholtz energy A is defined as 𝐴𝐴 = 𝑈𝑈 − 𝑇𝑇𝑇𝑇 entropy S 𝑆𝑆 = 𝑘𝑘𝛽𝛽 𝑈𝑈 − 𝑈𝑈0 + 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 1 𝑈𝑈 − 𝑈𝑈0 𝛽𝛽 = 𝑆𝑆 = + 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑘𝑘𝑘𝑘 𝑇𝑇 𝑈𝑈 − 𝑈𝑈0 𝐴𝐴 = 𝑈𝑈 − 𝑇𝑇 + 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑇𝑇 𝐴𝐴 = 𝑈𝑈 − 𝑈𝑈 − 𝑈𝑈0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐴𝐴 = 𝑈𝑈 − 𝑈𝑈 + 𝑈𝑈0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐴𝐴 = 𝑈𝑈0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 / 𝑇𝑇 = 0 𝐴𝐴 = 𝑈𝑈 − 𝑇𝑇 𝑆𝑆 𝐴𝐴0 = 𝑈𝑈0 − 0 𝑆𝑆 𝐴𝐴0 = 𝑈𝑈0 entropy S 𝑆𝑆 = 𝑘𝑘𝛽𝛽 𝑈𝑈 − 𝑈𝑈0 + 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 22 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Helmholtz energy A -S U V H A -p G T The Helmholtz energy A is defined as 𝐴𝐴 = 𝑈𝑈 − 𝑇𝑇𝑇𝑇 entropy S 𝑆𝑆 = 𝑘𝑘𝛽𝛽 𝑈𝑈 − 𝑈𝑈0 + 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 1 𝑈𝑈 − 𝑈𝑈0 𝛽𝛽 = 𝑆𝑆 = + 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑘𝑘𝑘𝑘 𝑇𝑇 𝑈𝑈 − 𝑈𝑈0 𝐴𝐴 = 𝑈𝑈 − 𝑇𝑇 + 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑇𝑇 𝐴𝐴 = 𝑈𝑈 − 𝑈𝑈 − 𝑈𝑈0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐴𝐴 = 𝑈𝑈 − 𝑈𝑈 + 𝑈𝑈0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐴𝐴 = 𝑈𝑈0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 / 𝑇𝑇 = 0 𝐴𝐴 = 𝑈𝑈 − 𝑇𝑇 𝑆𝑆 𝐴𝐴0 = 𝑈𝑈0 − 0 𝑆𝑆 𝐴𝐴0 = 𝑈𝑈0 𝐴𝐴 = 𝐴𝐴0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 23 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Pressure p -S U V H A -p G T Thermodynamic square Good Physicists Have Studied Under Very Ambitious Teachers 𝜕𝜕𝜕𝜕 𝜕𝜕𝐴𝐴 𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑑𝑑 + 𝑑𝑑𝑑𝑑 𝜕𝜕𝜕𝜕 𝜕𝜕𝑇𝑇 -S U V 𝑇𝑇 𝑉𝑉 𝑑𝑑𝑑𝑑 = −𝑝𝑝 𝑑𝑑𝑑𝑑 − 𝑆𝑆 𝑑𝑑𝑑𝑑 H A 𝜕𝜕𝐴𝐴 𝑝𝑝 = − / 𝐴𝐴 = 𝐴𝐴0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 -p G T 𝜕𝜕𝑉𝑉 𝑇𝑇 24 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Pressure p -S U V H A -p G T Thermodynamic square Good Physicists Have Studied Under Very Ambitious Teachers 𝜕𝜕𝜕𝜕 𝜕𝜕𝐴𝐴 𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑑𝑑 + 𝑑𝑑𝑑𝑑 𝜕𝜕𝜕𝜕 𝜕𝜕𝑇𝑇 -S U V 𝑇𝑇 𝑉𝑉 𝑑𝑑𝑑𝑑 = −𝑝𝑝 𝑑𝑑𝑑𝑑 − 𝑆𝑆 𝑑𝑑𝑑𝑑 H A 𝜕𝜕𝐴𝐴 𝑝𝑝 = − / 𝐴𝐴 = 𝐴𝐴0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 -p G T 𝜕𝜕𝑉𝑉 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑝𝑝 = 𝑘𝑘𝑘𝑘 𝜕𝜕𝑉𝑉 𝑇𝑇 25 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Enthalpy H -S U V H A -p G T Enthalpy H is defined as 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐻𝐻 = 𝑈𝑈 + 𝑝𝑝𝑝𝑝 𝑈𝑈 = 𝑈𝑈0 − 𝑝𝑝 = 𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝜕𝜕𝑉𝑉 𝑇𝑇 𝑉𝑉 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐻𝐻 = 𝑈𝑈0 − + 𝑘𝑘𝑘𝑘𝑘𝑘 / 𝐻𝐻0 = 𝑈𝑈0 𝜕𝜕𝜕𝜕 𝑉𝑉 𝜕𝜕𝜕𝜕 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑝𝑝 = 𝑘𝑘𝑘𝑘 𝜕𝜕𝑉𝑉 𝑇𝑇 26 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Enthalpy H -S U V H A -p G T Enthalpy H is defined as 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐻𝐻 = 𝑈𝑈 + 𝑝𝑝𝑝𝑝 𝑈𝑈 = 𝑈𝑈0 − 𝑝𝑝 = 𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝜕𝜕𝑉𝑉 𝑇𝑇 𝑉𝑉 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐻𝐻 = 𝑈𝑈0 − + 𝑘𝑘𝑘𝑘𝑘𝑘 / 𝐻𝐻0 = 𝑈𝑈0 𝜕𝜕𝜕𝜕 𝑉𝑉 𝜕𝜕𝜕𝜕 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐻𝐻 = 𝐻𝐻0 − + 𝑘𝑘𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝑉𝑉 𝜕𝜕𝜕𝜕 𝑇𝑇 