Boltzmanns Equation and Microstates

The science of thermodynamics developed as a means of describing the properties of matter in our macroscopic world without regard to microscopic structure. In fact, thermodynamics was a well-developed field before the modern view of atomic and molecular structure was even known. The thermodynamic properties of water, for example, addressed the behavior of bulk water (or ice or water vapor) as a substance without considering any specific properties of individual H2O molecules. 

To connect the microscopic and macroscopic descriptions of matter, scientists have developed the field of statistical thermodynamicSy which uses the took of statistics and probability to link the microscopic and macroscopic worlds. Here we show how entropy, which is a property of bulk matter, can be connected to the behavior of atoms and molecules. Because the mathematics of statistical thermodynamics is complex, our discussion will be largely conceptual. 

Suppose we now consider one mole of an ideal gas in a particular thermodynamic state, which we can define by specifying the temperature, 

and volume, V, of the gas. What is happening to this gas at the microscopic level, and how does what is going on at the microscopic level relate to the entropy of the gas  

As you no doubt see, there would be such a sta eringly large number of microstates that taking individual snapshots of all of them is not feasible. Because we are examining such a large number of particles, however, we can use the tools of statistics and probability to determine the total number of microstates for the thermodynamic state. (That is where the statistical part of the name statistical ermodynamics comes in.) Each thermodynamic state has a characteristic number of microstates associated with it, and we will use the symbol W for that number. 

In this equation, k is the Boltzmann constant, 1.38 X 10 J/K. Thus, entropy is a measure of how many microstates are associated with a particular macroscopic state. 

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