Plz look at the picture below and explain what does everything mean

It is very complicated.

Plz explain as easy as possible $($ but explain every equation line, term etc$)$

Entropy of mixing

We evaluate the entropy change due to mixing of $N$ individual gaseous species, which are identical pressure and temperature

hat$(S{)}_1={\displaystyle \underset{i=1}{\overset{N}{}}}{n}_i$hat$(s{)}_i(T,p)={\displaystyle \underset{i=1}{\overset{N}{}}}{n}_i[hat(s{)}_i^o(T)-ln(\frac{p}{p^o})]$

Since per total mole species $n_{\text{t}\text{o}\text{t}}={\displaystyle \underset{i=1}{\overset{N}{}}}{n}_i,$ we have

hat$(s{)}_1={\displaystyle \underset{i=1}{\overset{N}{}}}{x}_i$hat$(s{)}_i(T,p)={\displaystyle \underset{i=1}{\overset{N}{}}}{x}_i[hat(s{)}_i^o(T)-ln(\frac{p}{p^o})]$

The specific entropy after mixing is

hat$(s{)}_2={\displaystyle \underset{i=1}{\overset{N}{}}}{x}_i$hat$(s{)}_i(T,p)={\displaystyle \underset{i=1}{\overset{N}{}}}{x}_i$hat$(s{)}_i^o(T)-ln(\frac{p}{p^o}){\displaystyle \underset{i=1}{\overset{N}{}}}{x}_{i}ln{x}_i$

Thus entropy generation due to this adiabatic, constant temperature, constant pressure mixing process, measured per mole of mixture, is given by

hat$(s{)}_g=(hat(s{)}_2-$hat$(s{)}_1)=-{\displaystyle \underset{i=1}{\overset{N}{}}}{x}_{i}ln{x}_i$