Lecture Notes in Advanced Thermodynamics

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Example: ideal gas – the well-known equations of the ideal gas (R is the gas constant and c is the isochoric specific heat, they are real constants): E = cMT ; pV = MRT – from that the state functions are: T (E, V, M) = E cM p(E, V, M) = MRT V = RE cV – let us choose the chemical potential function as: µ(E, V, M) = c ln E M + R ln V M – for these functions an appropriate entropy function: S(E, V, M) = Mc ln E M + MR ln V M + M(c + R) – it can be easily checked that for c > 0 this entropy function fulfils properties (P 1 ), (P 2 ) and (P 3 ), for proving the validity of (P 4 ) a longer calculation is required 2.1.3 Important relations from the model Gibbs relation – the total differential of the entropy function (d can be imagined as a very small change in the quantity) dS = ∂S ∂E ∣ dE + ∂S V,M ∂V ∣ dV + ∂S E,M ∂M ∣ dM E,V – using the (P 1 ) property: dS = 1 T dE + p T dV − µ dM (5) T – after rearranging the Gibbs relation is obtained: dE = T dS − pdV + µdM (6) 6
intensive property of T ,p and µ: – let us calculate the resizing of (5): dS ◦ λ = 1 T ◦ λ dE ◦ λ + p ◦ λ T ◦ λ dV ◦ λ − µ ◦ λ T ◦ λ dM ◦ λ – E,V ,M are extensive (trivial) and S is extensive too due to property (P 3 ): λdS = 1 T ◦ λ λdE + p ◦ λ T ◦ λ λdV − µ ◦ λ T ◦ λ λdM – from that: ∂S ∂E ∣ = 1 V,M T ◦ λ ; ∂S ∂V ∣ = p ◦ λ E,M T ◦ λ ; – comparing them with property (P 1 ): ∂S ∂M ∣ = − µ ◦ λ E,V T ◦ λ T ◦ λ = T ; p ◦ λ = p; µ ◦ λ = µ (7) – thus T , p and µ are intensive state functions Potential relation – differentiating property (P 1 ) respect to λ: ( ) ( ) ( ) ∂S ∂S ∂E ∣ ◦ λ E + ∂S V,M ∂V ∣ ◦ λ V + E,M ∂M ∣ ◦ λ M = S E,V – using property (P 1 ): – using (7): 1 T ◦ λ E + p ◦ λ T ◦ λ V − µ ◦ λ T ◦ λ M = S 1 T E + p T V − µ T M = S – after rearranging we get the potential relation: E = T S − pV + µM (8) 7
Page 1 and 2: Lecture Notes in Advanced Thermodyn
Page 3 and 4: thermostatics - thermodynamics - th
Page 5: - for that there exists a S(E, V, M
Page 9 and 10: Example: internal energy - let us c
Page 11 and 12: - substituting to the potential rel
Page 13 and 14: - considering an intensive state fu
Page 15 and 16: extensive quantity (A) V M E S F G
Page 17 and 18: Condition of the concave property o
Page 19 and 20: - the condition (27): - the conditi

Lecture Notes in Advanced Thermodynamics

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