(Solved): (1) A kindergarten class is divided into two groups, with girls making up 20% of the students in gr ...

(1) A kindergarten class is divided into two groups, with girls making up $20\%$ of the students in group 1 and $80\%$ in group 2 . Overall, $40\%$ of the students are girls in the class. (a) What percentage of students, boy plus girls, will be in group 1 and what percentage in group $2?$ (b) Suppose the class consists of 50 children; other numbers are as in part (a). How many children are in group 1 and how many in group 2? Verify your answer by computing the percentages of girls in each group. (2) Consider the Gibbs free energy for a $c$-component system: $G=G\left(T,P,N_1,\dots ,N_c\right)$. (a) Recall that $G=U?TS+PV.$ Show that $G$ is a first order homogenous function. (b) Show that $G={?}_1{N}_1+{?}_2{N}_2+?+{?}_c{N}_c.$ (c) Show that only $c?1$ chemical potentials are independent of each other. (d) Construct a first order homogeneous function and verify Euler's theorem for it.