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If the system does work what is the sign of w and why?

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My question is If the system does work what is the sign of w and why? Also in an exothermic and endothermic reaction what is the sign of w and why?

Oct 16th, 2017

In this respect it should be noted that entropy has been related to the statistical probability of a system, W (the number of equivalent way molecules can be arranged) 

by the expression: S = κ*lnW   (κ is the Boltzman constant = R/Avogadro's Number)

S=k log W

Some consider S = k ln W to be easily the second most important formula of physics, next to E = mc² or at par with it. [1] 

This formula is sometime called "Boltzmann formula" (or Boltzmann entropy formula) and entropy calculated from this formula is sometimes called Boltzmann entropy.

In 1877 Ludwig Boltzmann provided a basis for answering this question when he introduced the concept of the entropy of a system as a measure of the amount of disorder in the system. 

A deck of cards fresh from the manufacturer is perfectly ordered and the entropy of this system is zero. When the deck is shuffled, the entropy of the system increases as the deck becomes more disordered.

There are 8.066 x 1067 different ways of organizing a deck of cards. The probability of obtaining any particular sequence of cards when the deck is shuffled is therefore 1 part in 8.066 x 1067. In theory, it is possible to shuffle a deck of cards until the cards fall into perfect order. But it isn't very likely!

Boltzmann proposed the following equation to describe the relationship between entropy and the amount of disorder in a system.

S = k ln W

In this equation, S is the entropy of the system, k is a proportionality constant equal to the ideal gas constant divided by Avogadro's constant, ln represents a logarithm to the base e, and W is the number of equivalent ways of describing the state of the system. According to this equation, the entropy of a system increases as the number of equivalent ways of describing the state of the system increases.

The relationship between the number of equivalent ways of describing a system and the amount of disorder in the system can be demonstrated with another analogy based on a deck of cards. 

There are 2,598,960 different hands that could be dealt in a game of five-card poker. More than half of these hands are essentially worthless. 



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