As an example consider two phases denoted by A and B respectively contained

# As an example consider two phases denoted by a and b

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As an example, consider two phases denoted by A and B respectively contained within a compos- ite system held at constant ¯ S , ¯ V and N . For the system we will only consider exchanges in entropy between the phases under the constraint δ ¯ S = 0 = δ ¯ S ( A ) + δ ¯ S ( B ) (2.12) Now we consider the second order displacement of the internal energy from the taylor series expan- sion where the first order term has been set to zero ° δ 2 ¯ U ¢ = 1 2 µ 2 ¯ U ¯ S 2 ( A ) ¯ V,N ° δ ¯ S ( A ) ¢ 2 + 1 2 µ 2 ¯ U ¯ S 2 ( B ) ¯ V,N ° δ ¯ S ( B ) ¢ 2 . (2.13) From the definition of heat capacity, we have µ 2 ¯ U ¯ S 2 ¯ V,N = µ T ¯ S ¯ V,N = T ¯ C V (2.14) so that we have ° δ 2 ¯ U ¢ N, ¯ V, ¯ S = 1 2 ° δ ¯ S ( A ) ¢ 2 " T ( A ) ¯ C ( A ) V + T ( B ) ¯ C ( B ) V # . (2.15)

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Hamza J. Magnier At equilibrium, the temperatures of phases denoted by A and B are equal to each other. Therefore applying the stability criteria given by 2.11, we find that T h 1 / ¯ C ( A ) V + 1 / ¯ C ( B ) V i 0 (2.16) or since the fluctuations in entropy are arbitrary, the result implies that ¯ C V 0 . (2.17) Note the consequences of having a positive value of C v . If a system has a negative value of C v , the flow of heat from a hot body to a cold body would cause the temperatue gradient between the bodies to grow. That is, the temperature of the body being heated would decrease, whereas the temperature of the body losing heat would increase. Stability criteria can also be used for determining the sign of other properties such as the isother- mal compressibility κ T = 1 V µ V p T > 0 , (2.18) which is based on the relation ° δ 2 ¯ A ¢ N, ¯ V,T 0 . (2.19) 2.5 The Clapeyron equation In this section, we derive the Clapeyron equation. This equation relates changes in the pressure to changes in the temperature along a two-phase coexistence. Note that the condition for equilibrium between two phases is given by μ ( A ) = μ ( B ) G ( A ) = G ( B ) dG ( A ) = dG ( B ) S ( A ) dT + V ( A ) dp = S ( B ) dT + V ( B ) dp dp dT = S ( A ) S ( B ) V ( A ) V ( B ) (2.20) This is one form of the Clapeyron equation. It relates the slope of the coexistence curve to the entropy change and volume change of the phase transition.
• Fall '19
• Salvatore Ziccone

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