Question $3$

$($i$)$ The diagram below shows a lattice. Count the number of sites $N$ available and calculate how many configurations are possible by placing $4$ identical atoms on this lattice. Show the steps in your calculation.

$[20\%]$

$($ii$)$ Explain why the configurational entropy of mixing cannot ever be negative. $[20\%]$

$($iii$)$ The heat capacity of gold is a function of temperature as follows:

$300$KFe,$Mn,Cu,Ni,W{C}_P=21\mathrm{.}05+0\mathrm{.}02T,Jmo{l}^{-1}{K}^{-1},$ for $100$

There are $no$ phase $or$ magnetic transitions $in$ gold over the stated temperature range. Showing the steps $in$ your method, calculate the enthalpy and entropy change when the sample $is$ heated from $100to300K.$

$[40\%]$

$(iv)$ What are the concentrations $ofFe,Mn,Cu,Ni,W$ that $on$ mixing $to$ form a single solid solution, would give the maximul configurational$-$entropy $of$ mixing? $[20\%]$