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# calculating microstates thermodynamics

Boltzmann proved that this expression for S is equivalent to the definition $\Delta{S}=\frac{Q}{T}\\$, which we have used extensively. We get the new number of microstates and the entropy to be, $$S_2 = k_B \ln((2M)^N) = k_BN \ln(2M) = k_BN [ln 2 + \ln M] = k_BN \ln 2 + S_1$$. Therefore. Now, if we start with an orderly macrostate like 100 heads and toss the coins, there is a virtual certainty that we will get a less orderly macrostate.
Well, you can calculate the NUMBER of microstates at "298.15 K" with tabulated standard molar entropies. If you tossed the coins once each second, you could expect to get either 100 heads or 100 tails once in 2 × 1022 years!

Given that a ΔS 10−21 J/K corresponds to about a 1 in 1030 chance, a decrease of this size (103 J/K) is an utter impossibility. Noting that the number of microstates is labeled W in Table 2 for the 100-coin toss, we can use ΔS = Sf − Si = k lnWf – klnWi to calculate the change in entropy. (b) What percent of the total possibilities is this?

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), The macroscopic state of a system is characterized by a distribution on the microstates. Changes in the entropy caused by changes in the external constraints are then given by: where we have twice used the conservation of probability, ∑ dpi = 0. The macrostate of the system is a description of its thermodynamic variables.

The macrostates are specified by the total number of heads and tails, whereas the microstates are specified by the facings of each individual coin.

Even when a system is entirely isolated from external influences, its microstate is constantly changing. The microstate of the system is a description of the positions and momenta of all the atoms. behavior, we are going beyond traditional thermodynamics. As such, according to the second law of thermodynamics, it is the equilibrium configuration of an isolated system. This means that no matter how cold the temperature gets, the lattice will always vibrate. Reference: wikipedia A microstate is a specific microscopic configuration of a thermodynamic system,that the system may occupy with certain probability in …

These probabilities imply, again, that for a macroscopic system, a decrease in entropy is impossible. There are three important points to note. 0. We can do this for each molecule, so the total number of ways we can put the molecules into the bins is $M \times M \times M ... \times M$ (N times) so. It is overwhelmingly probable for the gas to spread out to fill the container evenly, which is the new equilibrium macrostate of the system. (They are the least structured.)

Now pull out the partition so the molecules spread to both parts.

Some macrostates are more likely to occur than others. Thus the second law of thermodynamics is explained on a very basic level: entropy either remains the same or increases in every process.
This is an example illustrating the second law of thermodynamics: Since its discovery, this idea has been the focus of a great deal of thought, some of it confused. Each sequence is called a microstate—a detailed description of every element of a system.

If we start with an orderly array like 5 heads and toss the coins, it is very likely that we will get a less orderly array as a result, since 30 out of the 32 possibilities are less orderly. Neglecting correlations (or, more generally, statistical dependencies) between the states of individual particles will lead to an incorrect probability distribution on the microstates and hence to an overestimate of the entropy.

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