27 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Gibbs energy G -S U V H A -p G T Gibbs energy G is defined as 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐺𝐺 = 𝐻𝐻 − 𝑇𝑇𝑇𝑇 / 𝐻𝐻 = 𝑈𝑈 + 𝑝𝑝𝑝𝑝 𝑝𝑝 = 𝑘𝑘𝑘𝑘 𝜕𝜕𝑉𝑉 𝑇𝑇 𝐺𝐺 = 𝑈𝑈 + 𝑝𝑝𝑝𝑝 − 𝑇𝑇𝑇𝑇 𝐺𝐺 = 𝑈𝑈 − 𝑇𝑇𝑇𝑇 + 𝑝𝑝𝑝𝑝 / 𝐴𝐴 = 𝑈𝑈 − 𝑇𝑇𝑇𝑇 𝐺𝐺 = 𝐴𝐴 + 𝑝𝑝𝑝𝑝 / 𝐴𝐴 = 𝐴𝐴0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐺𝐺 = 𝐴𝐴0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 + 𝑘𝑘𝑘𝑘𝑘𝑘 / 𝐴𝐴0 = 𝐺𝐺0 𝜕𝜕𝜕𝜕 𝑇𝑇 28 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Gibbs energy G -S U V H A -p G T Gibbs energy G is defined as 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐺𝐺 = 𝐻𝐻 − 𝑇𝑇𝑇𝑇 / 𝐻𝐻 = 𝑈𝑈 + 𝑝𝑝𝑝𝑝 𝑝𝑝 = 𝑘𝑘𝑘𝑘 𝜕𝜕𝑉𝑉 𝑇𝑇 𝐺𝐺 = 𝑈𝑈 + 𝑝𝑝𝑝𝑝 − 𝑇𝑇𝑇𝑇 𝐺𝐺 = 𝑈𝑈 − 𝑇𝑇𝑇𝑇 + 𝑝𝑝𝑝𝑝 / 𝐴𝐴 = 𝑈𝑈 − 𝑇𝑇𝑇𝑇 𝐺𝐺 = 𝐴𝐴 + 𝑝𝑝𝑝𝑝 / 𝐴𝐴 = 𝐴𝐴0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐺𝐺 = 𝐴𝐴0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 + 𝑘𝑘𝑘𝑘𝑘𝑘 / 𝐴𝐴0 = 𝐺𝐺0 𝜕𝜕𝜕𝜕 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐺𝐺 = 𝐺𝐺0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 + 𝑘𝑘𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝑇𝑇 29 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Summary (canonical ensemble) -S U V H A -p G T V V A T T 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 constants 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕𝜕𝜕 𝜕𝜕𝑉𝑉 𝑇𝑇 𝑉𝑉 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑈𝑈 = 𝑈𝑈0 − 𝜕𝜕𝜕𝜕 𝑉𝑉 𝑈𝑈 − 𝑈𝑈0 1 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑆𝑆 = + 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 = − + 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑇𝑇 𝑇𝑇 𝜕𝜕𝜕𝜕 𝑉𝑉 𝐴𝐴 = 𝐴𝐴0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑝𝑝 = 𝑘𝑘𝑘𝑘 𝜕𝜕𝑉𝑉 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐻𝐻 = 𝐻𝐻0 − + 𝑘𝑘𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝑉𝑉 𝜕𝜕𝜕𝜕 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐺𝐺 = 𝐺𝐺0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 + 𝑘𝑘𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝑇𝑇 30 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Summary (canonical ensemble) -S U V H A -p G T V U0 V A T T 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 constants 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕𝜕𝜕 𝜕𝜕𝑉𝑉 𝑇𝑇 𝑉𝑉 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑈𝑈 = 𝑈𝑈0 − 𝜕𝜕𝜕𝜕 𝑉𝑉 𝑈𝑈 − 𝑈𝑈0 1 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑆𝑆 = + 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 = − + 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑇𝑇 𝑇𝑇 𝜕𝜕𝜕𝜕 𝑉𝑉 𝐴𝐴 = 𝐴𝐴0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑝𝑝 = 𝑘𝑘𝑘𝑘 𝜕𝜕𝑉𝑉 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐻𝐻 = 𝐻𝐻0 − + 𝑘𝑘𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝑉𝑉 𝜕𝜕𝜕𝜕 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐺𝐺 = 𝐺𝐺0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 + 𝑘𝑘𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝑇𝑇 31 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Summary (canonical ensemble) -S U V H A -p G T V U0 V A T T 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 constants 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕𝜕𝜕 