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The_third_law - The third law ChE210A A bsolute entropies...

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last modified 11/2/2009 © M. S. Shell 2009 1/9 The third law ChE210A Absolute entropies and absolute zero In this lecture we address two questions: Is there such thing as an absolute value of the entropy? That is, is there a way and does it make sense to identify an exact numerical value of g1845 for a particular system and a particular state point, rather than a change in entropy between two state points? What is the behavior of the entropy and other thermodynamic functions as g2176 ap- proaches absolute zero? Recall the microscopic definition of the entropy, g1845 g3404 g1863 g3003 ln Ω As we have seen, the interpretation of is that it counts the number of microscopic states of a system. The behavior of depends on whether or not we have a quantum or classical view- point of the world: classical description – In this approximation, we cannot count exactly the number of microstates, since we can continuously change the positions and velocities of all the atoms. Therefore, there are an infinite amount of configurations. While we cannot count absolute numbers of configurations, we can count relative numbers, and hence obtain entropy differences. For example, if a volume g1848 containing a single particle is expanded to 2g1848 , the number of microstates doubles such that Δg1845 g3404 g1863 g3003 lng46662g1848 g1848 g4667 g3404 g1863 g3003 ln 2 . quantum description – Quantum mechanics says that there are discrete quantum states a system can occupy. For example, for a particle in volume g1848 g3404 g1838 g2871 , only states whose energy is given by g2035 g3404 g3035 g3118 g2876g3040g3013 g3118 g3435g1866 g3051 g2870 g3397 g1866 g3052 g2870 g3397 g1866 g3053 g2870 g3439 are allowed, where g1866 g3051 , g1866 g3052 , and g1866 g3053 are all positive integers. Therefore, in a quantum description and in reality, it is possible to count microstates exactly. These considerations show that a quantum description is required if we are to have an absolute entropy, whereby we can count exactly. Is this enough to give us an absolute entropy? What if, in contrast to the above equation, we had defined the entropy using
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last modified 11/2/2009 © M. S. Shell 2009 2/9 g1845 g3404 g1863 g3003 ln Ω g3397 g1845 ref where g1845 ref is some universal, material-independent constant? Would this re-defining of the entropy imply any physical changes to the equilibrium behavior of a system, as given by the relationships that the entropy function implies? It turns out that this addition would not change any aspect of the physical behavior of our system. Recall that g1846, g1842, and g2020 all derive from derivatives of g1845 ; therefore, the g1845 ref constant vanishes in the relationship between g1831, g1848, g1840 and these properties, for any substance. It is for the same reason, in fact, that classical descriptions, which cannot have an absolute entropy, can describe the properties of many systems. There is no reason why we could not define the entropy with a nonzero value for g1845 ref . Recall that the original justification for the existence of the entropy was that the rule of equal a priori probabilities said that the macrostate with the most microstates dominated the equilibrium behavior.
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