Problem $4\mathrm{.}22$ :

Sometimes it is desirable to be able to determine the weight percent of one

element, $C_1,$ that will produce a specified concentration in terms of the number of

atoms per cubic centimeter, $N_1,$ for an alloy composed of two types of atoms. This

computation is possible using the following expression:

$C_1=\frac{100}{1+\frac{N_A{}_2}{N_1{A}_1}-\frac{{}_2}{{}_1}},\{\begin{array}{ccc}\text{:}[A_1& \text{a}\text{n}\text{d}{A}_2\text{a}\text{r}\text{e}\text{a}\text{t}\text{o}\text{m}\text{i}\text{c}\text{w}\text{e}\text{i}\text{g}\text{h}\text{t}of\text{t}\text{w}\text{o}\text{e}\text{l}\text{e}\text{m}\text{e}\text{n}\text{t}\text{s}& ]\\ N_A& \text{:}\text{A}\text{v}\text{o}\text{g}\text{a}\text{d}\text{r}\text{o}\text{n}\text{u}\text{m}\text{b}\text{e}\text{r}& \\ {}_1& \text{a}\text{n}\text{d}{}_2\text{a}\text{r}\text{e}\text{d}\text{e}\text{n}\text{s}\text{i}\text{t}\text{i}\text{e}\text{s}of\text{t}\text{h}\text{e}\text{t}\text{w}\text{o}\text{e}\text{l}\text{e}\text{m}\text{e}\text{n}\text{t}\text{s}& \end{array}$

Derive this expression using equation $4\mathrm{.}2$ and expressions contained in section $4\mathrm{.}4$