- #1

Buzz Bloom

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All of the references I cite are from Wikipedia.

Ex 1. Does the following isolated system have a calculable value for its entropy?

Assume a spherical volume of radius R

_{1}containing N

_{1}moles of hydrogen H

_{2}molecules at a temperature T

_{1}(Kelvin), at which the hydrogen has a gaseous state (except perhaps for an extremely small fraction of molecules). Also assume that the boundary of the sphere consists of a perfect insulator which maintains the temperature at T

_{1}, and that sufficient time has elapsed for the system to be in a state of equilibrium. To the extent that H

_{2}is close to being an ideal gas, the temperature, volume, and pressure are related by

https://en.wikipedia.org/wiki/Ideal_gas_law

PV=nRT

where P, V and T are the pressure, volume and absolute temperature; n is the number of moles of gas; and R is the ideal gas constant.

The volume iswhere P, V and T are the pressure, volume and absolute temperature; n is the number of moles of gas; and R is the ideal gas constant.

V

https://en.wikipedia.org/wiki/Entropy_{1}= (4/3) π R_{1}^{3}.In statistical mechanics, entropy is an extensive property of a thermodynamic system. It is closely related to the number Ω of microscopic configurations (known as microstates) that are consistent with the macroscopic quantities that characterize the system (such as its volume, pressure and temperature). Under the assumption that each microstate is equally probable, the entropy S is the natural logarithm of the number of microstates, multiplied by the Boltzmann constant k

My understanding is that the answer to (1) is YES. My confusion at this point is that the concept of "microstate" is unclear. Here is a definition._{B}.

https://en.wikipedia.org/wiki/Microstate_(statistical_mechanics)

Treatments on statistical mechanics, define a macrostate as follows: a particular set of values of energy, the number of particles, and the volume of an isolated thermodynamic system is said to specify a particular macrostate of it. In this description, microstates appear as different possible ways the system can achieve a particular macrostate.

This definition does not help me understand how many microstates exist in the (1) system. One interpretation might be that because the (1) system is assumed to be in equilibrium, there is only one microstate, the equilibrium state, and therefore it's entropy is zero. This implies that for all non-equilibrium macrostates, the entropy is always negative, since as the system changes moving closer to equilibrium its entropy will increase towards zero. (Deeper considerations that I plan to discuss later have convinced me that this interpretation makes plausible sense. Although I find this interpretation to be plausible, I have no confidence that it is correct.)

If my YES answer above is incorrect and/or my interpretation that the entropy of an equilibrium system is zero is wrong, I would much appreciate someone helping me understand the correct answers, especially regarding microstates.