From Eq 26 we know that the left hand side of Eq 27 must be greater than or

From eq 26 we know that the left hand side of eq 27

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From Eq. (2.6), we know that the left-hand side of Eq. (2.7) must be greater than or equal to zero. The only manner in which to guarantee this condition of Eq. (2.7) is if coefficients of the δ ¯ U ( A ) , δ ¯ V ( A ) , and δ N ( A ) terms are each equal to zero. As a result, this implies the relations: T ( A ) = T ( B ) p ( A ) = p ( B ) μ ( A ) = μ ( B ) (2.8)
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Hamza J. Magnier Therefore, we find that the temperatures, pressures, and chemical potentials are equal for coexisting phases in equilibrium. We will later extend this derivation to multicomponent system. One interesting point to note is that the labels A and B which we used to derive Eqs. (2.8) do not have to refer to different phases. For example, A and B can refer to different parts of a one phase system. Therefore, Eqs. (2.8) can be interpreted as stating the the temperature, pressure, and chemical potential of a system at equilibrium are uniform (N.B., we did not include the influence of external fields, such as gravitional or electrostatic fields). 2.4 Stability criteria The condition for stable equilibrium is that all displacements from the equilibrium state must satisfy the condition that ( ¯ S ) N, ¯ V, ¯ U < 0 , where the subscripts denote those quantities are held constant for the system. We also found that the first order displacement from equilibrium is equal to zero, that is ( δ ¯ S ) N, ¯ V, ¯ U = 0 . As a consequence, if we consider ( ¯ S ) N, ¯ V, ¯ U = ( δ ¯ S ) N, ¯ V, ¯ U + ° δ 2 ¯ S ¢ N, ¯ V, ¯ U + ... (2.9) the condition for equilibrium implies that the second order variational displacement must satisfy the criteria ° δ 2 ¯ S ¢ N, ¯ V, ¯ U 0 (2.10) Conditions derived from this relation are termed stability criteria. We also could use a similar ap- proach to show an analogous relation for the change in internal energy associated with a change from the equilibrium state should correspond to ° δ 2 ¯ U ¢ N, ¯ V, ¯ S 0 (2.11) If the inequality ( δ 2 ¯ U ) N, ¯ V, ¯ S > 0 holds, the system is stable to fluctuations and will remain in a stable equilibrium state. However, if ( δ 2 ¯ U ) N, ¯ V, ¯ S = 0 , then higher order terms would need to be considered for establishing the equilibrium.
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  • Fall '19
  • Salvatore Ziccone

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