CHM2104 Hour 15 v2 PDF

Maximum work and free energy Combination of first and second laws. Maxwell relations Overview ▪ Derive the Clausius inequality; ▪ Express the inequality in system only terms ▪ Define Gibbs (G) & Helmholtz (H) state functions ▪ Derive the proof that Gibbs energy is a measure of ma...

Maximum work and free energy Combination of first and second laws. Maxwell relations Overview ▪ Derive the Clausius inequality; ▪ Express the inequality in system only terms ▪ Define Gibbs (G) & Helmholtz (H) state functions ▪ Derive the proof that Gibbs energy is a measure of maximum non-expansion work ▪ Derive the Fundamental equation by Combining the First and Second Laws ▪ Use it to derive a Maxwell relation and discuss their importance 2 The Clausius inequality ▪ For a system in thermal and mechanical contact with the surroundings (not necessarily in equilibrium) undergoing spontaneous change we recall: 𝒅𝑺𝒔𝒚𝒔 + 𝒅𝑺𝒔𝒖𝒓𝒓 ≥ 𝟎 Which implies: 𝒅𝑺𝒔𝒚𝒔 ≥ −𝒅𝑺𝒔𝒖𝒓𝒓 ▪ Recall dS surr = -dqsys/T 𝒅𝒒𝒔𝒚𝒔 𝒅𝑺𝒔𝒚𝒔 ≥ 𝑻 ▪ This is the Clausius inequality. Thinking question: What happens to dq if the system is isolated? 3 Concentrating on the system only ▪ S does indeed measure spontaneity but we need to see the system and the surroundings simultaneously. This can become tedious. ▪ We now seek “system-only based” criteria for spontaneity ▪ For a system in thermal equilibrium with surroundings at temperature T we invoke the Clausius inequality: 4 System-only parameters ▪ When heat is transferred at constant volume dU = dq since no work is done (dw = 0): ▪ So now we get: ▪ Which rearranges to: ▪ Now we have system-only parameters to describe spontaneous change. ▪ What happens at constant U? At constant S? 5 System-only parameters ▪ When heat transfer occurs at constant pressure, we know dqp = dH so we get: ▪ ▪ This gives: ▪ Once again we have system only parameters. ▪ What happens at constant H? At constant S? 6 Gibbs and Helmholtz functions ▪ We now have two ways of describing spontaneous change without directly considering the surroundings: 𝑑𝑈 − 𝑇𝑑𝑆 ≤ 0 and 𝑑𝐻 − 𝑇𝑑𝑆 ≤ 0 ▪ These two relationships have been used to define two “new” thermodynamic functions: ▪Helmholtz energy, A defined as 𝑨 = 𝑼 − 𝑻𝑺 such that 𝑑𝐴 = 𝑑𝑈 − 𝑇𝑑𝑆 or ∆𝐴 = ∆𝑈 − 𝑇∆𝑆 ▪Gibbs energy, G defined as G = H – TS such that 𝑑𝐺 = 𝑑𝐻 − 𝑇𝑑𝑆 or ∆𝐺 = ∆𝐻 − 𝑇∆𝑆 ▪ Now spontaneity simply becomes: 𝒅𝑨𝑽,𝑻 ≤ 𝟎 or 𝒅𝑮𝒑,𝑻 ≤ 𝟎 7 Why focus on Gibbs energy? ▪ Which is more popular – Gibbs energy (G) or Helmholtz energy (A)? Why? ▪ So at constant T and P, spontaneous changes and reactions are those with decreasing Gibbs energy. ▪ We can use dG = dH – TdS to determine whether a spontaneous change is “enthalpy driven” or “entropy- driven” 8 ∆G and Maximum non-expansion work ▪ Non-expansion work is called additional work. ▪ Eg. Work done in moving electrons through a circuit or raising a mass ▪ It can be shown that G is the maximum non-expansion work that the system can do at constant T and P. ▪ This is stated: 9 Atkins pg 99 10 ∆A and Maximum work ▪ The change in Helmholtz energy is equal to the maximum work accompanying a process 𝒅𝑨 = 𝒅𝒘𝒎𝒂𝒙 ▪ Helmholtz function A is sometimes referred to as the “maximum work function” or “work function” 11 Atkins pg 97 12 Combining First and Second Laws ▪ First law states: 𝒅𝑼 = 𝒅𝒒 + 𝒅𝒘 ▪ Second Law states: 𝒅𝒒𝒓𝒆𝒗 = 𝑻𝒅𝑺 ▪ Recall: 𝒅𝒘𝒓𝒆𝒗 = −𝒑𝒅𝑽 ▪ Combining the First and Second Laws of Thermodynamics gives the Fundamental equation: 𝒅𝑼 = 𝑻𝒅𝑺 − 𝒑𝒅𝑽 ▪ NB. We derived it for reversible changes in heat and work, but since U is a state function… 13 Maxwell Relations The Fundamental equation: ▪ Applies to any change of a closed system with constant composition that does no additional work ▪ Defined reversibly but since U is a state function (it does not have to be a reversible path) Since U is a function of S and V we can write dU and a sum of the partial derivatives: 𝝏𝑼 𝝏𝑼 𝒅𝑼 = 𝒅𝑺 + 𝒅𝑽 𝝏𝑺 𝑽 𝝏𝑽 𝑺 14 Maxwell Relations cont’d 𝝏𝑼 𝝏𝑼 𝒅𝑼 = 𝒅𝑺 + 𝒅𝑽 𝝏𝑺 𝑽 𝝏𝑽 𝑺 ▪ However: 𝒅𝑼 = 𝑻𝒅𝑺 − 𝒑𝒅𝑽 𝜕𝑈 𝜕𝑈 ▪ This implies that: 𝑇 = and −𝑝 = 𝜕𝑆 𝑉 𝜕𝑉 𝑆 It can be shown that if df = gdx + hdy then for df to be an exact differential: 𝜕𝑔 𝜕ℎ = 𝜕𝑦 𝑥 𝜕𝑥 𝑦 Since U is an exact differential then: 𝜕𝑇 𝜕𝑝 =− 𝜕𝑉 𝑆 𝜕𝑆 𝑉 This is a Maxwell Relation. 15 Maxwell Relations cont’d ▪ Maxwell relations allow us to derive unusual and useful relationships between quantities that may not seem related ▪ They gives us the flexibility to pursue changes in the system along paths that are most convenient to us ▪ Try deriving Maxwell relations for H, G and A 16 Maxwell Relations cont’d 17