Eigenvalues

# Eigenvalues - EIGENVALUES AND THERMODYNAMIC STABILITY...

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• MohsinMMK
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EIGENVALUES AND THERMODYNAMIC STABILITY CHEMISTRY 213 When considering thermodynamic stability for an isolated system, we had to show that E (2) 1 2 2 E S 2 V , N i S 2 + 2 E S V N i S V + 1 2 2 E V 2 S , N i E 2 + ... (1) was positive, where terms involving changes of N i have been dropped. Moreover, by using the expressions for the derivatives of the internal energy, Eq. (1) can be rewritten as: E (2) T 2 C V S 2 + T V S , N i S V - 1 2 P V S , N i V 2 + ... , (2) or in matrix notation: E (2) = 1 2 S V T C V T V S , N i T V S , N i - P V S , N i S V . (3) The matrix in Eq. (3) is symmetric. From linear algebra, we know that symmetric matrix can be diagonalized; i.e., a basis in the [ S ,

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