𝜕𝜕𝑉𝑉 𝑇𝑇 𝑉𝑉 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑈𝑈 = 𝑈𝑈0 − 𝜕𝜕𝜕𝜕 𝑉𝑉 1 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑆𝑆 = 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 − 𝑇𝑇 𝜕𝜕𝜕𝜕 𝑉𝑉 𝐴𝐴 = 𝐴𝐴0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑝𝑝 = 𝑘𝑘𝑘𝑘 𝜕𝜕𝑉𝑉 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐻𝐻 = 𝐻𝐻0 − + 𝑘𝑘𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝑉𝑉 𝜕𝜕𝜕𝜕 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐺𝐺 = 𝐺𝐺0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 + 𝑘𝑘𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝑇𝑇 32 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Summary (canonical ensemble) -S U V H A -p G T V U0 V A A0 T T 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 constants 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕𝜕𝜕 𝜕𝜕𝑉𝑉 𝑇𝑇 𝑉𝑉 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑈𝑈 = 𝑈𝑈0 − 𝜕𝜕𝜕𝜕 𝑉𝑉 1 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑆𝑆 = 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 − 𝑇𝑇 𝜕𝜕𝜕𝜕 𝑉𝑉 𝐴𝐴 = 𝐴𝐴0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑝𝑝 = 𝑘𝑘𝑘𝑘 𝜕𝜕𝑉𝑉 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐻𝐻 = 𝐻𝐻0 − + 𝑘𝑘𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝑉𝑉 𝜕𝜕𝜕𝜕 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐺𝐺 = 𝐺𝐺0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 + 𝑘𝑘𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝑇𝑇 33 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Summary (canonical ensemble) -S U V H A -p G T V U0 V A A0 T T 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 constants 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕𝜕𝜕 𝜕𝜕𝑉𝑉 𝑇𝑇 𝑉𝑉 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑈𝑈 = 𝑈𝑈0 − 𝜕𝜕𝜕𝜕 𝑉𝑉 1 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑆𝑆 = 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 − 𝑇𝑇 𝜕𝜕𝜕𝜕 𝑉𝑉 𝐴𝐴 = 𝐴𝐴0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑝𝑝 = 𝑘𝑘𝑘𝑘 𝜕𝜕𝑉𝑉 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐻𝐻 = 𝐻𝐻0 − + 𝑘𝑘𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝑉𝑉 𝜕𝜕𝜕𝜕 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐺𝐺 = 𝐺𝐺0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 + 𝑘𝑘𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝑇𝑇 34 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Summary (canonical ensemble) -S U V H A -p G T V U0 V A H0 A0 T T 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 constants 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕𝜕𝜕 𝜕𝜕𝑉𝑉 𝑇𝑇 𝑉𝑉 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑈𝑈 = 𝑈𝑈0 − 𝜕𝜕𝜕𝜕 𝑉𝑉 1 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑆𝑆 = 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 − 𝑇𝑇 𝜕𝜕𝜕𝜕 𝑉𝑉 𝐴𝐴 = 𝐴𝐴0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑝𝑝 = 𝑘𝑘𝑘𝑘 𝜕𝜕𝑉𝑉 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐻𝐻 = 𝐻𝐻0 − + 𝑘𝑘𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝑉𝑉 𝜕𝜕𝜕𝜕 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐺𝐺 = 𝐺𝐺0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 + 𝑘𝑘𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝑇𝑇 