4. (30 points) An ideal gas containing N=61023 molecules is in a box of volume V. The weight (or number of ways) j of having j molecules in the left half and Nj molecules in the right half of the box is given by the formula j=j!(Nj)!N!, and the entropy is Sj=kBlnj. The Boltzmann constant kB=1.41023JK1. For large numbers, lnN!=NlnNN. ln2=0.7,ln5=1.6, approximately. For small numbers f,ln(1+f)=f,ln(1f)=f. a) (10 points) Initially j=(N/2)(1+f) molecules are in the left half and Nj=(N/2)(1f) molecules are in the right half. Here f is a small fraction. What are the values of and S in this initial state? b) (5 points) What is the entropy of the initial state if f=0.2 ? c) (5 points) In which state is the entropy maximum? What are the values of and S in this maximum entropy state state? This is the final (equilibrium) state. c) (10 points) What is the entropy change S from the initial state in b) to the final state in c)? Evaluate S in Joules