Question

a hypothetical a - b alloy of composition 52 wt% b - 48 wt% a at some temperature is found to consist of mass fractions of 0.5 for both α and β phases. if the composition of the β phase is 83 wt% b - 17 wt% a, what is the composition of the α phase?
17 wt% b - 83 wt% a
21 wt% b - 79 wt% a
32 wt% b - 68 wt% a
47 wt% b - 53 wt% a

Explanation:

Step1: Apply lever rule formula

The lever rule formula for mass fraction \(W_{\alpha}=\frac{C_{\beta}-C_{0}}{C_{\beta}-C_{\alpha}}\). Here \(W_{\alpha} = W_{\beta}= 0.5\) (since mass fractions of \(\alpha\) and \(\beta\) are equal), \(C_{0}=48\) wt% A (composition of the alloy). Let \(C_{\alpha}\) be the composition of \(\alpha\) phase (wt% A) and \(C_{\beta} = 17\) wt% A (given composition of \(\beta\) phase).
Substituting into the formula \(0.5=\frac{17 - 48}{17 - C_{\alpha}}\)

Step2: Solve for \(C_{\alpha}\)

Cross - multiply: \(0.5\times(17 - C_{\alpha})=17 - 48\)
\(8.5-0.5C_{\alpha}=- 31\)
\(-0.5C_{\alpha}=-31 - 8.5\)
\(-0.5C_{\alpha}=-39.5\)
\(C_{\alpha}=\frac{-39.5}{-0.5}=79\) wt% A

Answer:

21 wt% B - 79 wt% A