35 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Summary (canonical ensemble) -S U V H A -p G T V U0 V A H0 A0 T G0 T 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 constants 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕𝜕𝜕 𝜕𝜕𝑉𝑉 𝑇𝑇 𝑉𝑉 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑈𝑈 = 𝑈𝑈0 − 𝜕𝜕𝜕𝜕 𝑉𝑉 1 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑆𝑆 = 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 − 𝑇𝑇 𝜕𝜕𝜕𝜕 𝑉𝑉 𝐴𝐴 = 𝐴𝐴0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑝𝑝 = 𝑘𝑘𝑘𝑘 𝜕𝜕𝑉𝑉 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐻𝐻 = 𝐻𝐻0 − + 𝑘𝑘𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝑉𝑉 𝜕𝜕𝜕𝜕 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐺𝐺 = 𝐺𝐺0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 + 𝑘𝑘𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝑇𝑇 36 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Summary (canonical ensemble) -S U V H A -p G T V U0 V A H0 A0 T G0 T 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 constants 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕𝜕𝜕 𝜕𝜕𝑉𝑉 𝑇𝑇 𝑉𝑉 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑈𝑈 = 𝑈𝑈0 − 𝜕𝜕𝜕𝜕 𝑉𝑉 1 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑆𝑆 = 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 − / 𝑥𝑥 𝑇𝑇 𝑇𝑇 𝜕𝜕𝜕𝜕 𝑉𝑉 𝐴𝐴 = 𝐴𝐴0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑝𝑝 = 𝑘𝑘𝑘𝑘 / 𝑥𝑥 𝑉𝑉 𝜕𝜕𝑉𝑉 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐻𝐻 = 𝐻𝐻0 − + 𝑘𝑘𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝑉𝑉 𝜕𝜕𝜕𝜕 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐺𝐺 = 𝐺𝐺0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 + 𝑘𝑘𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝑇𝑇 37 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Summary (canonical ensemble) -S U V H A -p G T V U0 V A H0 A0 T G0 T 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 constants 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕𝜕𝜕 𝜕𝜕𝑉𝑉 𝑇𝑇 𝑉𝑉 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑈𝑈 = 𝑈𝑈0 − 𝜕𝜕𝜕𝜕 𝑉𝑉 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑆𝑆𝑆𝑆 = 𝑘𝑘𝑇𝑇 𝑙𝑙𝑙𝑙 𝑄𝑄 − 𝜕𝜕𝜕𝜕 𝑉𝑉 𝐴𝐴 = 𝐴𝐴0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑝𝑝𝑝𝑝 = 𝑘𝑘𝑘𝑘𝑘𝑘 𝜕𝜕𝑉𝑉 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐻𝐻 = 𝐻𝐻0 − + 𝑘𝑘𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝑉𝑉 𝜕𝜕𝜕𝜕 𝑇𝑇 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝐺𝐺 = 𝐺𝐺0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 + 𝑘𝑘𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝑇𝑇 38 Materials Chemistry II – Kinetics – Lecture 2 Thermodynamics Stanislav Mráz | Winter semester 2024 / 25 Summary (canonical ensemble) -S U V H A -p G T V U0 V A H0 A0 T G0 T 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 constants 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕𝜕𝜕 𝜕𝜕𝑉𝑉 𝑇𝑇 𝑉𝑉 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑈𝑈 = 𝑈𝑈0 − 𝜕𝜕𝜕𝜕 𝑉𝑉 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑆𝑆𝑆𝑆 = 𝑘𝑘𝑇𝑇 𝑙𝑙𝑙𝑙 𝑄𝑄 − 𝜕𝜕𝜕𝜕 𝑉𝑉 𝐴𝐴 = 𝐴𝐴0 − 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝜕𝜕 𝑙𝑙𝑙𝑙 𝑄𝑄 𝑋𝑋 = (𝑋𝑋0 ) ± 𝑘𝑘𝑘𝑘 𝑙𝑙𝑙𝑙 𝑄𝑄 − + ?