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Unformatted text preview: Fundamental equation of thermodynamics
The ﬁrst and second laws of thermodynamics imply that dU = T dS + Y dX + µdN with ∂U ∂S ∂U ∂X ∂U ∂N [tln16] (1) = T,
X,N = Y,
S,X is the exact diﬀerential of a function U (S, X, N ). Here X stands for V, M, . . . and Y stands for −p, H, . . .. Note: for irreversible processes dU < T dS + Y dX + µdN holds. U, S, X, N are extensive state variables. U (S, X, N ) is a 1st order homogeneous function: U (λS, λX, λN ) = λU (S, X, N ). U [(1 + )S, (1 + )X, (1 + )N ] = U + Euler equation: U = T S + Y X + µN. (2) ∂U ∂U ∂U S+ X+ N = (1 + )U. ∂S ∂X ∂N Total diﬀerential of (2): dU = T dS + SdT + Y dX + XdY + µdN + N dµ Subtract (1) from (3): Gibbs-Duhem equation: SdT + XdY + N dµ = 0. The Gibbs-Duhem equation expresses a relationship between the intensive variables T, Y, µ. It can be integrated, for example, into a function µ(T, Y ). Note: a system speciﬁed by m independent extensive variables possesses m − 1 independent intensive variables. Example for m = 3: S, V, N (extensive); S/N, V /N or p, T (intensive). Complete speciﬁcation of a thermodynamic system must involve at least one extensive variable. (3) ...
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This note was uploaded on 09/24/2010 for the course MMAE 510 taught by Professor Hassannagib during the Spring '10 term at Illinois Tech.
- Spring '10