### Statistical Mechanics Lecture 10 of 29

April 18, 2012 by Antonello Scardicchio

A. Scardicchio, ICTP

The first part of this lesson was focused in the classifications of the phase transitions and the main differences between the first order and the second order phase transitions. It was shown that the first derivative of the Gibbs free energy (G) function is discontinuous in both planes the G vs T/Tc for P< Pc and G vs P/Pc for T< Tc, where Tc and Pc the critical temperature and pressure respectively (first order phase transition). In the case of T?Tc and P ? Pc the first derivative of G is continuous and the discontinuity is found in the second derivative of G (second order phase transition). It was emphasized that in the case of the first order phase transition the thermodynamic potential has two local minimum and the role of global minima is exchange when passing from one phase to the other while in the second order transition G has always just one local-global minimum.

In the second part of the lesson the chemical potential was introduced, it was define as the partial derivative of the thermodynamic potential with respect to the number of particles. Then as an example it was calculated the chemical potential of the ideal gas and it was considered the case of the Earth atmosphere where with the help of the chemical potential it was possible to obtain an expression for the pressure as a function of the height (h).

What is the pressure at h=4km from the surface of the Earth assuming that the pressure at the surface h=0 is P0=1 atm, the temperature is T=290 K and the air is formed just by nitrogen N2?

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